id sid tid token lemma pos iajs-1955 1 1 microsoft microsoft PROPN iajs-1955 1 2 word word NOUN iajs-1955 1 3 164 164 NUM iajs-1955 1 4 - - SYM iajs-1955 1 5 178 178 NUM iajs-1955 1 6             SPACE iajs-1955 1 7 164 164 NUM iajs-1955 1 8     SPACE iajs-1955 1 9 | | ADV iajs-1955 1 10     SPACE iajs-1955 1 11 mathmatics mathmatic NOUN iajs-1955 1 12 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 1 13     SPACE iajs-1955 1 14 2018 2018 NUM iajs-1955 1 15 ) ) PUNCT iajs-1955 1 16 عام عام ADP iajs-1955 1 17 2العدد 2العدد NUM iajs-1955 1 18 ( ( PUNCT iajs-1955 1 19 31المجلد 31المجلد NUM iajs-1955 1 20 مجلة مجلة VERB iajs-1955 1 21 إبن إبن VERB iajs-1955 1 22 الهيثم الهيثم ADJ iajs-1955 1 23 للعلوم للعلوم NOUN iajs-1955 1 24 الصرفة الصرفة NOUN iajs-1955 2 1 و و PRON iajs-1955 2 2 التطبيقية التطبيقية ADJ iajs-1955 2 3 ibn ibn PROPN iajs-1955 2 4 al al PROPN iajs-1955 2 5 - - PUNCT iajs-1955 2 6 haitham haitham PROPN iajs-1955 2 7 jour jour X iajs-1955 2 8 . . PROPN iajs-1955 2 9 for for ADP iajs-1955 2 10 pure pure ADJ iajs-1955 2 11 & & CCONJ iajs-1955 2 12 appl appl PROPN iajs-1955 2 13 . . PUNCT iajs-1955 3 1 sci sci PROPN iajs-1955 3 2 . . PUNCT iajs-1955 3 3 vol vol NOUN iajs-1955 3 4 . . PROPN iajs-1955 4 1 31 31 NUM iajs-1955 4 2 ( ( PUNCT iajs-1955 4 3 2 2 NUM iajs-1955 4 4 ) ) PUNCT iajs-1955 4 5 2018 2018 NUM iajs-1955 4 6 w w ADV iajs-1955 4 7 - - PUNCT iajs-1955 4 8 closed close VERB iajs-1955 4 9 submodule submodule NOUN iajs-1955 4 10 and and CCONJ iajs-1955 4 11 related related ADJ iajs-1955 4 12 concepts concept NOUN iajs-1955 4 13 haibat haibat PROPN iajs-1955 4 14 k. k. PROPN iajs-1955 4 15 mohammad mohammad PROPN iajs-1955 4 16 ali ali PROPN iajs-1955 4 17 mohammad mohammad PROPN iajs-1955 4 18 e. e. PROPN iajs-1955 4 19 dahsh dahsh PROPN iajs-1955 4 20 dept dept PROPN iajs-1955 4 21 . . PROPN iajs-1955 4 22 of of ADP iajs-1955 4 23 mathematics mathematics PROPN iajs-1955 4 24 / / SYM iajs-1955 4 25 college college NOUN iajs-1955 4 26 of of ADP iajs-1955 4 27 computer computer NOUN iajs-1955 4 28 science science NOUN iajs-1955 4 29 and and CCONJ iajs-1955 4 30 mathematics mathematics PROPN iajs-1955 4 31 / / SYM iajs-1955 4 32 tikrit tikrit NOUN iajs-1955 4 33 university university PROPN iajs-1955 4 34 dr dr PROPN iajs-1955 4 35 .mohammadali2013@gmail.com .mohammadali2013@gmail.com PUNCT iajs-1955 5 1 mohmad.alduri90@gmail.com mohmad.alduri90@gmail.com PROPN iajs-1955 5 2 received receive VERB iajs-1955 5 3 in:30 in:30 PROPN iajs-1955 5 4 / / SYM iajs-1955 5 5 january/2018 january/2018 NOUN iajs-1955 5 6 , , PUNCT iajs-1955 5 7 accepted accept VERB iajs-1955 5 8 in:7 in:7 PROPN iajs-1955 5 9 / / SYM iajs-1955 5 10 march/2018 march/2018 NOUN iajs-1955 5 11 abstract abstract ADV iajs-1955 5 12 let let VERB iajs-1955 5 13 r r PRON iajs-1955 5 14 be be AUX iajs-1955 5 15 a a DET iajs-1955 5 16 commutative commutative ADJ iajs-1955 5 17 ring ring NOUN iajs-1955 5 18 with with ADP iajs-1955 5 19 identity identity NOUN iajs-1955 5 20 , , PUNCT iajs-1955 5 21 and and CCONJ iajs-1955 5 22 m m VERB iajs-1955 5 23 be be AUX iajs-1955 5 24 a a DET iajs-1955 5 25 left left ADJ iajs-1955 5 26 untial untial ADJ iajs-1955 5 27 module module NOUN iajs-1955 5 28 . . PUNCT iajs-1955 6 1 in in ADP iajs-1955 6 2 this this DET iajs-1955 6 3 paper paper NOUN iajs-1955 6 4 we we PRON iajs-1955 6 5 introduce introduce VERB iajs-1955 6 6 and and CCONJ iajs-1955 6 7 study study VERB iajs-1955 6 8 the the DET iajs-1955 6 9 concept concept NOUN iajs-1955 6 10 w w NOUN iajs-1955 6 11 - - PUNCT iajs-1955 6 12 closed closed ADJ iajs-1955 6 13 submodules submodule NOUN iajs-1955 6 14 , , PUNCT iajs-1955 6 15 that that PRON iajs-1955 6 16 is be AUX iajs-1955 6 17 stronger strong ADJ iajs-1955 6 18 form form NOUN iajs-1955 6 19 of of ADP iajs-1955 6 20 the the DET iajs-1955 6 21 concept concept NOUN iajs-1955 6 22 of of ADP iajs-1955 6 23 closed closed ADJ iajs-1955 6 24 submodules submodule NOUN iajs-1955 6 25 , , PUNCT iajs-1955 6 26 where where SCONJ iajs-1955 6 27 asubmodule asubmodule PROPN iajs-1955 6 28 k k PROPN iajs-1955 6 29 of of ADP iajs-1955 6 30 a a DET iajs-1955 6 31 module module NOUN iajs-1955 6 32 m m VERB iajs-1955 6 33 is be AUX iajs-1955 6 34 called call VERB iajs-1955 6 35 w w NOUN iajs-1955 6 36 - - PUNCT iajs-1955 6 37 closed closed ADJ iajs-1955 6 38 in in ADP iajs-1955 6 39 m m PROPN iajs-1955 6 40 , , PUNCT iajs-1955 6 41 " " PUNCT iajs-1955 6 42 if if SCONJ iajs-1955 6 43 it it PRON iajs-1955 6 44 has have VERB iajs-1955 6 45 no no DET iajs-1955 6 46 proper proper ADJ iajs-1955 6 47 weak weak ADJ iajs-1955 6 48 essential essential ADJ iajs-1955 6 49 extension extension NOUN iajs-1955 6 50 in in ADP iajs-1955 6 51 m m NOUN iajs-1955 6 52 " " PUNCT iajs-1955 6 53 , , PUNCT iajs-1955 6 54 that that PRON iajs-1955 6 55 is be AUX iajs-1955 6 56 if if SCONJ iajs-1955 6 57 there there PRON iajs-1955 6 58 exists exist VERB iajs-1955 6 59 a a DET iajs-1955 6 60 submodule submodule NOUN iajs-1955 6 61 l l NOUN iajs-1955 6 62 of of ADP iajs-1955 6 63 m m PROPN iajs-1955 6 64 with with ADP iajs-1955 6 65 k k PROPN iajs-1955 6 66 is be AUX iajs-1955 6 67 weak weak ADJ iajs-1955 6 68 essential essential ADJ iajs-1955 6 69 submodule submodule NOUN iajs-1955 6 70 of of ADP iajs-1955 6 71 l l PROPN iajs-1955 6 72 then then ADV iajs-1955 6 73 k k X iajs-1955 6 74 = = PROPN iajs-1955 6 75 l. l. NOUN iajs-1955 6 76 some some DET iajs-1955 6 77 basic basic ADJ iajs-1955 6 78 properties property NOUN iajs-1955 6 79 , , PUNCT iajs-1955 6 80 examples example NOUN iajs-1955 6 81 of of ADP iajs-1955 6 82 w w NOUN iajs-1955 6 83 - - PUNCT iajs-1955 6 84 closed closed ADJ iajs-1955 6 85 submodules submodule NOUN iajs-1955 6 86 are be AUX iajs-1955 6 87 investigated investigate VERB iajs-1955 6 88 , , PUNCT iajs-1955 6 89 and and CCONJ iajs-1955 6 90 some some DET iajs-1955 6 91 relationships relationship NOUN iajs-1955 6 92 between between ADP iajs-1955 6 93 w w NOUN iajs-1955 6 94 - - PUNCT iajs-1955 6 95 closed closed ADJ iajs-1955 6 96 submodules submodule NOUN iajs-1955 6 97 and and CCONJ iajs-1955 6 98 other other ADJ iajs-1955 6 99 related related ADJ iajs-1955 6 100 modules module NOUN iajs-1955 6 101 are be AUX iajs-1955 6 102 studied study VERB iajs-1955 6 103 . . PUNCT iajs-1955 7 1 furthermore furthermore ADV iajs-1955 7 2 , , PUNCT iajs-1955 7 3 modules module NOUN iajs-1955 7 4 with with ADP iajs-1955 7 5 chain chain NOUN iajs-1955 7 6 condition condition NOUN iajs-1955 7 7 on on ADP iajs-1955 7 8 w w ADJ iajs-1955 7 9 - - PUNCT iajs-1955 7 10 closed closed ADJ iajs-1955 7 11 submodules submodule NOUN iajs-1955 7 12 are be AUX iajs-1955 7 13 studied study VERB iajs-1955 7 14 . . PUNCT iajs-1955 8 1 keywords keyword NOUN iajs-1955 8 2 : : PUNCT iajs-1955 8 3 closed closed ADJ iajs-1955 8 4 submodules submodule NOUN iajs-1955 8 5 , , PUNCT iajs-1955 8 6 weak weak ADJ iajs-1955 8 7 essential essential ADJ iajs-1955 8 8 submodules submodule NOUN iajs-1955 8 9 , , PUNCT iajs-1955 8 10 w w NOUN iajs-1955 8 11 - - PUNCT iajs-1955 8 12 closed closed ADJ iajs-1955 8 13 submodules submodule NOUN iajs-1955 8 14 , , PUNCT iajs-1955 8 15 completely completely ADV iajs-1955 8 16 essential essential ADJ iajs-1955 8 17 modules module NOUN iajs-1955 8 18 , , PUNCT iajs-1955 8 19 y y NOUN iajs-1955 8 20 - - PUNCT iajs-1955 8 21 closed close VERB iajs-1955 8 22 submodules submodule NOUN iajs-1955 8 23 , , PUNCT iajs-1955 8 24 minimal minimal ADJ iajs-1955 8 25 semi semi ADJ iajs-1955 8 26 - - ADJ iajs-1955 8 27 prime prime ADJ iajs-1955 8 28 submodules submodule NOUN iajs-1955 8 29 . . PUNCT iajs-1955 8 30             SPACE iajs-1955 9 1 165 165 NUM iajs-1955 9 2     SPACE iajs-1955 9 3 | | ADV iajs-1955 9 4     SPACE iajs-1955 9 5 mathmatics mathmatic NOUN iajs-1955 9 6 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 9 7     SPACE iajs-1955 9 8 2018 2018 NUM iajs-1955 9 9 ) ) PUNCT iajs-1955 10 1 عام عام ADP iajs-1955 10 2 2العدد 2العدد NUM iajs-1955 10 3 ( ( PUNCT iajs-1955 10 4 31لمجلد 31لمجلد NUM iajs-1955 10 5 ا ا NOUN iajs-1955 10 6 مجلة مجلة NOUN iajs-1955 10 7 إبن إبن VERB iajs-1955 10 8 الهيثم الهيثم ADJ iajs-1955 10 9 للعلوم للعلوم NOUN iajs-1955 10 10 الصرفة الصرفة NOUN iajs-1955 11 1 و و PRON iajs-1955 11 2 التطبيقية التطبيقية ADJ iajs-1955 11 3 ibn ibn PROPN iajs-1955 11 4 al al PROPN iajs-1955 11 5 - - PUNCT iajs-1955 11 6 haitham haitham PROPN iajs-1955 11 7 jour jour X iajs-1955 11 8 . . PROPN iajs-1955 11 9 for for ADP iajs-1955 11 10 pure pure ADJ iajs-1955 11 11 & & CCONJ iajs-1955 11 12 appl appl PROPN iajs-1955 11 13 . . PUNCT iajs-1955 12 1 sci sci PROPN iajs-1955 12 2 . . PUNCT iajs-1955 12 3 vol vol NOUN iajs-1955 12 4 . . PROPN iajs-1955 13 1 31 31 NUM iajs-1955 13 2 ( ( PUNCT iajs-1955 13 3 2 2 NUM iajs-1955 13 4 ) ) SYM iajs-1955 13 5 2018 2018 NUM iajs-1955 13 6 introduction introduction NOUN iajs-1955 13 7 in in ADP iajs-1955 13 8 this this DET iajs-1955 13 9 note note NOUN iajs-1955 13 10 , , PUNCT iajs-1955 13 11 we we PRON iajs-1955 13 12 shall shall AUX iajs-1955 13 13 assume assume VERB iajs-1955 13 14 that that SCONJ iajs-1955 13 15 all all DET iajs-1955 13 16 rings ring NOUN iajs-1955 13 17 are be AUX iajs-1955 13 18 commutative commutative ADJ iajs-1955 13 19 with with ADP iajs-1955 13 20 unity unity NOUN iajs-1955 13 21 and and CCONJ iajs-1955 13 22 all all DET iajs-1955 13 23 modules module NOUN iajs-1955 13 24 are be AUX iajs-1955 13 25 unital unital ADJ iajs-1955 13 26 left leave VERB iajs-1955 13 27 modules module NOUN iajs-1955 13 28 , , PUNCT iajs-1955 13 29 and and CCONJ iajs-1955 13 30 all all DET iajs-1955 13 31 r r NOUN iajs-1955 13 32 - - PUNCT iajs-1955 13 33 modules module NOUN iajs-1955 13 34 under under ADP iajs-1955 13 35 study study NOUN iajs-1955 13 36 contains contain VERB iajs-1955 13 37 semi semi ADJ iajs-1955 13 38 - - ADJ iajs-1955 13 39 prime prime ADJ iajs-1955 13 40 submodules submodule NOUN iajs-1955 13 41 . . PUNCT iajs-1955 14 1 " " PUNCT iajs-1955 14 2 a a DET iajs-1955 14 3 submodule submodule NOUN iajs-1955 14 4 l l NOUN iajs-1955 14 5 of of ADP iajs-1955 14 6 a a DET iajs-1955 14 7 module module NOUN iajs-1955 14 8 m m VERB iajs-1955 14 9 is be AUX iajs-1955 14 10 called call VERB iajs-1955 14 11 closed close VERB iajs-1955 14 12 in in ADP iajs-1955 14 13 m m AUX iajs-1955 14 14 provided provide VERB iajs-1955 14 15 that that SCONJ iajs-1955 14 16 l l NOUN iajs-1955 14 17 has have VERB iajs-1955 14 18 no no DET iajs-1955 14 19 proper proper ADJ iajs-1955 14 20 essential essential ADJ iajs-1955 14 21 extension extension NOUN iajs-1955 14 22 in in ADP iajs-1955 14 23 m m PROPN iajs-1955 14 24 [ [ X iajs-1955 14 25 1 1 NUM iajs-1955 14 26 ] ] PUNCT iajs-1955 14 27 " " PUNCT iajs-1955 14 28 , , PUNCT iajs-1955 14 29 " " PUNCT iajs-1955 14 30 where where SCONJ iajs-1955 14 31 a a DET iajs-1955 14 32 non non ADJ iajs-1955 14 33 - - ADJ iajs-1955 14 34 zero zero NUM iajs-1955 14 35 submodule submodule NOUN iajs-1955 14 36 n n PROPN iajs-1955 14 37 of of ADP iajs-1955 14 38 m m PROPN iajs-1955 14 39 is be AUX iajs-1955 14 40 called call VERB iajs-1955 14 41 essential essential ADJ iajs-1955 14 42 if if SCONJ iajs-1955 14 43 n n NOUN iajs-1955 14 44 ∩ ∩ NOUN iajs-1955 14 45 e e X iajs-1955 14 46 0 0 NUM iajs-1955 14 47 for for ADP iajs-1955 14 48 all all DET iajs-1955 14 49 non non ADJ iajs-1955 14 50 - - ADJ iajs-1955 14 51 zero zero NUM iajs-1955 14 52 submodule submodule NOUN iajs-1955 14 53 e e PROPN iajs-1955 14 54 of of ADP iajs-1955 14 55 m m PROPN iajs-1955 15 1 [ [ X iajs-1955 15 2 1 1 NUM iajs-1955 15 3 ] ] NUM iajs-1955 15 4 " " PUNCT iajs-1955 15 5 , , PUNCT iajs-1955 15 6 " " PUNCT iajs-1955 15 7 and and CCONJ iajs-1955 15 8 a a DET iajs-1955 15 9 non non ADJ iajs-1955 15 10 - - ADJ iajs-1955 15 11 zero zero NUM iajs-1955 15 12 submodule submodule NOUN iajs-1955 15 13 n n PROPN iajs-1955 15 14 of of ADP iajs-1955 15 15 m m PROPN iajs-1955 15 16 is be AUX iajs-1955 15 17 called call VERB iajs-1955 15 18 weak weak ADJ iajs-1955 15 19 essential essential ADJ iajs-1955 15 20 if if SCONJ iajs-1955 15 21 n n NOUN iajs-1955 15 22 ∩ ∩ NOUN iajs-1955 15 23 s s PART iajs-1955 15 24 0 0 NUM iajs-1955 15 25 ∀ ∀ X iajs-1955 15 26 non non X iajs-1955 15 27 zero zero NUM iajs-1955 15 28 semi semi ADJ iajs-1955 15 29 - - ADJ iajs-1955 15 30 prime prime ADJ iajs-1955 15 31 submodule submodule NOUN iajs-1955 15 32 s s PROPN iajs-1955 15 33 of of ADP iajs-1955 15 34 m m PROPN iajs-1955 15 35 [ [ X iajs-1955 15 36 2 2 NUM iajs-1955 15 37 ] ] PUNCT iajs-1955 15 38 " " PUNCT iajs-1955 15 39 . . PUNCT iajs-1955 16 1 " " PUNCT iajs-1955 16 2 equivalently equivalently ADV iajs-1955 16 3 , , PUNCT iajs-1955 16 4 a a DET iajs-1955 16 5 submodule submodule NOUN iajs-1955 16 6 n n PROPN iajs-1955 16 7 of of ADP iajs-1955 16 8 a a DET iajs-1955 16 9 module module NOUN iajs-1955 16 10 m m VERB iajs-1955 16 11 is be AUX iajs-1955 16 12 called call VERB iajs-1955 16 13 weak weak ADJ iajs-1955 16 14 essential essential ADJ iajs-1955 16 15 if if SCONJ iajs-1955 16 16 whenever whenever SCONJ iajs-1955 16 17 n n X iajs-1955 16 18 ∩ ∩ NOUN iajs-1955 16 19 s s PART iajs-1955 16 20 0 0 NUM iajs-1955 16 21 , , PUNCT iajs-1955 16 22 then then ADV iajs-1955 16 23 s=(o s=(o NOUN iajs-1955 16 24 ) ) PUNCT iajs-1955 16 25 for for ADP iajs-1955 16 26 every every DET iajs-1955 16 27 semi semi ADJ iajs-1955 16 28 - - ADJ iajs-1955 16 29 prime prime ADJ iajs-1955 16 30 submodule submodule NOUN iajs-1955 16 31 s s PROPN iajs-1955 16 32 of of ADP iajs-1955 16 33 m m PROPN iajs-1955 16 34 [ [ X iajs-1955 16 35 3]","where 3]","where NUM iajs-1955 16 36 a a DET iajs-1955 16 37 submodule submodule NOUN iajs-1955 16 38 s s NOUN iajs-1955 16 39 of of ADP iajs-1955 16 40 a a DET iajs-1955 16 41 module module NOUN iajs-1955 16 42 m m VERB iajs-1955 16 43 is be AUX iajs-1955 16 44 called call VERB iajs-1955 16 45 semi semi ADJ iajs-1955 16 46 - - ADJ iajs-1955 16 47 prime prime ADJ iajs-1955 16 48 if if SCONJ iajs-1955 16 49 for for ADP iajs-1955 16 50 each each DET iajs-1955 16 51 𝑟 𝑟 PRON iajs-1955 16 52 ∈ ∈ PROPN iajs-1955 16 53 𝑅 𝑅 PROPN iajs-1955 16 54 𝑎𝑛𝑑 𝑎𝑛𝑑 ADJ iajs-1955 16 55 𝑦 𝑦 PROPN iajs-1955 16 56 ∈ ∈ PROPN iajs-1955 16 57 𝑀 𝑀 PROPN iajs-1955 16 58 𝑤𝑖𝑡ℎ 𝑤𝑖𝑡ℎ VERB iajs-1955 16 59 𝑟 𝑟 X iajs-1955 16 60 𝑦 𝑦 PROPN iajs-1955 16 61 ∈ ∈ PROPN iajs-1955 16 62 𝑆 𝑆 PROPN iajs-1955 16 63 , , PUNCT iajs-1955 16 64 𝑘 𝑘 DET iajs-1955 16 65 ∈ ∈ PROPN iajs-1955 16 66 𝑍 𝑍 NOUN iajs-1955 16 67 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒𝑛 NOUN iajs-1955 16 68 𝑟𝑦 𝑟𝑦 X iajs-1955 16 69 ∈ ∈ PROPN iajs-1955 16 70 𝑆 𝑆 PROPN iajs-1955 17 1 [ [ X iajs-1955 17 2 4]" 4]" NUM iajs-1955 17 3 . . PUNCT iajs-1955 18 1 "equivalently "equivalently ADV iajs-1955 18 2 if if SCONJ iajs-1955 18 3 𝑟 𝑟 X iajs-1955 18 4 𝑦 𝑦 PROPN iajs-1955 18 5 ∈ ∈ PROPN iajs-1955 18 6 𝑆 𝑆 PROPN iajs-1955 18 7 , , PUNCT iajs-1955 18 8 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒𝑛 VERB iajs-1955 18 9 𝑟𝑦 𝑟𝑦 X iajs-1955 18 10 ∈ ∈ PROPN iajs-1955 18 11 𝑆 𝑆 PROPN iajs-1955 19 1 [ [ X iajs-1955 19 2 5]".in 5]".in PRON iajs-1955 19 3 this this PRON iajs-1955 19 4 proper proper ADJ iajs-1955 19 5 , , PUNCT iajs-1955 19 6 " " PUNCT iajs-1955 19 7 we we PRON iajs-1955 19 8 introduce introduce VERB iajs-1955 19 9 the the DET iajs-1955 19 10 concept concept NOUN iajs-1955 19 11 of of ADP iajs-1955 19 12 w w NOUN iajs-1955 19 13 - - PUNCT iajs-1955 19 14 closed closed ADJ iajs-1955 19 15 submodule submodule NOUN iajs-1955 19 16 " " PUNCT iajs-1955 19 17 which which PRON iajs-1955 19 18 is be AUX iajs-1955 19 19 stronger strong ADJ iajs-1955 19 20 than than ADP iajs-1955 19 21 the the DET iajs-1955 19 22 concept concept NOUN iajs-1955 19 23 of of ADP iajs-1955 19 24 closed closed ADJ iajs-1955 19 25 submodule submodule NOUN iajs-1955 19 26 " " PUNCT iajs-1955 19 27 , , PUNCT iajs-1955 19 28 where where SCONJ iajs-1955 19 29 a a DET iajs-1955 19 30 submodule submodule NOUN iajs-1955 19 31 k k NOUN iajs-1955 19 32 of of ADP iajs-1955 19 33 an an DET iajs-1955 19 34 r r NOUN iajs-1955 19 35 - - PUNCT iajs-1955 19 36 module module NOUN iajs-1955 19 37 m m NOUN iajs-1955 19 38 is be AUX iajs-1955 19 39 called call VERB iajs-1955 19 40 w w ADV iajs-1955 19 41 - - PUNCT iajs-1955 19 42 closed closed ADJ iajs-1955 19 43 " " PUNCT iajs-1955 19 44 if if SCONJ iajs-1955 19 45 k k PROPN iajs-1955 19 46 has have VERB iajs-1955 19 47 no no DET iajs-1955 19 48 proper proper ADJ iajs-1955 19 49 weak weak ADJ iajs-1955 19 50 essential essential ADJ iajs-1955 19 51 extension extension NOUN iajs-1955 19 52 in in ADP iajs-1955 19 53 m m NOUN iajs-1955 19 54 " " PUNCT iajs-1955 19 55 . . PUNCT iajs-1955 20 1 that that PRON iajs-1955 20 2 is be AUX iajs-1955 20 3 if if SCONJ iajs-1955 20 4 k k PROPN iajs-1955 20 5 is be AUX iajs-1955 20 6 weak weak ADJ iajs-1955 20 7 essential essential ADJ iajs-1955 20 8 in in ADP iajs-1955 20 9 l l NOUN iajs-1955 20 10 , , PUNCT iajs-1955 20 11 where where SCONJ iajs-1955 20 12 l l NOUN iajs-1955 20 13 is be AUX iajs-1955 20 14 a a DET iajs-1955 20 15 submodule submodule NOUN iajs-1955 20 16 of of ADP iajs-1955 20 17 m m PROPN iajs-1955 20 18 , , PUNCT iajs-1955 20 19 then then ADV iajs-1955 20 20 k k X iajs-1955 20 21 = = NOUN iajs-1955 20 22 l l NOUN iajs-1955 20 23 . . PUNCT iajs-1955 21 1 a a DET iajs-1955 21 2 module module NOUN iajs-1955 21 3 m m VERB iajs-1955 21 4 is be AUX iajs-1955 21 5 called call VERB iajs-1955 21 6 chaine chaine ADJ iajs-1955 21 7 if if SCONJ iajs-1955 21 8 for for ADP iajs-1955 21 9 each each DET iajs-1955 21 10 submodules submodule NOUN iajs-1955 21 11 e e NOUN iajs-1955 21 12 and and CCONJ iajs-1955 21 13 d d PROPN iajs-1955 21 14 of of ADP iajs-1955 21 15 m m PROPN iajs-1955 21 16 either either CCONJ iajs-1955 21 17 𝐸 𝐸 PROPN iajs-1955 22 1 ⊆ ⊆ NUM iajs-1955 22 2 𝐷 𝐷 NOUN iajs-1955 22 3 𝑜𝑟 𝑜𝑟 PRON iajs-1955 22 4 𝐷 𝐷 PROPN iajs-1955 22 5 ⊆ ⊆ NUM iajs-1955 22 6 𝐸 𝐸 PROPN iajs-1955 23 1 [ [ X iajs-1955 23 2 6 6 NUM iajs-1955 23 3 ] ] PUNCT iajs-1955 23 4 . . PUNCT iajs-1955 24 1 an an DET iajs-1955 24 2 r r NOUN iajs-1955 24 3 - - PUNCT iajs-1955 24 4 module module NOUN iajs-1955 24 5 m m NOUN iajs-1955 24 6 is be AUX iajs-1955 24 7 called call VERB iajs-1955 24 8 fully fully ADV iajs-1955 24 9 semi semi ADJ iajs-1955 24 10 - - ADJ iajs-1955 24 11 prime prime ADJ iajs-1955 24 12 , , PUNCT iajs-1955 24 13 if if SCONJ iajs-1955 24 14 every every DET iajs-1955 24 15 proper proper ADJ iajs-1955 24 16 submodule submodule NOUN iajs-1955 24 17 of of ADP iajs-1955 24 18 m m PROPN iajs-1955 24 19 is be AUX iajs-1955 24 20 semi semi ADJ iajs-1955 24 21 - - ADJ iajs-1955 24 22 prime prime ADJ iajs-1955 24 23 submodule submodule NOUN iajs-1955 25 1 [ [ X iajs-1955 25 2 3].a 3].a NUM iajs-1955 25 3 semi semi ADJ iajs-1955 25 4 - - ADJ iajs-1955 25 5 prime prime ADJ iajs-1955 25 6 radical radical NOUN iajs-1955 25 7 of of ADP iajs-1955 25 8 a a DET iajs-1955 25 9 module module NOUN iajs-1955 25 10 m m VERB iajs-1955 25 11 denoted denote VERB iajs-1955 25 12 by by ADP iajs-1955 25 13 srad srad PROPN iajs-1955 25 14 ( ( PUNCT iajs-1955 25 15 m m PROPN iajs-1955 25 16 ) ) PUNCT iajs-1955 25 17 , , PUNCT iajs-1955 25 18 and and CCONJ iajs-1955 25 19 it it PRON iajs-1955 25 20 is be AUX iajs-1955 25 21 the the DET iajs-1955 25 22 intersection intersection NOUN iajs-1955 25 23 of of ADP iajs-1955 25 24 all all DET iajs-1955 25 25 semi semi ADJ iajs-1955 25 26 - - ADJ iajs-1955 25 27 prime prime ADJ iajs-1955 25 28 submodule submodule NOUN iajs-1955 25 29 of of ADP iajs-1955 25 30 m m PROPN iajs-1955 25 31 [ [ X iajs-1955 25 32 3 3 NUM iajs-1955 25 33 ] ] PUNCT iajs-1955 25 34 . . PUNCT iajs-1955 26 1 a a DET iajs-1955 26 2 submodule submodule PROPN iajs-1955 26 3 n n PROPN iajs-1955 26 4 of of ADP iajs-1955 26 5 a a DET iajs-1955 26 6 module module NOUN iajs-1955 26 7 m m VERB iajs-1955 26 8 is be AUX iajs-1955 26 9 called call VERB iajs-1955 26 10 y y NOUN iajs-1955 26 11 - - PUNCT iajs-1955 26 12 closed close VERB iajs-1955 26 13 submodule submodule NOUN iajs-1955 26 14 in in ADP iajs-1955 26 15 m m PROPN iajs-1955 26 16 , , PUNCT iajs-1955 26 17 if if SCONJ iajs-1955 26 18 is be AUX iajs-1955 26 19 a a DET iajs-1955 26 20 non non ADJ iajs-1955 26 21 - - ADJ iajs-1955 26 22 singular singular ADJ iajs-1955 26 23 module module NOUN iajs-1955 26 24 [ [ X iajs-1955 26 25 1],"where 1],"where PROPN iajs-1955 26 26 an an DET iajs-1955 26 27 rmodule rmodule NOUN iajs-1955 26 28 m m VERB iajs-1955 26 29 is be AUX iajs-1955 26 30 called call VERB iajs-1955 26 31 non non ADJ iajs-1955 26 32 - - ADJ iajs-1955 26 33 singular singular ADJ iajs-1955 26 34 if if SCONJ iajs-1955 26 35 𝑍 𝑍 ADP iajs-1955 26 36 𝑀 𝑀 PROPN iajs-1955 26 37 𝑥 𝑥 NOUN iajs-1955 26 38 ∈ ∈ PROPN iajs-1955 26 39 𝑀 𝑀 PROPN iajs-1955 26 40 : : PUNCT iajs-1955 26 41 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-1955 26 42 𝑥 𝑥 PRON iajs-1955 26 43 𝑖𝑠 𝑖𝑠 INTJ iajs-1955 26 44 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 ADJ iajs-1955 26 45 𝑖𝑑𝑒𝑎𝑙 𝑖𝑑𝑒𝑎𝑙 ADJ iajs-1955 26 46 𝑖𝑛 𝑖𝑛 PRON iajs-1955 26 47 𝑅 𝑅 PROPN iajs-1955 26 48 =( =( NOUN iajs-1955 26 49 0 0 NUM iajs-1955 26 50 ) ) PUNCT iajs-1955 27 1 [ [ X iajs-1955 27 2 3 3 NUM iajs-1955 27 3 ] ] X iajs-1955 27 4 " " PUNCT iajs-1955 27 5 . . PUNCT iajs-1955 28 1 a a DET iajs-1955 28 2 module module NOUN iajs-1955 28 3 m m VERB iajs-1955 28 4 is be AUX iajs-1955 28 5 called call VERB iajs-1955 28 6 multiplication multiplication NOUN iajs-1955 28 7 module module NOUN iajs-1955 28 8 , , PUNCT iajs-1955 28 9 if if SCONJ iajs-1955 28 10 every every DET iajs-1955 28 11 submodule submodule NOUN iajs-1955 28 12 n n PROPN iajs-1955 28 13 of of ADP iajs-1955 28 14 m m PROPN iajs-1955 28 15 is be AUX iajs-1955 28 16 equal equal ADJ iajs-1955 28 17 i i PRON iajs-1955 28 18 m. m. NOUN iajs-1955 28 19 i.e i.e PROPN iajs-1955 28 20 n n NOUN iajs-1955 28 21 = = NOUN iajs-1955 28 22 im im NOUN iajs-1955 28 23 for for ADP iajs-1955 28 24 some some DET iajs-1955 28 25 ideal ideal NOUN iajs-1955 28 26 i i PRON iajs-1955 28 27 of of ADP iajs-1955 28 28 r r NOUN iajs-1955 29 1 [ [ X iajs-1955 29 2 7 7 NUM iajs-1955 29 3 ] ] PUNCT iajs-1955 29 4 . . PUNCT iajs-1955 30 1 basic basic ADJ iajs-1955 30 2 properties property NOUN iajs-1955 30 3 of of ADP iajs-1955 30 4 w w NOUN iajs-1955 30 5 - - PUNCT iajs-1955 30 6 closed closed ADJ iajs-1955 30 7 submodules submodule NOUN iajs-1955 30 8 " " PUNCT iajs-1955 30 9 in in ADP iajs-1955 30 10 this this DET iajs-1955 30 11 section section NOUN iajs-1955 30 12 , , PUNCT iajs-1955 30 13 we we PRON iajs-1955 30 14 introduce introduce VERB iajs-1955 30 15 the the DET iajs-1955 30 16 definition definition NOUN iajs-1955 30 17 of of ADP iajs-1955 30 18 " " PUNCT iajs-1955 30 19 w w NOUN iajs-1955 30 20 - - PUNCT iajs-1955 30 21 closed closed ADJ iajs-1955 30 22 submodule submodule NOUN iajs-1955 30 23 , , PUNCT iajs-1955 30 24 and and CCONJ iajs-1955 30 25 we we PRON iajs-1955 30 26 will will AUX iajs-1955 30 27 give give VERB iajs-1955 30 28 basic basic ADJ iajs-1955 30 29 properties property NOUN iajs-1955 30 30 , , PUNCT iajs-1955 30 31 examples example NOUN iajs-1955 30 32 of of ADP iajs-1955 30 33 w w NOUN iajs-1955 30 34 - - PUNCT iajs-1955 30 35 closed closed ADJ iajs-1955 30 36 submodule submodule NOUN iajs-1955 30 37 . . PUNCT iajs-1955 31 1 definition definition NOUN iajs-1955 31 2 ( ( PUNCT iajs-1955 31 3 2.1 2.1 NUM iajs-1955 31 4 ) ) PUNCT iajs-1955 31 5 asubmodule asubmodule NOUN iajs-1955 31 6 k k PROPN iajs-1955 31 7 of of ADP iajs-1955 31 8 a a DET iajs-1955 31 9 module module NOUN iajs-1955 31 10 m m VERB iajs-1955 31 11 is be AUX iajs-1955 31 12 called call VERB iajs-1955 31 13 w w NOUN iajs-1955 31 14 - - PUNCT iajs-1955 31 15 closed closed ADJ iajs-1955 31 16 in in ADP iajs-1955 31 17 m m PROPN iajs-1955 31 18 , , PUNCT iajs-1955 31 19 " " PUNCT iajs-1955 31 20 if if SCONJ iajs-1955 31 21 k k PROPN iajs-1955 31 22 has have VERB iajs-1955 31 23 no no DET iajs-1955 31 24 proper proper ADJ iajs-1955 31 25 weak weak ADJ iajs-1955 31 26 essential essential ADJ iajs-1955 31 27 extension extension NOUN iajs-1955 31 28 in in ADP iajs-1955 31 29 m m NOUN iajs-1955 31 30 " " PUNCT iajs-1955 31 31 . . PUNCT iajs-1955 32 1 that that PRON iajs-1955 32 2 is be AUX iajs-1955 32 3 if if SCONJ iajs-1955 32 4 there there PRON iajs-1955 32 5 exists exist VERB iajs-1955 32 6 asubmodule asubmodule NOUN iajs-1955 32 7 l l NOUN iajs-1955 32 8 of of ADP iajs-1955 32 9 m m PROPN iajs-1955 32 10 with with ADP iajs-1955 32 11 k k PROPN iajs-1955 32 12 " " PUNCT iajs-1955 32 13 is be AUX iajs-1955 32 14 a a DET iajs-1955 32 15 weak weak ADJ iajs-1955 32 16 essential essential ADJ iajs-1955 32 17 submodule submodule NOUN iajs-1955 32 18 of of ADP iajs-1955 32 19 l l PROPN iajs-1955 32 20 " " PUNCT iajs-1955 32 21 , , PUNCT iajs-1955 32 22 then then ADV iajs-1955 32 23 k k X iajs-1955 32 24 = = NOUN iajs-1955 32 25 l l NOUN iajs-1955 32 26 . . PUNCT iajs-1955 33 1 an an DET iajs-1955 33 2 ideal ideal ADJ iajs-1955 33 3 j j PROPN iajs-1955 33 4 of of ADP iajs-1955 33 5 r r NOUN iajs-1955 33 6 is be AUX iajs-1955 33 7 called call VERB iajs-1955 33 8 w w NOUN iajs-1955 33 9 - - PUNCT iajs-1955 33 10 closed closed ADJ iajs-1955 33 11 , , PUNCT iajs-1955 33 12 if if SCONJ iajs-1955 33 13 it it PRON iajs-1955 33 14 is be AUX iajs-1955 33 15 w w NOUN iajs-1955 33 16 - - PUNCT iajs-1955 33 17 closed closed ADJ iajs-1955 33 18 rsubmodule rsubmodule NOUN iajs-1955 33 19 . . PUNCT iajs-1955 34 1 remark remark NOUN iajs-1955 34 2 ( ( PUNCT iajs-1955 34 3 2.2 2.2 NUM iajs-1955 34 4 ) ) PUNCT iajs-1955 34 5 every every DET iajs-1955 34 6 w w NOUN iajs-1955 34 7 - - PUNCT iajs-1955 34 8 closed closed ADJ iajs-1955 34 9 submodule submodule NOUN iajs-1955 34 10 in in ADP iajs-1955 34 11 a a DET iajs-1955 34 12 module module NOUN iajs-1955 34 13 m m NOUN iajs-1955 34 14 is be AUX iajs-1955 34 15 a a DET iajs-1955 34 16 closed closed ADJ iajs-1955 34 17 submodule submodule NOUN iajs-1955 34 18 in in ADP iajs-1955 34 19 m m PROPN iajs-1955 34 20 , , PUNCT iajs-1955 34 21 but but CCONJ iajs-1955 34 22 the the DET iajs-1955 34 23 converse converse NOUN iajs-1955 34 24 is be AUX iajs-1955 34 25 not not PART iajs-1955 34 26 true true ADJ iajs-1955 34 27 in in ADP iajs-1955 34 28 general general ADJ iajs-1955 34 29 . . PUNCT iajs-1955 35 1 proof proof NOUN iajs-1955 35 2 let let VERB iajs-1955 35 3 k k PROPN iajs-1955 35 4 be be AUX iajs-1955 35 5 a a DET iajs-1955 35 6 w w NOUN iajs-1955 35 7 - - PUNCT iajs-1955 35 8 closed closed ADJ iajs-1955 35 9 submodule submodule NOUN iajs-1955 35 10 in in ADP iajs-1955 35 11 m m PROPN iajs-1955 35 12 and and CCONJ iajs-1955 35 13 l l NOUN iajs-1955 35 14 is be AUX iajs-1955 35 15 a a DET iajs-1955 35 16 submodule submodule NOUN iajs-1955 35 17 in in ADP iajs-1955 35 18 m m PROPN iajs-1955 35 19 with with ADP iajs-1955 35 20 k k PROPN iajs-1955 35 21 is be AUX iajs-1955 35 22 essential essential ADJ iajs-1955 35 23 in in ADP iajs-1955 35 24 l l NOUN iajs-1955 35 25 , , PUNCT iajs-1955 35 26 then then ADV iajs-1955 35 27 by by ADP iajs-1955 35 28 [ [ X iajs-1955 35 29 2 2 NUM iajs-1955 35 30 ] ] X iajs-1955 35 31 k k X iajs-1955 35 32 is be AUX iajs-1955 35 33 weak weak ADJ iajs-1955 35 34 essential essential ADJ iajs-1955 35 35 in in ADP iajs-1955 35 36 l. l. PROPN iajs-1955 35 37 but but CCONJ iajs-1955 35 38 k k PROPN iajs-1955 35 39 is be AUX iajs-1955 35 40 w w NOUN iajs-1955 35 41 - - PUNCT iajs-1955 35 42 closed closed ADJ iajs-1955 35 43 in in ADP iajs-1955 35 44 m m PROPN iajs-1955 35 45 , , PUNCT iajs-1955 35 46 thus thus ADV iajs-1955 35 47 k k X iajs-1955 35 48 = = PROPN iajs-1955 35 49 l. l. NOUN iajs-1955 35 50 hence hence ADV iajs-1955 35 51 k k PROPN iajs-1955 35 52 is be AUX iajs-1955 35 53 closed close VERB iajs-1955 35 54 submodule submodule NOUN iajs-1955 35 55 in in ADP iajs-1955 35 56 m. m. NOUN iajs-1955 35 57 for for ADP iajs-1955 35 58 the the DET iajs-1955 35 59 converse converse NOUN iajs-1955 35 60 , , PUNCT iajs-1955 35 61 we we PRON iajs-1955 35 62 give give VERB iajs-1955 35 63 the the DET iajs-1955 35 64 following follow VERB iajs-1955 35 65 example example NOUN iajs-1955 35 66 : : PUNCT iajs-1955 35 67 3)(2.example 3)(2.example NUM iajs-1955 35 68 let let VERB iajs-1955 35 69 m=𝑍 m=𝑍 NOUN iajs-1955 35 70 as as ADP iajs-1955 35 71 a a DET iajs-1955 35 72 z z NOUN iajs-1955 35 73 - - PUNCT iajs-1955 35 74 module module NOUN iajs-1955 35 75 , , PUNCT iajs-1955 35 76 and and CCONJ iajs-1955 35 77 𝐾 𝐾 PROPN iajs-1955 35 78 〈 〈 NOUN iajs-1955 35 79 3 3 NUM iajs-1955 35 80 〉 〉 NOUN iajs-1955 35 81 is be AUX iajs-1955 35 82 closed close VERB iajs-1955 35 83 submodule submodule NOUN iajs-1955 35 84 in in ADP iajs-1955 35 85 𝑍 𝑍 PROPN iajs-1955 35 86 , , PUNCT iajs-1955 35 87 since since SCONJ iajs-1955 35 88 k k PROPN iajs-1955 35 89 is be AUX iajs-1955 35 90 a a DET iajs-1955 35 91 direct direct ADJ iajs-1955 35 92 summand summand NOUN iajs-1955 35 93 of of ADP iajs-1955 35 94 the the DET iajs-1955 35 95 z z NOUN iajs-1955 35 96 - - PUNCT iajs-1955 35 97 module module NOUN iajs-1955 35 98 𝑍 𝑍 NOUN iajs-1955 35 99 , , PUNCT iajs-1955 35 100 but but CCONJ iajs-1955 35 101 k k PROPN iajs-1955 35 102 is be AUX iajs-1955 35 103 not not PART iajs-1955 35 104 w w NOUN iajs-1955 35 105 - - PUNCT iajs-1955 35 106 closed closed ADJ iajs-1955 35 107 submodule submodule NOUN iajs-1955 35 108 in in ADP iajs-1955 35 109 𝑍 𝑍 PROPN iajs-1955 35 110 because because SCONJ iajs-1955 35 111 k k PROPN iajs-1955 35 112 is be AUX iajs-1955 35 113 weak weak ADJ iajs-1955 35 114 essential essential ADJ iajs-1955 35 115 submodule submodule NOUN iajs-1955 35 116 in in ADP iajs-1955 35 117 𝑍 𝑍 PROPN iajs-1955 35 118 . . PUNCT iajs-1955 36 1 proposition proposition NOUN iajs-1955 36 2 ( ( PUNCT iajs-1955 36 3 2.4 2.4 NUM iajs-1955 36 4 ) ) PUNCT iajs-1955 36 5 if if SCONJ iajs-1955 36 6 m m NOUN iajs-1955 36 7 is be AUX iajs-1955 36 8 a a DET iajs-1955 36 9 module module NOUN iajs-1955 36 10 , , PUNCT iajs-1955 36 11 and and CCONJ iajs-1955 36 12 e e NOUN iajs-1955 36 13 is be AUX iajs-1955 36 14 a a DET iajs-1955 36 15 submodule submodule NOUN iajs-1955 36 16 of of ADP iajs-1955 36 17 m m PRON iajs-1955 36 18 such such ADJ iajs-1955 36 19 that that SCONJ iajs-1955 36 20 e e NOUN iajs-1955 36 21 is be AUX iajs-1955 36 22 weak weak ADJ iajs-1955 36 23 essential essential ADJ iajs-1955 36 24 and and CCONJ iajs-1955 36 25 w w NOUN iajs-1955 36 26 - - PUNCT iajs-1955 36 27 closed closed ADJ iajs-1955 36 28 in in ADP iajs-1955 36 29 m m PROPN iajs-1955 36 30 , , PUNCT iajs-1955 36 31 then then ADV iajs-1955 36 32 e e NOUN iajs-1955 36 33 = = NOUN iajs-1955 36 34 m. m. NOUN iajs-1955 36 35             SPACE iajs-1955 36 36 166 166 NUM iajs-1955 36 37     SPACE iajs-1955 36 38 | | ADV iajs-1955 36 39     SPACE iajs-1955 36 40 mathmatics mathmatic NOUN iajs-1955 36 41 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 36 42     SPACE iajs-1955 36 43 2018 2018 NUM iajs-1955 36 44 ) ) PUNCT iajs-1955 36 45 عام عام ADP iajs-1955 36 46 2العدد 2العدد NUM iajs-1955 36 47 ( ( PUNCT iajs-1955 36 48 31لمجلد 31لمجلد NUM iajs-1955 36 49 ا ا NOUN iajs-1955 36 50 مجلة مجلة NOUN iajs-1955 36 51 إبن إبن VERB iajs-1955 36 52 الهيثم الهيثم ADJ iajs-1955 36 53 للعلوم للعلوم NOUN iajs-1955 36 54 الصرفة الصرفة NOUN iajs-1955 37 1 و و PRON iajs-1955 37 2 التطبيقية التطبيقية ADJ iajs-1955 37 3 ibn ibn PROPN iajs-1955 37 4 al al PROPN iajs-1955 37 5 - - PUNCT iajs-1955 37 6 haitham haitham PROPN iajs-1955 37 7 jour jour X iajs-1955 37 8 . . PROPN iajs-1955 37 9 for for ADP iajs-1955 37 10 pure pure ADJ iajs-1955 37 11 & & CCONJ iajs-1955 37 12 appl appl PROPN iajs-1955 37 13 . . PUNCT iajs-1955 38 1 sci sci PROPN iajs-1955 38 2 . . PUNCT iajs-1955 38 3 vol vol NOUN iajs-1955 38 4 . . PROPN iajs-1955 39 1 31 31 NUM iajs-1955 39 2 ( ( PUNCT iajs-1955 39 3 2 2 NUM iajs-1955 39 4 ) ) PUNCT iajs-1955 39 5 2018 2018 NUM iajs-1955 39 6 proof proof NOUN iajs-1955 39 7 follows follow VERB iajs-1955 39 8 from from ADP iajs-1955 39 9 definition definition NOUN iajs-1955 39 10 of of ADP iajs-1955 39 11 w w NOUN iajs-1955 39 12 - - PUNCT iajs-1955 39 13 closed closed ADJ iajs-1955 39 14 submodule submodule NOUN iajs-1955 39 15 . . PUNCT iajs-1955 40 1 remark remark NOUN iajs-1955 40 2 ( ( PUNCT iajs-1955 40 3 2.5 2.5 NUM iajs-1955 40 4 ) ) PUNCT iajs-1955 40 5 ( ( PUNCT iajs-1955 40 6 1 1 X iajs-1955 40 7 ) ) PUNCT iajs-1955 40 8 every every DET iajs-1955 40 9 module module NOUN iajs-1955 40 10 m m VERB iajs-1955 40 11 is be AUX iajs-1955 40 12 a a DET iajs-1955 40 13 w w NOUN iajs-1955 40 14 - - PUNCT iajs-1955 40 15 closed closed ADJ iajs-1955 40 16 submodule submodule NOUN iajs-1955 40 17 in in ADP iajs-1955 40 18 itself itself PRON iajs-1955 40 19 . . PUNCT iajs-1955 41 1 ( ( PUNCT iajs-1955 41 2 2 2 X iajs-1955 41 3 ) ) PUNCT iajs-1955 41 4 the the DET iajs-1955 41 5 trivial trivial ADJ iajs-1955 41 6 submodule submodule NOUN iajs-1955 41 7 < < X iajs-1955 41 8 0 0 NUM iajs-1955 41 9 > > X iajs-1955 41 10 may may AUX iajs-1955 41 11 not not PART iajs-1955 41 12 be be AUX iajs-1955 41 13 w w NOUN iajs-1955 41 14 - - PUNCT iajs-1955 41 15 closed closed ADJ iajs-1955 41 16 submodule submodule NOUN iajs-1955 41 17 of of ADP iajs-1955 41 18 an an DET iajs-1955 41 19 r r NOUN iajs-1955 41 20 - - PUNCT iajs-1955 41 21 module module NOUN iajs-1955 41 22 m m NOUN iajs-1955 41 23 , , PUNCT iajs-1955 41 24 for for ADP iajs-1955 41 25 example example NOUN iajs-1955 41 26 : : PUNCT iajs-1955 41 27 𝑀 𝑀 PROPN iajs-1955 41 28 𝑍 𝑍 VERB iajs-1955 41 29 as as ADP iajs-1955 41 30 a a DET iajs-1955 41 31 z z NOUN iajs-1955 41 32 - - PUNCT iajs-1955 41 33 module module NOUN iajs-1955 41 34 , , PUNCT iajs-1955 41 35 𝐾 𝐾 PROPN iajs-1955 41 36 〈 〈 NOUN iajs-1955 41 37 0 0 NUM iajs-1955 41 38 〉 〉 NOUN iajs-1955 41 39 is be AUX iajs-1955 41 40 not not PART iajs-1955 41 41 w w NOUN iajs-1955 41 42 - - PUNCT iajs-1955 41 43 closed closed ADJ iajs-1955 41 44 submodule submodule NOUN iajs-1955 41 45 in in ADP iajs-1955 41 46 m. m. NOUN iajs-1955 41 47 proposition(2.6 proposition(2.6 PROPN iajs-1955 41 48 ) ) PUNCT iajs-1955 41 49 if if SCONJ iajs-1955 41 50 m m NOUN iajs-1955 41 51 is be AUX iajs-1955 41 52 a a DET iajs-1955 41 53 module module NOUN iajs-1955 41 54 , , PUNCT iajs-1955 41 55 and and CCONJ iajs-1955 41 56 let let VERB iajs-1955 41 57 u u PRON iajs-1955 41 58 be be AUX iajs-1955 41 59 a a DET iajs-1955 41 60 non non ADJ iajs-1955 41 61 - - ADJ iajs-1955 41 62 zero zero NUM iajs-1955 41 63 submodule submodule NOUN iajs-1955 41 64 of of ADP iajs-1955 41 65 m m PROPN iajs-1955 41 66 , , PUNCT iajs-1955 41 67 then then ADV iajs-1955 41 68 ∃ ∃ PROPN iajs-1955 41 69 a a DET iajs-1955 41 70 w w NOUN iajs-1955 41 71 - - PUNCT iajs-1955 41 72 closed close VERB iajs-1955 41 73 submodule submodule NOUN iajs-1955 41 74 t t PROPN iajs-1955 41 75 in in ADP iajs-1955 41 76 m m PROPN iajs-1955 41 77 with with ADP iajs-1955 41 78 u u NOUN iajs-1955 41 79 is be AUX iajs-1955 41 80 weak weak ADJ iajs-1955 41 81 essential essential ADJ iajs-1955 41 82 in in ADP iajs-1955 41 83 t. t. NOUN iajs-1955 41 84 proof proof NOUN iajs-1955 41 85 let let VERB iajs-1955 41 86 𝒜 𝒜 NOUN iajs-1955 41 87 = = NOUN iajs-1955 41 88 { { PUNCT iajs-1955 41 89 q q NOUN iajs-1955 41 90 : : PUNCT iajs-1955 41 91 q q X iajs-1955 41 92 " " PUNCT iajs-1955 41 93 is be AUX iajs-1955 41 94 a a DET iajs-1955 41 95 submodule submodule NOUN iajs-1955 41 96 of of ADP iajs-1955 41 97 m m PRON iajs-1955 41 98 such such ADJ iajs-1955 41 99 that that SCONJ iajs-1955 41 100 " " PUNCT iajs-1955 41 101 u u NOUN iajs-1955 41 102 is be AUX iajs-1955 41 103 weak weak ADJ iajs-1955 41 104 essential essential ADJ iajs-1955 41 105 in in ADP iajs-1955 41 106 q q PROPN iajs-1955 41 107 } } PUNCT iajs-1955 41 108 . . PUNCT iajs-1955 42 1 clearly clearly ADV iajs-1955 42 2 𝒜 𝒜 NOUN iajs-1955 42 3 is be AUX iajs-1955 42 4 a a DET iajs-1955 42 5 non non ADJ iajs-1955 42 6 - - ADJ iajs-1955 42 7 empty empty ADJ iajs-1955 42 8 . . PUNCT iajs-1955 43 1 𝒜 𝒜 NOUN iajs-1955 43 2 has have VERB iajs-1955 43 3 maximal maximal ADJ iajs-1955 43 4 element element NOUN iajs-1955 43 5 say say VERB iajs-1955 43 6 t t NOUN iajs-1955 43 7 " " PUNCT iajs-1955 43 8 by by ADP iajs-1955 43 9 zorn zorn PROPN iajs-1955 43 10 's 's PART iajs-1955 43 11 lemma lemma PROPN iajs-1955 43 12 ” " PUNCT iajs-1955 43 13 . . PUNCT iajs-1955 44 1 " " PUNCT iajs-1955 44 2 to to PART iajs-1955 44 3 prove prove VERB iajs-1955 44 4 that that SCONJ iajs-1955 44 5 " " PUNCT iajs-1955 44 6 t t PROPN iajs-1955 44 7 is be AUX iajs-1955 44 8 a a DET iajs-1955 44 9 wclosed wclosed ADJ iajs-1955 44 10 submodule submodule NOUN iajs-1955 44 11 in in ADP iajs-1955 44 12 m. m. NOUN iajs-1955 44 13 assume assume VERB iajs-1955 44 14 that that SCONJ iajs-1955 44 15 there there PRON iajs-1955 44 16 exists exist VERB iajs-1955 44 17 a a DET iajs-1955 44 18 submodule submodule NOUN iajs-1955 44 19 l l NOUN iajs-1955 44 20 of of ADP iajs-1955 44 21 m m PROPN iajs-1955 44 22 with with ADP iajs-1955 44 23 t t PROPN iajs-1955 44 24 weak weak ADJ iajs-1955 44 25 essential essential ADJ iajs-1955 44 26 in in ADP iajs-1955 44 27 l. l. NOUN iajs-1955 44 28 since since SCONJ iajs-1955 44 29 u u NOUN iajs-1955 44 30 is be AUX iajs-1955 44 31 weak weak ADJ iajs-1955 44 32 essential essential ADJ iajs-1955 44 33 in in ADP iajs-1955 44 34 t t PROPN iajs-1955 44 35 and and CCONJ iajs-1955 44 36 t t PROPN iajs-1955 44 37 is be AUX iajs-1955 44 38 weak weak ADJ iajs-1955 44 39 essential essential ADJ iajs-1955 44 40 in in ADP iajs-1955 44 41 l l NOUN iajs-1955 44 42 so so ADV iajs-1955 44 43 by by ADP iajs-1955 44 44 [ [ X iajs-1955 44 45 3 3 NUM iajs-1955 44 46 , , PUNCT iajs-1955 44 47 prop prop NOUN iajs-1955 44 48 ( ( PUNCT iajs-1955 44 49 1.4 1.4 NUM iajs-1955 44 50 ) ) PUNCT iajs-1955 44 51 ] ] PUNCT iajs-1955 44 52 . . PUNCT iajs-1955 45 1 u u NOUN iajs-1955 45 2 is be AUX iajs-1955 45 3 weak weak ADJ iajs-1955 45 4 essential essential ADJ iajs-1955 45 5 in in ADP iajs-1955 45 6 l. l. PROPN iajs-1955 45 7 but but CCONJ iajs-1955 45 8 this this PRON iajs-1955 45 9 is be AUX iajs-1955 45 10 a a DET iajs-1955 45 11 contradicts contradict VERB iajs-1955 45 12 the the DET iajs-1955 45 13 maximality maximality NOUN iajs-1955 45 14 of of ADP iajs-1955 45 15 t.thus t.thus NOUN iajs-1955 45 16 t t PROPN iajs-1955 45 17 = = SYM iajs-1955 45 18 l. l. PROPN iajs-1955 45 19 hence hence ADV iajs-1955 45 20 t t PROPN iajs-1955 45 21 is be AUX iajs-1955 45 22 wclosed wclose VERB iajs-1955 45 23 submodule submodule NOUN iajs-1955 45 24 in in ADP iajs-1955 45 25 m m PROPN iajs-1955 45 26 , , PUNCT iajs-1955 45 27 with with SCONJ iajs-1955 45 28 u u NOUN iajs-1955 45 29 is be AUX iajs-1955 45 30 weak weak ADJ iajs-1955 45 31 essential essential ADJ iajs-1955 45 32 in in ADP iajs-1955 45 33 t. t. PROPN iajs-1955 45 34 the the DET iajs-1955 45 35 following follow VERB iajs-1955 45 36 remark remark NOUN iajs-1955 45 37 shows show VERB iajs-1955 45 38 that that SCONJ iajs-1955 45 39 w w NOUN iajs-1955 45 40 - - PUNCT iajs-1955 45 41 closed closed ADJ iajs-1955 45 42 property property NOUN iajs-1955 45 43 is be AUX iajs-1955 45 44 not not PART iajs-1955 45 45 hereditary hereditary ADJ iajs-1955 45 46 property property NOUN iajs-1955 45 47 . . PUNCT iajs-1955 46 1 remark(2.7 remark(2.7 PROPN iajs-1955 46 2 ) ) PUNCT iajs-1955 46 3 if if SCONJ iajs-1955 46 4 𝑄 𝑄 PRON iajs-1955 46 5 and and CCONJ iajs-1955 46 6 𝑄 𝑄 PROPN iajs-1955 46 7 are be AUX iajs-1955 46 8 submodules submodule NOUN iajs-1955 46 9 of of ADP iajs-1955 46 10 an an DET iajs-1955 46 11 r r NOUN iajs-1955 46 12 - - PUNCT iajs-1955 46 13 module module NOUN iajs-1955 46 14 m m NOUN iajs-1955 46 15 with with ADP iajs-1955 46 16 𝑄 𝑄 PROPN iajs-1955 46 17 is be AUX iajs-1955 46 18 a a DET iajs-1955 46 19 submodule submodule NOUN iajs-1955 46 20 of of ADP iajs-1955 46 21 𝑄 𝑄 PROPN iajs-1955 46 22 , , PUNCT iajs-1955 46 23 and and CCONJ iajs-1955 46 24 𝑄 𝑄 PRON iajs-1955 46 25 is be AUX iajs-1955 46 26 a a DET iajs-1955 46 27 w w NOUN iajs-1955 46 28 - - PUNCT iajs-1955 46 29 closed closed ADJ iajs-1955 46 30 submodule submodule NOUN iajs-1955 46 31 in in ADP iajs-1955 46 32 m m PROPN iajs-1955 46 33 then then ADV iajs-1955 46 34 𝑄 𝑄 PRON iajs-1955 46 35 need need VERB iajs-1955 46 36 not not PART iajs-1955 46 37 to to PART iajs-1955 46 38 be be AUX iajs-1955 46 39 w w NOUN iajs-1955 46 40 - - PUNCT iajs-1955 46 41 closed closed ADJ iajs-1955 46 42 submodule submodule NOUN iajs-1955 46 43 in in ADP iajs-1955 46 44 m. m. NOUN iajs-1955 46 45 for for ADP iajs-1955 46 46 example example NOUN iajs-1955 46 47 : : PUNCT iajs-1955 46 48 m m PUNCT iajs-1955 46 49 = = NOUN iajs-1955 46 50 z z PROPN iajs-1955 46 51 the the DET iajs-1955 46 52 z z NOUN iajs-1955 46 53 - - PUNCT iajs-1955 46 54 module module NOUN iajs-1955 46 55 , , PUNCT iajs-1955 46 56 m m VERB iajs-1955 46 57 is be AUX iajs-1955 46 58 a a DET iajs-1955 46 59 w w NOUN iajs-1955 46 60 - - PUNCT iajs-1955 46 61 closed closed ADJ iajs-1955 46 62 submodule submodule NOUN iajs-1955 46 63 of of ADP iajs-1955 46 64 m m PROPN iajs-1955 46 65 , , PUNCT iajs-1955 46 66 and and CCONJ iajs-1955 46 67 2z 2z PRON iajs-1955 46 68 is be AUX iajs-1955 46 69 a a DET iajs-1955 46 70 submodule submodule NOUN iajs-1955 46 71 of of ADP iajs-1955 46 72 m m PROPN iajs-1955 46 73 is be AUX iajs-1955 46 74 not not PART iajs-1955 46 75 wclosed wclose VERB iajs-1955 46 76 submodule submodule NOUN iajs-1955 46 77 in in ADP iajs-1955 46 78 m m PROPN iajs-1955 46 79 , , PUNCT iajs-1955 46 80 since since SCONJ iajs-1955 46 81 2z 2z NUM iajs-1955 46 82 has have VERB iajs-1955 46 83 a a DET iajs-1955 46 84 proper proper ADJ iajs-1955 46 85 weak weak ADJ iajs-1955 46 86 essential essential ADJ iajs-1955 46 87 extension extension NOUN iajs-1955 46 88 . . PUNCT iajs-1955 47 1 the the DET iajs-1955 47 2 converse converse NOUN iajs-1955 47 3 of of ADP iajs-1955 47 4 remark remark NOUN iajs-1955 47 5 ( ( PUNCT iajs-1955 47 6 2.7 2.7 NUM iajs-1955 47 7 ) ) PUNCT iajs-1955 47 8 is be AUX iajs-1955 47 9 not not PART iajs-1955 47 10 true true ADJ iajs-1955 47 11 . . PUNCT iajs-1955 48 1 that that PRON iajs-1955 48 2 is be AUX iajs-1955 48 3 if if SCONJ iajs-1955 48 4 𝑄 𝑄 PRON iajs-1955 48 5 is be AUX iajs-1955 48 6 w w NOUN iajs-1955 48 7 - - PUNCT iajs-1955 48 8 closed closed ADJ iajs-1955 48 9 in in ADP iajs-1955 48 10 m m PROPN iajs-1955 48 11 , , PUNCT iajs-1955 48 12 then then ADV iajs-1955 48 13 𝑄 𝑄 PRON iajs-1955 48 14 need need VERB iajs-1955 48 15 not not PART iajs-1955 48 16 to to PART iajs-1955 48 17 be be AUX iajs-1955 48 18 w w NOUN iajs-1955 48 19 - - PUNCT iajs-1955 48 20 closed closed ADJ iajs-1955 48 21 in in ADP iajs-1955 48 22 m. m. NOUN iajs-1955 48 23 as as SCONJ iajs-1955 48 24 the the DET iajs-1955 48 25 next next ADJ iajs-1955 48 26 example example NOUN iajs-1955 48 27 explain explain VERB iajs-1955 48 28 : : PUNCT iajs-1955 48 29 example(2.8 example(2.8 ADV iajs-1955 48 30 ) ) PUNCT iajs-1955 48 31 take take VERB iajs-1955 48 32 the the DET iajs-1955 48 33 z z NOUN iajs-1955 48 34 - - PUNCT iajs-1955 48 35 module module NOUN iajs-1955 48 36 z z NOUN iajs-1955 48 37 and and CCONJ iajs-1955 49 1 𝑁 𝑁 PROPN iajs-1955 49 2 = = SYM iajs-1955 49 3 < < X iajs-1955 49 4 0 0 NUM iajs-1955 49 5 > > PUNCT iajs-1955 49 6 , , PUNCT iajs-1955 49 7 𝑁 𝑁 PROPN iajs-1955 49 8 = = PUNCT iajs-1955 49 9 2z 2z NOUN iajs-1955 49 10 are be AUX iajs-1955 49 11 z z NOUN iajs-1955 49 12 - - PUNCT iajs-1955 49 13 submodules submodule NOUN iajs-1955 49 14 of of ADP iajs-1955 49 15 z z NOUN iajs-1955 49 16 we we PRON iajs-1955 49 17 notes note VERB iajs-1955 49 18 that that SCONJ iajs-1955 49 19 𝑁 𝑁 PROPN iajs-1955 49 20 is be AUX iajs-1955 49 21 w w NOUN iajs-1955 49 22 - - PUNCT iajs-1955 49 23 closed closed ADJ iajs-1955 49 24 submodule submodule NOUN iajs-1955 49 25 in in ADP iajs-1955 49 26 z. z. PROPN iajs-1955 50 1 but but CCONJ iajs-1955 50 2 𝑁 𝑁 PROPN iajs-1955 50 3 is be AUX iajs-1955 50 4 not not PART iajs-1955 50 5 w w NOUN iajs-1955 50 6 - - PUNCT iajs-1955 50 7 closed closed ADJ iajs-1955 50 8 submodule submodule NOUN iajs-1955 50 9 in in ADP iajs-1955 50 10 z. z. PROPN iajs-1955 51 1 the the DET iajs-1955 51 2 following follow VERB iajs-1955 51 3 propositions proposition NOUN iajs-1955 51 4 show show VERB iajs-1955 51 5 that that SCONJ iajs-1955 51 6 the the DET iajs-1955 51 7 transitive transitive ADJ iajs-1955 51 8 property property NOUN iajs-1955 51 9 for for ADP iajs-1955 51 10 w w NOUN iajs-1955 51 11 - - PUNCT iajs-1955 51 12 closed closed ADJ iajs-1955 51 13 submodule submodule NOUN iajs-1955 51 14 hold hold NOUN iajs-1955 51 15 under under ADP iajs-1955 51 16 certain certain ADJ iajs-1955 51 17 conditions condition NOUN iajs-1955 51 18 . . PUNCT iajs-1955 52 1 proposition proposition NOUN iajs-1955 52 2 ( ( PUNCT iajs-1955 52 3 2.9 2.9 NUM iajs-1955 52 4 ) ) PUNCT iajs-1955 52 5 if if SCONJ iajs-1955 52 6 e e PROPN iajs-1955 52 7 and and CCONJ iajs-1955 52 8 d d PROPN iajs-1955 52 9 are be AUX iajs-1955 52 10 submodules submodule NOUN iajs-1955 52 11 of of ADP iajs-1955 52 12 a a DET iajs-1955 52 13 module module NOUN iajs-1955 52 14 m m NOUN iajs-1955 52 15 , , PUNCT iajs-1955 52 16 provided provide VERB iajs-1955 52 17 that that SCONJ iajs-1955 52 18 d d NOUN iajs-1955 52 19 contained contain VERB iajs-1955 52 20 in in ADP iajs-1955 52 21 any any DET iajs-1955 52 22 weak weak ADJ iajs-1955 52 23 essential essential ADJ iajs-1955 52 24 extensions extension NOUN iajs-1955 52 25 of of ADP iajs-1955 52 26 e e NOUN iajs-1955 52 27 , , PUNCT iajs-1955 52 28 and and CCONJ iajs-1955 52 29 e e NOUN iajs-1955 52 30 is be AUX iajs-1955 52 31 a a DET iajs-1955 52 32 w w NOUN iajs-1955 52 33 - - PUNCT iajs-1955 52 34 closed closed ADJ iajs-1955 52 35 submodule submodule NOUN iajs-1955 52 36 in in ADP iajs-1955 52 37 d d PROPN iajs-1955 52 38 and and CCONJ iajs-1955 52 39 d d PROPN iajs-1955 52 40 is be AUX iajs-1955 52 41 a a DET iajs-1955 52 42 w w NOUN iajs-1955 52 43 - - PUNCT iajs-1955 52 44 closed closed ADJ iajs-1955 52 45 submodule submodule NOUN iajs-1955 52 46 in in ADP iajs-1955 52 47 m m PROPN iajs-1955 52 48 , , PUNCT iajs-1955 52 49 then then ADV iajs-1955 52 50 e e PROPN iajs-1955 52 51 is be AUX iajs-1955 52 52 a a DET iajs-1955 52 53 w w ADV iajs-1955 52 54 - - PUNCT iajs-1955 52 55 closwed closwe VERB iajs-1955 52 56 submodule submodule NOUN iajs-1955 52 57 in in ADP iajs-1955 52 58 m. m. NOUN iajs-1955 52 59 proof proof NOUN iajs-1955 52 60 assume assume VERB iajs-1955 52 61 that that SCONJ iajs-1955 52 62 k k PROPN iajs-1955 52 63 is be AUX iajs-1955 52 64 a a DET iajs-1955 52 65 submodule submodule NOUN iajs-1955 52 66 of of ADP iajs-1955 52 67 m m PRON iajs-1955 52 68 such such ADJ iajs-1955 52 69 that that SCONJ iajs-1955 52 70 e e NOUN iajs-1955 52 71 is be AUX iajs-1955 52 72 weak weak ADJ iajs-1955 52 73 essential essential ADJ iajs-1955 52 74 in in ADP iajs-1955 52 75 k. k. NOUN iajs-1955 52 76 by by ADP iajs-1955 52 77 hypothesis hypothesis NOUN iajs-1955 52 78 d d NOUN iajs-1955 52 79 is be AUX iajs-1955 52 80 a a DET iajs-1955 52 81 submodule submodule NOUN iajs-1955 52 82 of of ADP iajs-1955 52 83 k. k. PROPN iajs-1955 52 84 since since SCONJ iajs-1955 52 85 e e PROPN iajs-1955 52 86 " " PUNCT iajs-1955 52 87 is be AUX iajs-1955 52 88 weak weak ADJ iajs-1955 52 89 essential essential ADJ iajs-1955 52 90 in in ADP iajs-1955 52 91 k k PROPN iajs-1955 52 92 and and CCONJ iajs-1955 52 93 e e PROPN iajs-1955 52 94 is be AUX iajs-1955 52 95 a a DET iajs-1955 52 96 submodule submodule NOUN iajs-1955 52 97 of of ADP iajs-1955 52 98 d d NOUN iajs-1955 52 99 " " PUNCT iajs-1955 52 100 then then ADV iajs-1955 52 101 by by ADP iajs-1955 52 102 [ [ X iajs-1955 52 103 2 2 NUM iajs-1955 52 104 , , PUNCT iajs-1955 52 105 rem(1.5)(2 rem(1.5)(2 PROPN iajs-1955 52 106 ) ) PUNCT iajs-1955 52 107 ] ] PUNCT iajs-1955 52 108 we we PRON iajs-1955 52 109 get get VERB iajs-1955 52 110 d d NOUN iajs-1955 52 111 is be AUX iajs-1955 52 112 weak weak ADJ iajs-1955 52 113 essential essential ADJ iajs-1955 52 114 in in ADP iajs-1955 52 115 k. k. PROPN iajs-1955 53 1 but but CCONJ iajs-1955 53 2 d d PROPN iajs-1955 53 3 is be AUX iajs-1955 53 4 w w NOUN iajs-1955 53 5 - - PUNCT iajs-1955 53 6 closed closed ADJ iajs-1955 53 7 submodule submodule NOUN iajs-1955 53 8 in in ADP iajs-1955 53 9 m m PROPN iajs-1955 53 10 , , PUNCT iajs-1955 53 11 then then ADV iajs-1955 53 12 d d X iajs-1955 53 13 = = PROPN iajs-1955 53 14 k. k. PROPN iajs-1955 53 15 that that PRON iajs-1955 53 16 is be AUX iajs-1955 53 17 e e NOUN iajs-1955 53 18 weak weak ADJ iajs-1955 53 19 essential essential ADJ iajs-1955 53 20 in in ADP iajs-1955 53 21 d. d. PROPN iajs-1955 54 1 but but CCONJ iajs-1955 54 2 e e PROPN iajs-1955 54 3 is be AUX iajs-1955 54 4 w w NOUN iajs-1955 54 5 - - PUNCT iajs-1955 54 6 closed closed ADJ iajs-1955 54 7 submodule submodule NOUN iajs-1955 54 8 in in ADP iajs-1955 54 9 d d PROPN iajs-1955 54 10 , , PUNCT iajs-1955 54 11 so so SCONJ iajs-1955 54 12 e e X iajs-1955 54 13 = = NOUN iajs-1955 54 14 d. d. NOUN iajs-1955 54 15 hence hence ADV iajs-1955 54 16 e e PROPN iajs-1955 54 17 is be AUX iajs-1955 54 18 a a DET iajs-1955 54 19 wclosed wclosed ADJ iajs-1955 54 20 submodule submodule NOUN iajs-1955 54 21 in in ADP iajs-1955 54 22 m. m. NOUN iajs-1955 54 23             SPACE iajs-1955 54 24 167 167 NUM iajs-1955 54 25     SPACE iajs-1955 54 26 | | ADV iajs-1955 54 27     SPACE iajs-1955 54 28 mathmatics mathmatic NOUN iajs-1955 54 29 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 54 30     SPACE iajs-1955 54 31 2018 2018 NUM iajs-1955 54 32 ) ) PUNCT iajs-1955 55 1 عام عام ADP iajs-1955 55 2 2العدد 2العدد NUM iajs-1955 55 3 ( ( PUNCT iajs-1955 55 4 31لمجلد 31لمجلد NUM iajs-1955 55 5 ا ا NOUN iajs-1955 55 6 مجلة مجلة NOUN iajs-1955 55 7 إبن إبن VERB iajs-1955 55 8 الهيثم الهيثم ADJ iajs-1955 55 9 للعلوم للعلوم NOUN iajs-1955 55 10 الصرفة الصرفة NOUN iajs-1955 56 1 و و PRON iajs-1955 56 2 التطبيقية التطبيقية ADJ iajs-1955 56 3 ibn ibn PROPN iajs-1955 56 4 al al PROPN iajs-1955 56 5 - - PUNCT iajs-1955 56 6 haitham haitham PROPN iajs-1955 56 7 jour jour X iajs-1955 56 8 . . PROPN iajs-1955 56 9 for for ADP iajs-1955 56 10 pure pure ADJ iajs-1955 56 11 & & CCONJ iajs-1955 56 12 appl appl PROPN iajs-1955 56 13 . . PUNCT iajs-1955 57 1 sci sci PROPN iajs-1955 57 2 . . PUNCT iajs-1955 57 3 vol vol NOUN iajs-1955 57 4 . . PROPN iajs-1955 58 1 31 31 NUM iajs-1955 58 2 ( ( PUNCT iajs-1955 58 3 2 2 NUM iajs-1955 58 4 ) ) PUNCT iajs-1955 58 5 2018 2018 NUM iajs-1955 58 6 proposition(2.10 proposition(2.10 NOUN iajs-1955 58 7 ) ) PUNCT iajs-1955 58 8 if if SCONJ iajs-1955 58 9 𝑁 𝑁 PROPN iajs-1955 58 10 and and CCONJ iajs-1955 58 11 𝑁 𝑁 PROPN iajs-1955 58 12 are be AUX iajs-1955 58 13 submodules submodule NOUN iajs-1955 58 14 of of ADP iajs-1955 58 15 a a DET iajs-1955 58 16 module module NOUN iajs-1955 58 17 m m NOUN iajs-1955 58 18 , , PUNCT iajs-1955 58 19 provided provide VERB iajs-1955 58 20 that that SCONJ iajs-1955 58 21 𝑁 𝑁 PROPN iajs-1955 58 22 is be AUX iajs-1955 58 23 containing contain VERB iajs-1955 58 24 any any DET iajs-1955 58 25 weak weak ADJ iajs-1955 58 26 essential essential ADJ iajs-1955 58 27 extensions extension NOUN iajs-1955 58 28 of of ADP iajs-1955 58 29 𝑁 𝑁 PROPN iajs-1955 58 30 , , PUNCT iajs-1955 58 31 and and CCONJ iajs-1955 58 32 𝑁 𝑁 PROPN iajs-1955 58 33 is be AUX iajs-1955 58 34 a a DET iajs-1955 58 35 w w NOUN iajs-1955 58 36 - - PUNCT iajs-1955 58 37 closed closed ADJ iajs-1955 58 38 submodule submodule NOUN iajs-1955 58 39 in in ADP iajs-1955 58 40 𝑁 𝑁 PROPN iajs-1955 58 41 and and CCONJ iajs-1955 58 42 𝑁 𝑁 PROPN iajs-1955 58 43 is be AUX iajs-1955 58 44 a a DET iajs-1955 58 45 w w NOUN iajs-1955 58 46 - - PUNCT iajs-1955 58 47 closed closed ADJ iajs-1955 58 48 submodule submodule NOUN iajs-1955 58 49 in in ADP iajs-1955 58 50 m m PROPN iajs-1955 58 51 , , PUNCT iajs-1955 58 52 then then ADV iajs-1955 58 53 𝑁 𝑁 PROPN iajs-1955 58 54 is be AUX iajs-1955 58 55 a a DET iajs-1955 58 56 w w NOUN iajs-1955 58 57 - - PUNCT iajs-1955 58 58 closed closed ADJ iajs-1955 58 59 submodule submodule NOUN iajs-1955 58 60 in in ADP iajs-1955 58 61 m. m. NOUN iajs-1955 58 62 proof proof NOUN iajs-1955 58 63 assume assume VERB iajs-1955 58 64 that that SCONJ iajs-1955 58 65 𝑈 𝑈 PROPN iajs-1955 58 66 𝑀 𝑀 PROPN iajs-1955 58 67 with with ADP iajs-1955 58 68 𝑁 𝑁 PROPN iajs-1955 58 69 is be AUX iajs-1955 58 70 weak weak ADJ iajs-1955 58 71 essential essential ADJ iajs-1955 58 72 submodule submodule NOUN iajs-1955 58 73 in in ADP iajs-1955 58 74 u u PROPN iajs-1955 58 75 , , PUNCT iajs-1955 58 76 then then ADV iajs-1955 58 77 by by ADP iajs-1955 58 78 hybothesis hybothesis NOUN iajs-1955 58 79 we we PRON iajs-1955 58 80 get get VERB iajs-1955 58 81 u u NOUN iajs-1955 58 82 is be AUX iajs-1955 58 83 a a DET iajs-1955 58 84 submodule submodule NOUN iajs-1955 58 85 in in ADP iajs-1955 58 86 𝑁 𝑁 PROPN iajs-1955 58 87 . . PUNCT iajs-1955 59 1 since since SCONJ iajs-1955 59 2 𝑁 𝑁 PROPN iajs-1955 59 3 is be AUX iajs-1955 59 4 a a DET iajs-1955 59 5 w w NOUN iajs-1955 59 6 - - PUNCT iajs-1955 59 7 closed closed ADJ iajs-1955 59 8 in in ADP iajs-1955 59 9 𝑁 𝑁 PROPN iajs-1955 59 10 , , PUNCT iajs-1955 59 11 then then ADV iajs-1955 59 12 𝑁 𝑁 PROPN iajs-1955 59 13 = = PUNCT iajs-1955 59 14 u. u. NOUN iajs-1955 59 15 thus thus ADV iajs-1955 59 16 𝑁 𝑁 PROPN iajs-1955 59 17 is be AUX iajs-1955 59 18 a a DET iajs-1955 59 19 w w NOUN iajs-1955 59 20 - - PUNCT iajs-1955 59 21 closed closed ADJ iajs-1955 59 22 .submodule .submodule NOUN iajs-1955 59 23 in in ADP iajs-1955 59 24 m m NOUN iajs-1955 59 25 proposition(2.11 proposition(2.11 NUM iajs-1955 59 26 ) ) PUNCT iajs-1955 59 27 if if SCONJ iajs-1955 59 28 m m NOUN iajs-1955 59 29 is be AUX iajs-1955 59 30 a a DET iajs-1955 59 31 chained chain VERB iajs-1955 59 32 module module NOUN iajs-1955 59 33 , , PUNCT iajs-1955 59 34 and and CCONJ iajs-1955 59 35 e e NOUN iajs-1955 59 36 , , PUNCT iajs-1955 59 37 d d X iajs-1955 59 38 are be AUX iajs-1955 59 39 submodules submodule NOUN iajs-1955 59 40 of of ADP iajs-1955 59 41 m m PROPN iajs-1955 59 42 with with ADP iajs-1955 59 43 𝐸 𝐸 PROPN iajs-1955 59 44 𝐷 𝐷 NOUN iajs-1955 59 45 , , PUNCT iajs-1955 59 46 and and CCONJ iajs-1955 59 47 𝐸 𝐸 PROPN iajs-1955 59 48 𝐷 𝐷 NOUN iajs-1955 59 49 and and CCONJ iajs-1955 59 50 𝐷 𝐷 PROPN iajs-1955 59 51 𝑀 𝑀 PROPN iajs-1955 59 52 , , PUNCT iajs-1955 59 53 then then ADV iajs-1955 59 54 𝐸 𝐸 ADJ iajs-1955 59 55 𝑀. 𝑀. NOUN iajs-1955 59 56 proof proof NOUN iajs-1955 59 57 let let VERB iajs-1955 59 58 k k PROPN iajs-1955 59 59 be be AUX iajs-1955 59 60 a a DET iajs-1955 59 61 submodule submodule NOUN iajs-1955 59 62 of of ADP iajs-1955 59 63 m m PROPN iajs-1955 59 64 with with ADP iajs-1955 59 65 e e NOUN iajs-1955 59 66 is be AUX iajs-1955 59 67 weak weak ADJ iajs-1955 59 68 essential essential ADJ iajs-1955 59 69 in in ADP iajs-1955 59 70 k. k. PROPN iajs-1955 59 71 since since SCONJ iajs-1955 59 72 m m PROPN iajs-1955 59 73 is be AUX iajs-1955 59 74 chained chain VERB iajs-1955 59 75 module module NOUN iajs-1955 59 76 , , PUNCT iajs-1955 59 77 then then ADV iajs-1955 59 78 either either CCONJ iajs-1955 59 79 k k PROPN iajs-1955 59 80 is be AUX iajs-1955 59 81 a a DET iajs-1955 59 82 submodule submodule NOUN iajs-1955 59 83 in in ADP iajs-1955 59 84 d d PROPN iajs-1955 59 85 or or CCONJ iajs-1955 59 86 d d NOUN iajs-1955 59 87 is be AUX iajs-1955 59 88 a a DET iajs-1955 59 89 submodule submodule NOUN iajs-1955 59 90 in in ADP iajs-1955 59 91 k. k. PROPN iajs-1955 59 92 if if SCONJ iajs-1955 59 93 k k PROPN iajs-1955 59 94 is be AUX iajs-1955 59 95 a a DET iajs-1955 59 96 submodule submodule NOUN iajs-1955 59 97 in in ADP iajs-1955 59 98 d d PROPN iajs-1955 59 99 , , PUNCT iajs-1955 59 100 and and CCONJ iajs-1955 59 101 since since SCONJ iajs-1955 59 102 e e PROPN iajs-1955 59 103 is be AUX iajs-1955 59 104 a a DET iajs-1955 59 105 w w NOUN iajs-1955 59 106 - - PUNCT iajs-1955 59 107 closed closed ADJ iajs-1955 59 108 submodule submodule NOUN iajs-1955 59 109 in in ADP iajs-1955 59 110 d d PROPN iajs-1955 59 111 , , PUNCT iajs-1955 59 112 then then ADV iajs-1955 59 113 e e NOUN iajs-1955 59 114 = = NOUN iajs-1955 59 115 k.hence k.hence NOUN iajs-1955 59 116 e e NOUN iajs-1955 59 117 is be AUX iajs-1955 59 118 a a DET iajs-1955 59 119 w w NOUN iajs-1955 59 120 - - PUNCT iajs-1955 59 121 closed closed ADJ iajs-1955 59 122 submodule submodule NOUN iajs-1955 59 123 in in ADP iajs-1955 59 124 m. m. NOUN iajs-1955 59 125 if if SCONJ iajs-1955 59 126 d d PROPN iajs-1955 59 127 is be AUX iajs-1955 59 128 a a DET iajs-1955 59 129 submodule submodule NOUN iajs-1955 59 130 in in ADP iajs-1955 59 131 k k PROPN iajs-1955 59 132 , , PUNCT iajs-1955 59 133 and and CCONJ iajs-1955 59 134 since since SCONJ iajs-1955 59 135 e e NOUN iajs-1955 59 136 is be AUX iajs-1955 59 137 weak weak ADJ iajs-1955 59 138 essential essential ADJ iajs-1955 59 139 in in ADP iajs-1955 59 140 k k NOUN iajs-1955 59 141 , , PUNCT iajs-1955 59 142 then then ADV iajs-1955 59 143 by by ADP iajs-1955 59 144 [ [ PUNCT iajs-1955 59 145 2 2 NUM iajs-1955 59 146 , , PUNCT iajs-1955 59 147 rem(1.5)(2 rem(1.5)(2 PROPN iajs-1955 59 148 ) ) PUNCT iajs-1955 59 149 ] ] PUNCT iajs-1955 60 1 d d X iajs-1955 60 2 is be AUX iajs-1955 60 3 a a DET iajs-1955 60 4 weak weak ADJ iajs-1955 60 5 essential essential ADJ iajs-1955 60 6 submodule submodule NOUN iajs-1955 60 7 in in ADP iajs-1955 60 8 k. k. PROPN iajs-1955 61 1 but but CCONJ iajs-1955 61 2 d d PROPN iajs-1955 61 3 is be AUX iajs-1955 61 4 a a DET iajs-1955 61 5 w w NOUN iajs-1955 61 6 - - PUNCT iajs-1955 61 7 closed closed ADJ iajs-1955 61 8 submodule submodule NOUN iajs-1955 61 9 in in ADP iajs-1955 61 10 m m PROPN iajs-1955 61 11 , , PUNCT iajs-1955 61 12 hence hence ADV iajs-1955 61 13 d d NOUN iajs-1955 61 14 = = NOUN iajs-1955 61 15 k.thus k.thus NOUN iajs-1955 61 16 , , PUNCT iajs-1955 61 17 e e PROPN iajs-1955 61 18 is be AUX iajs-1955 61 19 a a DET iajs-1955 61 20 weak weak ADJ iajs-1955 61 21 essential essential ADJ iajs-1955 61 22 submodule submodule NOUN iajs-1955 61 23 in in ADP iajs-1955 61 24 d. d. PROPN iajs-1955 61 25 but but CCONJ iajs-1955 61 26 e e PROPN iajs-1955 61 27 is be AUX iajs-1955 61 28 a a DET iajs-1955 61 29 w w NOUN iajs-1955 61 30 - - PUNCT iajs-1955 61 31 closed closed ADJ iajs-1955 61 32 submodule submodule NOUN iajs-1955 61 33 in in ADP iajs-1955 61 34 d d PROPN iajs-1955 61 35 , , PUNCT iajs-1955 61 36 then then ADV iajs-1955 61 37 e e NOUN iajs-1955 61 38 = = NOUN iajs-1955 61 39 d.hence d.hence NOUN iajs-1955 61 40 e e NOUN iajs-1955 61 41 is be AUX iajs-1955 61 42 a a DET iajs-1955 61 43 wclosed wclosed ADJ iajs-1955 61 44 submodule submodule NOUN iajs-1955 61 45 in in ADP iajs-1955 61 46 m. m. NOUN iajs-1955 61 47 before before SCONJ iajs-1955 61 48 we we PRON iajs-1955 61 49 give give VERB iajs-1955 61 50 the the DET iajs-1955 61 51 next next ADJ iajs-1955 61 52 proposition proposition NOUN iajs-1955 61 53 , , PUNCT iajs-1955 61 54 we we PRON iajs-1955 61 55 introduce introduce VERB iajs-1955 61 56 the the DET iajs-1955 61 57 following following ADJ iajs-1955 61 58 denifition denifition NOUN iajs-1955 61 59 . . PUNCT iajs-1955 62 1 " " PUNCT iajs-1955 62 2 definition(2.12 definition(2.12 X iajs-1955 62 3 ) ) PUNCT iajs-1955 62 4 a a DET iajs-1955 62 5 module module NOUN iajs-1955 62 6 m m VERB iajs-1955 62 7 is be AUX iajs-1955 62 8 called call VERB iajs-1955 62 9 completly completly ADV iajs-1955 62 10 essential essential ADJ iajs-1955 62 11 if if SCONJ iajs-1955 62 12 every every DET iajs-1955 62 13 non non ADJ iajs-1955 62 14 zero zero NUM iajs-1955 62 15 weak weak ADJ iajs-1955 62 16 essential essential ADJ iajs-1955 62 17 submodule submodule NOUN iajs-1955 62 18 of of ADP iajs-1955 62 19 m m PROPN iajs-1955 62 20 is be AUX iajs-1955 62 21 an an DET iajs-1955 62 22 essential essential ADJ iajs-1955 62 23 submodule submodule NOUN iajs-1955 62 24 of of ADP iajs-1955 62 25 m m PROPN iajs-1955 62 26 " " PUNCT iajs-1955 62 27 . . PUNCT iajs-1955 63 1 completely completely ADV iajs-1955 63 2 essential essential ADJ iajs-1955 63 3 in in ADP iajs-1955 63 4 [ [ X iajs-1955 63 5 3 3 X iajs-1955 63 6 ] ] PUNCT iajs-1955 63 7 is be AUX iajs-1955 63 8 called call VERB iajs-1955 63 9 fully fully ADV iajs-1955 63 10 essential essential ADJ iajs-1955 63 11 . . PUNCT iajs-1955 64 1 the the DET iajs-1955 64 2 following follow VERB iajs-1955 64 3 proposition proposition NOUN iajs-1955 64 4 show show VERB iajs-1955 64 5 that that SCONJ iajs-1955 64 6 closed close VERB iajs-1955 64 7 submodules submodule NOUN iajs-1955 64 8 and and CCONJ iajs-1955 64 9 w w NOUN iajs-1955 64 10 - - PUNCT iajs-1955 64 11 closed closed ADJ iajs-1955 64 12 submodules submodule NOUN iajs-1955 64 13 are be AUX iajs-1955 64 14 equivalents equivalent NOUN iajs-1955 64 15 under under ADP iajs-1955 64 16 certain certain ADJ iajs-1955 64 17 conditions condition NOUN iajs-1955 64 18 . . PUNCT iajs-1955 65 1 proposition(2.13 proposition(2.13 X iajs-1955 65 2 ) ) PUNCT iajs-1955 66 1 " " PUNCT iajs-1955 66 2 if if SCONJ iajs-1955 66 3 m m NOUN iajs-1955 66 4 is be AUX iajs-1955 66 5 a a DET iajs-1955 66 6 module module NOUN iajs-1955 66 7 , , PUNCT iajs-1955 66 8 and and CCONJ iajs-1955 66 9 e e NOUN iajs-1955 66 10 be be AUX iajs-1955 66 11 a a DET iajs-1955 66 12 non non ADJ iajs-1955 66 13 zero zero NUM iajs-1955 66 14 submodule submodule NOUN iajs-1955 66 15 of of ADP iajs-1955 66 16 m m PROPN iajs-1955 66 17 " " PUNCT iajs-1955 66 18 such such ADJ iajs-1955 66 19 that that SCONJ iajs-1955 66 20 every every DET iajs-1955 66 21 weak weak ADJ iajs-1955 66 22 essential essential ADJ iajs-1955 66 23 extensions extension NOUN iajs-1955 66 24 of of ADP iajs-1955 66 25 e e NOUN iajs-1955 66 26 is be AUX iajs-1955 66 27 a a DET iajs-1955 66 28 completly completly ADV iajs-1955 66 29 essential essential ADJ iajs-1955 66 30 , , PUNCT iajs-1955 66 31 then then ADV iajs-1955 66 32 e e PROPN iajs-1955 66 33 is be AUX iajs-1955 66 34 a a DET iajs-1955 66 35 closed closed ADJ iajs-1955 66 36 submodule submodule NOUN iajs-1955 66 37 in in ADP iajs-1955 66 38 m m PROPN iajs-1955 66 39 if if SCONJ iajs-1955 66 40 and and CCONJ iajs-1955 66 41 only only ADV iajs-1955 66 42 if if SCONJ iajs-1955 66 43 e e NOUN iajs-1955 66 44 is be AUX iajs-1955 66 45 a a DET iajs-1955 66 46 w w NOUN iajs-1955 66 47 - - PUNCT iajs-1955 66 48 closed closed ADJ iajs-1955 66 49 submodule submodule NOUN iajs-1955 66 50 in in ADP iajs-1955 66 51 m. m. NOUN iajs-1955 66 52 proof proof NOUN iajs-1955 66 53 let let VERB iajs-1955 66 54 e e PRON iajs-1955 66 55 be be AUX iajs-1955 66 56 a a DET iajs-1955 66 57 non non ADJ iajs-1955 66 58 zero zero NUM iajs-1955 66 59 closed closed ADJ iajs-1955 66 60 submodule submodule NOUN iajs-1955 66 61 in in ADP iajs-1955 66 62 m m PROPN iajs-1955 66 63 , , PUNCT iajs-1955 66 64 and and CCONJ iajs-1955 66 65 u u PRON iajs-1955 66 66 be be VERB iajs-1955 66 67 a a DET iajs-1955 66 68 submodule submodule NOUN iajs-1955 66 69 of of ADP iajs-1955 66 70 m m PRON iajs-1955 66 71 such such ADJ iajs-1955 66 72 that that SCONJ iajs-1955 66 73 e e NOUN iajs-1955 66 74 is be AUX iajs-1955 66 75 a a DET iajs-1955 66 76 weak weak ADJ iajs-1955 66 77 essential essential ADJ iajs-1955 66 78 in in ADP iajs-1955 66 79 u. u. NOUN iajs-1955 66 80 by by ADP iajs-1955 66 81 hypothesis hypothesis NOUN iajs-1955 66 82 u u NOUN iajs-1955 66 83 is be AUX iajs-1955 66 84 a a DET iajs-1955 66 85 completely completely ADV iajs-1955 66 86 essential essential ADJ iajs-1955 66 87 , , PUNCT iajs-1955 66 88 therefore therefore ADV iajs-1955 66 89 e e PROPN iajs-1955 66 90 is be AUX iajs-1955 66 91 an an DET iajs-1955 66 92 essential essential ADJ iajs-1955 66 93 submodule submodule NOUN iajs-1955 66 94 in in ADP iajs-1955 66 95 u. u. PROPN iajs-1955 66 96 but but CCONJ iajs-1955 66 97 e e NOUN iajs-1955 66 98 is be AUX iajs-1955 66 99 a a DET iajs-1955 66 100 closed closed ADJ iajs-1955 66 101 submodule submodule NOUN iajs-1955 66 102 in in ADP iajs-1955 66 103 m m PROPN iajs-1955 66 104 , , PUNCT iajs-1955 66 105 then then ADV iajs-1955 66 106 e e X iajs-1955 66 107 = = NOUN iajs-1955 66 108 u.that u.that PRON iajs-1955 66 109 is be AUX iajs-1955 66 110 e e NOUN iajs-1955 66 111 is be AUX iajs-1955 66 112 a a DET iajs-1955 66 113 w w NOUN iajs-1955 66 114 - - PUNCT iajs-1955 66 115 closed closed ADJ iajs-1955 66 116 submodule submodule NOUN iajs-1955 66 117 . . PUNCT iajs-1955 67 1 the the DET iajs-1955 67 2 converse converse NOUN iajs-1955 67 3 is be AUX iajs-1955 67 4 direct direct ADJ iajs-1955 67 5 . . PUNCT iajs-1955 68 1 proposition(2.14 proposition(2.14 NOUN iajs-1955 68 2 ) ) PUNCT iajs-1955 68 3 if if SCONJ iajs-1955 68 4 m m NOUN iajs-1955 68 5 is be AUX iajs-1955 68 6 a a DET iajs-1955 68 7 fully fully ADV iajs-1955 68 8 semi semi ADJ iajs-1955 68 9 - - ADJ iajs-1955 68 10 prime prime ADJ iajs-1955 68 11 module module NOUN iajs-1955 68 12 , , PUNCT iajs-1955 68 13 and and CCONJ iajs-1955 68 14 e e NOUN iajs-1955 68 15 be be AUX iajs-1955 68 16 a a DET iajs-1955 68 17 non non ADJ iajs-1955 68 18 zero zero NUM iajs-1955 68 19 submodule submodule NOUN iajs-1955 68 20 of of ADP iajs-1955 68 21 m m PROPN iajs-1955 68 22 , , PUNCT iajs-1955 68 23 then then ADV iajs-1955 68 24 e e PROPN iajs-1955 68 25 is be AUX iajs-1955 68 26 a a DET iajs-1955 68 27 closed closed ADJ iajs-1955 68 28 submodule submodule NOUN iajs-1955 68 29 in in ADP iajs-1955 68 30 m m PROPN iajs-1955 68 31 if if SCONJ iajs-1955 68 32 and and CCONJ iajs-1955 68 33 only only ADV iajs-1955 68 34 if if SCONJ iajs-1955 68 35 e e NOUN iajs-1955 68 36 is be AUX iajs-1955 68 37 a a DET iajs-1955 68 38 w w NOUN iajs-1955 68 39 - - PUNCT iajs-1955 68 40 closed closed ADJ iajs-1955 68 41 submodule submodule NOUN iajs-1955 68 42 in in ADP iajs-1955 68 43 m. m. NOUN iajs-1955 68 44             SPACE iajs-1955 68 45 168 168 NUM iajs-1955 68 46     SPACE iajs-1955 68 47 | | ADV iajs-1955 68 48     SPACE iajs-1955 68 49 mathmatics mathmatic NOUN iajs-1955 68 50 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 68 51     SPACE iajs-1955 68 52 2018 2018 NUM iajs-1955 68 53 ) ) PUNCT iajs-1955 68 54 عام عام ADP iajs-1955 68 55 2العدد 2العدد NUM iajs-1955 68 56 ( ( PUNCT iajs-1955 68 57 31لمجلد 31لمجلد NUM iajs-1955 68 58 ا ا NOUN iajs-1955 68 59 مجلة مجلة NOUN iajs-1955 68 60 إبن إبن VERB iajs-1955 68 61 الهيثم الهيثم ADJ iajs-1955 68 62 للعلوم للعلوم NOUN iajs-1955 68 63 الصرفة الصرفة NOUN iajs-1955 69 1 و و PRON iajs-1955 69 2 التطبيقية التطبيقية ADJ iajs-1955 69 3 ibn ibn PROPN iajs-1955 69 4 al al PROPN iajs-1955 69 5 - - PUNCT iajs-1955 69 6 haitham haitham PROPN iajs-1955 69 7 jour jour X iajs-1955 69 8 . . PROPN iajs-1955 69 9 for for ADP iajs-1955 69 10 pure pure ADJ iajs-1955 69 11 & & CCONJ iajs-1955 69 12 appl appl PROPN iajs-1955 69 13 . . PUNCT iajs-1955 70 1 sci sci PROPN iajs-1955 70 2 . . PUNCT iajs-1955 70 3 vol vol NOUN iajs-1955 70 4 . . PROPN iajs-1955 71 1 31 31 NUM iajs-1955 71 2 ( ( PUNCT iajs-1955 71 3 2 2 NUM iajs-1955 71 4 ) ) PUNCT iajs-1955 71 5 2018 2018 NUM iajs-1955 71 6 proof proof NOUN iajs-1955 71 7 assume assume VERB iajs-1955 71 8 that that SCONJ iajs-1955 71 9 e e PRON iajs-1955 71 10 is be AUX iajs-1955 71 11 a a DET iajs-1955 71 12 non non ADJ iajs-1955 71 13 zero zero NUM iajs-1955 71 14 closed closed ADJ iajs-1955 71 15 submodule submodule NOUN iajs-1955 71 16 in in ADP iajs-1955 71 17 m m PROPN iajs-1955 71 18 , , PUNCT iajs-1955 71 19 and and CCONJ iajs-1955 71 20 u u NOUN iajs-1955 71 21 is be AUX iajs-1955 71 22 a a DET iajs-1955 71 23 submodule submodule NOUN iajs-1955 71 24 of of ADP iajs-1955 71 25 m m PRON iajs-1955 71 26 such such ADJ iajs-1955 71 27 that that SCONJ iajs-1955 71 28 e e NOUN iajs-1955 71 29 is be AUX iajs-1955 71 30 a a DET iajs-1955 71 31 weak weak ADJ iajs-1955 71 32 essential essential ADJ iajs-1955 71 33 submodule submodule NOUN iajs-1955 71 34 in in ADP iajs-1955 71 35 u. u. PROPN iajs-1955 71 36 then then ADV iajs-1955 71 37 by by ADP iajs-1955 71 38 [ [ X iajs-1955 71 39 3 3 NUM iajs-1955 71 40 , , PUNCT iajs-1955 71 41 cor(2.5 cor(2.5 NUM iajs-1955 71 42 ) ) PUNCT iajs-1955 71 43 ] ] PUNCT iajs-1955 72 1 e e X iajs-1955 72 2 is be AUX iajs-1955 72 3 an an DET iajs-1955 72 4 essential essential ADJ iajs-1955 72 5 submodule submodule NOUN iajs-1955 72 6 in in ADP iajs-1955 72 7 u. u. PROPN iajs-1955 72 8 but but CCONJ iajs-1955 72 9 e e PROPN iajs-1955 72 10 is be AUX iajs-1955 72 11 a a DET iajs-1955 72 12 non non ADJ iajs-1955 72 13 - - ADJ iajs-1955 72 14 zero zero NUM iajs-1955 72 15 closed closed ADJ iajs-1955 72 16 submodule submodule NOUN iajs-1955 72 17 in in ADP iajs-1955 72 18 m m PROPN iajs-1955 72 19 , , PUNCT iajs-1955 72 20 hence hence ADV iajs-1955 72 21 e e NOUN iajs-1955 72 22 = = NOUN iajs-1955 72 23 u. u. NOUN iajs-1955 72 24 that that PRON iajs-1955 72 25 is be AUX iajs-1955 72 26 e e NOUN iajs-1955 72 27 is be AUX iajs-1955 72 28 a a DET iajs-1955 72 29 w w NOUN iajs-1955 72 30 - - PUNCT iajs-1955 72 31 closed closed ADJ iajs-1955 72 32 submodule submodule NOUN iajs-1955 72 33 in in ADP iajs-1955 72 34 m. m. NOUN iajs-1955 72 35 the the DET iajs-1955 72 36 converse converse NOUN iajs-1955 72 37 is be AUX iajs-1955 72 38 direct direct ADJ iajs-1955 72 39 . . PUNCT iajs-1955 73 1 corllary corllary ADJ iajs-1955 73 2 ( ( PUNCT iajs-1955 73 3 2.15 2.15 NUM iajs-1955 73 4 ) ) PUNCT iajs-1955 73 5 if if SCONJ iajs-1955 73 6 m m NOUN iajs-1955 73 7 is be AUX iajs-1955 73 8 a a DET iajs-1955 73 9 uniform uniform ADJ iajs-1955 73 10 module module NOUN iajs-1955 73 11 , , PUNCT iajs-1955 73 12 and and CCONJ iajs-1955 73 13 e e NOUN iajs-1955 73 14 be be AUX iajs-1955 73 15 a a DET iajs-1955 73 16 non non ADJ iajs-1955 73 17 zero zero NUM iajs-1955 73 18 submodule submodule NOUN iajs-1955 73 19 of of ADP iajs-1955 73 20 m m PROPN iajs-1955 73 21 , , PUNCT iajs-1955 73 22 then then ADV iajs-1955 73 23 e e PROPN iajs-1955 73 24 is be AUX iajs-1955 73 25 a a DET iajs-1955 73 26 closed closed ADJ iajs-1955 73 27 submodule submodule NOUN iajs-1955 73 28 in in ADP iajs-1955 73 29 m m PROPN iajs-1955 73 30 if if SCONJ iajs-1955 74 1 and and CCONJ iajs-1955 74 2 only only ADV iajs-1955 74 3 if if SCONJ iajs-1955 74 4 e e NOUN iajs-1955 74 5 is be AUX iajs-1955 74 6 a a DET iajs-1955 74 7 w w NOUN iajs-1955 74 8 - - PUNCT iajs-1955 74 9 closed closed ADJ iajs-1955 74 10 submodule submodule NOUN iajs-1955 74 11 in in ADP iajs-1955 74 12 m. m. NOUN iajs-1955 74 13 proof proof NOUN iajs-1955 74 14 assume assume VERB iajs-1955 74 15 that that SCONJ iajs-1955 74 16 e e PRON iajs-1955 74 17 is be AUX iajs-1955 74 18 a a DET iajs-1955 74 19 closed closed ADJ iajs-1955 74 20 submodule submodule NOUN iajs-1955 74 21 in in ADP iajs-1955 74 22 m m PROPN iajs-1955 74 23 and and CCONJ iajs-1955 74 24 let let VERB iajs-1955 74 25 e e PROPN iajs-1955 74 26 a a DET iajs-1955 74 27 weak weak ADJ iajs-1955 74 28 essential essential ADJ iajs-1955 74 29 in in ADP iajs-1955 74 30 u u NOUN iajs-1955 74 31 where where SCONJ iajs-1955 74 32 u u NOUN iajs-1955 74 33 is be AUX iajs-1955 74 34 a a DET iajs-1955 74 35 submodule submodule NOUN iajs-1955 74 36 of of ADP iajs-1955 74 37 m m PROPN iajs-1955 74 38 , , PUNCT iajs-1955 74 39 then then ADV iajs-1955 74 40 u u NOUN iajs-1955 74 41 is be AUX iajs-1955 74 42 a a DET iajs-1955 74 43 uniform uniform NOUN iajs-1955 74 44 . . PUNCT iajs-1955 75 1 hence hence ADV iajs-1955 75 2 by by ADP iajs-1955 75 3 [ [ X iajs-1955 75 4 3,prop(2.7 3,prop(2.7 NUM iajs-1955 75 5 ) ) PUNCT iajs-1955 75 6 ] ] PUNCT iajs-1955 75 7 u u NOUN iajs-1955 75 8 is be AUX iajs-1955 75 9 a a DET iajs-1955 75 10 completely completely ADV iajs-1955 75 11 essential essential ADJ iajs-1955 75 12 . . PUNCT iajs-1955 76 1 thus thus ADV iajs-1955 76 2 e e X iajs-1955 76 3 is be AUX iajs-1955 76 4 an an DET iajs-1955 76 5 essential essential ADJ iajs-1955 76 6 in in ADP iajs-1955 76 7 u. u. PROPN iajs-1955 76 8 but but CCONJ iajs-1955 76 9 e e NOUN iajs-1955 76 10 is be AUX iajs-1955 76 11 a a DET iajs-1955 76 12 closed closed ADJ iajs-1955 76 13 , , PUNCT iajs-1955 76 14 then then ADV iajs-1955 76 15 e e X iajs-1955 76 16 = = NOUN iajs-1955 76 17 u. u. NOUN iajs-1955 76 18 thus thus ADV iajs-1955 76 19 e e NOUN iajs-1955 76 20 is be AUX iajs-1955 76 21 a a DET iajs-1955 76 22 w w NOUN iajs-1955 76 23 - - PUNCT iajs-1955 76 24 closed closed ADJ iajs-1955 76 25 in in ADP iajs-1955 76 26 m. m. NOUN iajs-1955 76 27 the the DET iajs-1955 76 28 converse converse NOUN iajs-1955 76 29 is be AUX iajs-1955 76 30 direct direct ADJ iajs-1955 76 31 . . PUNCT iajs-1955 77 1 the the DET iajs-1955 77 2 following follow VERB iajs-1955 77 3 propositions proposition NOUN iajs-1955 77 4 show show VERB iajs-1955 77 5 that that SCONJ iajs-1955 77 6 the the DET iajs-1955 77 7 transitive transitive ADJ iajs-1955 77 8 property property NOUN iajs-1955 77 9 for for ADP iajs-1955 77 10 w w NOUN iajs-1955 77 11 - - PUNCT iajs-1955 77 12 closed closed ADJ iajs-1955 77 13 submodules submodule NOUN iajs-1955 77 14 hold hold VERB iajs-1955 77 15 under under ADP iajs-1955 77 16 conditions condition NOUN iajs-1955 77 17 fully fully ADV iajs-1955 77 18 semi semi ADJ iajs-1955 77 19 - - ADJ iajs-1955 77 20 prime prime ADJ iajs-1955 77 21 and and CCONJ iajs-1955 77 22 completely completely ADV iajs-1955 77 23 essential essential ADJ iajs-1955 77 24 . . PUNCT iajs-1955 78 1 proposition(2.16 proposition(2.16 X iajs-1955 78 2 ) ) PUNCT iajs-1955 78 3 let let VERB iajs-1955 78 4 m m PRON iajs-1955 78 5 be be AUX iajs-1955 78 6 a a DET iajs-1955 78 7 module module NOUN iajs-1955 78 8 , , PUNCT iajs-1955 78 9 and and CCONJ iajs-1955 78 10 e e NOUN iajs-1955 78 11 , , PUNCT iajs-1955 78 12 d d X iajs-1955 78 13 are be AUX iajs-1955 78 14 non non ADJ iajs-1955 78 15 - - ADJ iajs-1955 78 16 zero zero NUM iajs-1955 78 17 submodules submodule NOUN iajs-1955 78 18 of of ADP iajs-1955 78 19 m m PRON iajs-1955 78 20 such such ADJ iajs-1955 78 21 that that SCONJ iajs-1955 78 22 𝐸 𝐸 PROPN iajs-1955 78 23 𝐷 𝐷 NOUN iajs-1955 78 24 and and CCONJ iajs-1955 78 25 every every DET iajs-1955 78 26 weak weak ADJ iajs-1955 78 27 essential essential ADJ iajs-1955 78 28 extensions extension NOUN iajs-1955 78 29 of of ADP iajs-1955 78 30 e e NOUN iajs-1955 78 31 is be AUX iajs-1955 78 32 a a DET iajs-1955 78 33 completely completely ADV iajs-1955 78 34 essential essential ADJ iajs-1955 78 35 submodule submodule NOUN iajs-1955 78 36 of of ADP iajs-1955 78 37 m. m. NOUN iajs-1955 78 38 if if SCONJ iajs-1955 78 39 𝐸 𝐸 PROPN iajs-1955 78 40 𝐷 𝐷 NOUN iajs-1955 78 41 and and CCONJ iajs-1955 78 42 𝐷 𝐷 PROPN iajs-1955 78 43 𝑀 𝑀 PROPN iajs-1955 78 44 , , PUNCT iajs-1955 78 45 then then ADV iajs-1955 78 46 𝐸 𝐸 ADJ iajs-1955 78 47 𝑀. 𝑀. NOUN iajs-1955 78 48 proof proof NOUN iajs-1955 78 49 since since SCONJ iajs-1955 78 50 𝐸 𝐸 PROPN iajs-1955 78 51 𝐷 𝐷 NOUN iajs-1955 78 52 and and CCONJ iajs-1955 78 53 𝐷 𝐷 NOUN iajs-1955 78 54 𝑀. 𝑀. PROPN iajs-1955 78 55 then then ADV iajs-1955 78 56 by by ADP iajs-1955 78 57 remark(2.2 remark(2.2 NOUN iajs-1955 78 58 ) ) PUNCT iajs-1955 78 59 , , PUNCT iajs-1955 78 60 we we PRON iajs-1955 78 61 get get VERB iajs-1955 78 62 e e NOUN iajs-1955 78 63 is be AUX iajs-1955 78 64 a a DET iajs-1955 78 65 closed closed ADJ iajs-1955 78 66 submodule submodule NOUN iajs-1955 78 67 in in ADP iajs-1955 78 68 d d PROPN iajs-1955 79 1 and and CCONJ iajs-1955 79 2 d d PROPN iajs-1955 79 3 is be AUX iajs-1955 79 4 a a DET iajs-1955 79 5 closed closed ADJ iajs-1955 79 6 submodule submodule NOUN iajs-1955 79 7 in in ADP iajs-1955 79 8 m. m. NOUN iajs-1955 79 9 then then ADV iajs-1955 79 10 by by ADP iajs-1955 79 11 [ [ X iajs-1955 79 12 1,prop(1.5),p.18 1,prop(1.5),p.18 X iajs-1955 79 13 ] ] X iajs-1955 79 14 " " PUNCT iajs-1955 79 15 we we PRON iajs-1955 79 16 get get VERB iajs-1955 79 17 e e NOUN iajs-1955 79 18 is be AUX iajs-1955 79 19 a a DET iajs-1955 79 20 closed closed ADJ iajs-1955 79 21 submodule submodule NOUN iajs-1955 79 22 in in ADP iajs-1955 79 23 m m PROPN iajs-1955 79 24 " " PUNCT iajs-1955 79 25 , , PUNCT iajs-1955 79 26 then then ADV iajs-1955 79 27 by by ADP iajs-1955 79 28 prop(2.13 prop(2.13 NOUN iajs-1955 79 29 ) ) PUNCT iajs-1955 79 30 , , PUNCT iajs-1955 79 31 𝐸 𝐸 PROPN iajs-1955 79 32 𝑀. 𝑀. NOUN iajs-1955 79 33 proposition proposition NOUN iajs-1955 79 34 ( ( PUNCT iajs-1955 79 35 2.17 2.17 NUM iajs-1955 79 36 ) ) PUNCT iajs-1955 79 37 let let VERB iajs-1955 79 38 m m PRON iajs-1955 79 39 be be AUX iajs-1955 79 40 a a DET iajs-1955 79 41 fully fully ADV iajs-1955 79 42 semi semi ADJ iajs-1955 79 43 - - ADJ iajs-1955 79 44 prime prime ADJ iajs-1955 79 45 module module NOUN iajs-1955 79 46 , , PUNCT iajs-1955 79 47 and and CCONJ iajs-1955 79 48 let let VERB iajs-1955 79 49 e e PRON iajs-1955 79 50 be be AUX iajs-1955 79 51 a a DET iajs-1955 79 52 non non ADJ iajs-1955 79 53 - - ADJ iajs-1955 79 54 zero zero NUM iajs-1955 79 55 w w ADJ iajs-1955 79 56 - - PUNCT iajs-1955 79 57 closed close VERB iajs-1955 79 58 submodule submodule NOUN iajs-1955 79 59 in in ADP iajs-1955 79 60 d d PROPN iajs-1955 79 61 and and CCONJ iajs-1955 79 62 d d PROPN iajs-1955 79 63 is be AUX iajs-1955 79 64 a a DET iajs-1955 79 65 w w NOUN iajs-1955 79 66 - - PUNCT iajs-1955 79 67 closed closed ADJ iajs-1955 79 68 submodule submodule NOUN iajs-1955 79 69 in in ADP iajs-1955 79 70 m. m. NOUN iajs-1955 80 1 then then ADV iajs-1955 80 2 e e PROPN iajs-1955 80 3 is be AUX iajs-1955 80 4 a a DET iajs-1955 80 5 w w NOUN iajs-1955 80 6 - - PUNCT iajs-1955 80 7 closed closed ADJ iajs-1955 80 8 submodule submodule NOUN iajs-1955 80 9 in in ADP iajs-1955 80 10 m. m. NOUN iajs-1955 80 11 proof proof NOUN iajs-1955 80 12 since since SCONJ iajs-1955 80 13 e e PROPN iajs-1955 80 14 is be AUX iajs-1955 80 15 a a DET iajs-1955 80 16 w w NOUN iajs-1955 80 17 - - PUNCT iajs-1955 80 18 closed closed ADJ iajs-1955 80 19 submodule submodule NOUN iajs-1955 80 20 in in ADP iajs-1955 80 21 d d PROPN iajs-1955 80 22 and and CCONJ iajs-1955 80 23 d d PROPN iajs-1955 80 24 is be AUX iajs-1955 80 25 a a DET iajs-1955 80 26 w w NOUN iajs-1955 80 27 - - PUNCT iajs-1955 80 28 closed closed ADJ iajs-1955 80 29 submodule submodule NOUN iajs-1955 80 30 in in ADP iajs-1955 80 31 m m PROPN iajs-1955 80 32 , , PUNCT iajs-1955 80 33 then then ADV iajs-1955 80 34 by by ADP iajs-1955 80 35 remark(2.2 remark(2.2 NOUN iajs-1955 80 36 ) ) PUNCT iajs-1955 80 37 , , PUNCT iajs-1955 80 38 e e X iajs-1955 80 39 is be AUX iajs-1955 80 40 a a DET iajs-1955 80 41 closed closed ADJ iajs-1955 80 42 submodule submodule NOUN iajs-1955 80 43 in in ADP iajs-1955 80 44 d d PROPN iajs-1955 80 45 and and CCONJ iajs-1955 80 46 d d PROPN iajs-1955 80 47 is be AUX iajs-1955 80 48 a a DET iajs-1955 80 49 closed closed ADJ iajs-1955 80 50 submodule submodule NOUN iajs-1955 80 51 in in ADP iajs-1955 80 52 m. m. NOUN iajs-1955 80 53 hence hence ADV iajs-1955 80 54 by by ADP iajs-1955 80 55 [ [ X iajs-1955 80 56 1,prop(1.5 1,prop(1.5 NUM iajs-1955 80 57 ) ) PUNCT iajs-1955 80 58 , , PUNCT iajs-1955 80 59 p.18 p.18 PROPN iajs-1955 80 60 ] ] X iajs-1955 80 61 we we PRON iajs-1955 80 62 get get VERB iajs-1955 80 63 e e NOUN iajs-1955 80 64 is be AUX iajs-1955 80 65 a a DET iajs-1955 80 66 closed closed ADJ iajs-1955 80 67 submodule submodule NOUN iajs-1955 80 68 in in ADP iajs-1955 80 69 m. m. NOUN iajs-1955 80 70 thus thus ADV iajs-1955 80 71 by by ADP iajs-1955 80 72 prop(2.14 prop(2.14 NOUN iajs-1955 80 73 ) ) PUNCT iajs-1955 80 74 , , PUNCT iajs-1955 80 75 e e X iajs-1955 80 76 is be AUX iajs-1955 80 77 a a DET iajs-1955 80 78 w w NOUN iajs-1955 80 79 - - PUNCT iajs-1955 80 80 closed closed ADJ iajs-1955 80 81 submodule submodule NOUN iajs-1955 80 82 in in ADP iajs-1955 80 83 m. m. NOUN iajs-1955 80 84 remark remark NOUN iajs-1955 80 85 ( ( PUNCT iajs-1955 80 86 2.18 2.18 NUM iajs-1955 80 87 ) ) PUNCT iajs-1955 80 88 the the DET iajs-1955 80 89 intersection intersection NOUN iajs-1955 80 90 of of ADP iajs-1955 80 91 two two NUM iajs-1955 80 92 w w ADV iajs-1955 80 93 - - PUNCT iajs-1955 80 94 closed closed ADJ iajs-1955 80 95 submodule submodule NOUN iajs-1955 80 96 need need AUX iajs-1955 80 97 not not PART iajs-1955 80 98 to to PART iajs-1955 80 99 be be AUX iajs-1955 80 100 w w NOUN iajs-1955 80 101 - - PUNCT iajs-1955 80 102 closed closed ADJ iajs-1955 80 103 submodule submodule NOUN iajs-1955 80 104 as as SCONJ iajs-1955 80 105 the the DET iajs-1955 80 106 following follow VERB iajs-1955 80 107 example example NOUN iajs-1955 80 108 shows show VERB iajs-1955 80 109 : : PUNCT iajs-1955 80 110 in in ADP iajs-1955 80 111 the the DET iajs-1955 80 112 z z NOUN iajs-1955 80 113 - - PUNCT iajs-1955 80 114 module module NOUN iajs-1955 80 115 𝑍 𝑍 PROPN iajs-1955 80 116 ⨁ ⨁ PROPN iajs-1955 80 117 𝑍 𝑍 PROPN iajs-1955 80 118 , , PUNCT iajs-1955 80 119 the the DET iajs-1955 80 120 submodules submodule NOUN iajs-1955 81 1 𝑁 𝑁 PROPN iajs-1955 81 2 〈 〈 PROPN iajs-1955 81 3 0 0 NUM iajs-1955 81 4 , , PUNCT iajs-1955 81 5 1 1 NUM iajs-1955 81 6 〉 〉 NOUN iajs-1955 81 7 𝑎𝑛𝑑 𝑎𝑛𝑑 ADJ iajs-1955 81 8 𝐾 𝐾 PROPN iajs-1955 81 9 〈 〈 PROPN iajs-1955 81 10 4 4 NUM iajs-1955 81 11 , , PUNCT iajs-1955 81 12 1 1 NUM iajs-1955 81 13 〉 〉 NOUN iajs-1955 81 14 are be AUX iajs-1955 81 15 w w NOUN iajs-1955 81 16 - - PUNCT iajs-1955 81 17 closed closed ADJ iajs-1955 81 18 submodule submodule NOUN iajs-1955 81 19 in in ADP iajs-1955 81 20 𝑍 𝑍 PROPN iajs-1955 81 21 ⨁ ⨁ PROPN iajs-1955 81 22 𝑍 𝑍 NOUN iajs-1955 81 23 , , PUNCT iajs-1955 81 24 but but CCONJ iajs-1955 81 25 𝑁 𝑁 PROPN iajs-1955 81 26 ∩ ∩ ADJ iajs-1955 81 27 𝐾 𝐾 PROPN iajs-1955 81 28 0 0 NUM iajs-1955 81 29 , , PUNCT iajs-1955 81 30 0 0 NUM iajs-1955 81 31 is be AUX iajs-1955 81 32 not not PART iajs-1955 81 33 w w NOUN iajs-1955 81 34 - - PUNCT iajs-1955 81 35 closed closed ADJ iajs-1955 81 36 submodule submodule NOUN iajs-1955 81 37 in in ADP iajs-1955 81 38 𝑍 𝑍 PROPN iajs-1955 81 39 ⨁ ⨁ PROPN iajs-1955 81 40 𝑍 𝑍 NOUN iajs-1955 81 41 . . PUNCT iajs-1955 82 1 the the DET iajs-1955 82 2 following follow VERB iajs-1955 82 3 results result NOUN iajs-1955 82 4 give give VERB iajs-1955 82 5 more more ADJ iajs-1955 82 6 basic basic ADJ iajs-1955 82 7 properties property NOUN iajs-1955 82 8 of of ADP iajs-1955 82 9 w w NOUN iajs-1955 82 10 - - PUNCT iajs-1955 82 11 closed closed ADJ iajs-1955 82 12 submodules submodule NOUN iajs-1955 82 13 . . PUNCT iajs-1955 82 14             SPACE iajs-1955 83 1 169 169 NUM iajs-1955 83 2     SPACE iajs-1955 83 3 | | ADV iajs-1955 83 4     SPACE iajs-1955 83 5 mathmatics mathmatic NOUN iajs-1955 83 6 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 83 7     SPACE iajs-1955 83 8 2018 2018 NUM iajs-1955 83 9 ) ) PUNCT iajs-1955 84 1 عام عام ADP iajs-1955 84 2 2العدد 2العدد NUM iajs-1955 84 3 ( ( PUNCT iajs-1955 84 4 31لمجلد 31لمجلد NUM iajs-1955 84 5 ا ا NOUN iajs-1955 84 6 مجلة مجلة NOUN iajs-1955 84 7 إبن إبن VERB iajs-1955 84 8 الهيثم الهيثم ADJ iajs-1955 84 9 للعلوم للعلوم NOUN iajs-1955 84 10 الصرفة الصرفة NOUN iajs-1955 85 1 و و PRON iajs-1955 85 2 التطبيقية التطبيقية ADJ iajs-1955 85 3 ibn ibn PROPN iajs-1955 85 4 al al PROPN iajs-1955 85 5 - - PUNCT iajs-1955 85 6 haitham haitham PROPN iajs-1955 85 7 jour jour X iajs-1955 85 8 . . PROPN iajs-1955 85 9 for for ADP iajs-1955 85 10 pure pure ADJ iajs-1955 85 11 & & CCONJ iajs-1955 85 12 appl appl PROPN iajs-1955 85 13 . . PUNCT iajs-1955 86 1 sci sci PROPN iajs-1955 86 2 . . PUNCT iajs-1955 86 3 vol vol NOUN iajs-1955 86 4 . . PROPN iajs-1955 87 1 31 31 NUM iajs-1955 87 2 ( ( PUNCT iajs-1955 87 3 2 2 NUM iajs-1955 87 4 ) ) SYM iajs-1955 87 5 2018 2018 NUM iajs-1955 87 6 proposition proposition NOUN iajs-1955 87 7 ( ( PUNCT iajs-1955 87 8 2.19 2.19 NUM iajs-1955 87 9 ) ) PUNCT iajs-1955 87 10 if if SCONJ iajs-1955 87 11 every every DET iajs-1955 87 12 submodule submodule NOUN iajs-1955 87 13 of of ADP iajs-1955 87 14 a a DET iajs-1955 87 15 module module NOUN iajs-1955 87 16 m m NOUN iajs-1955 87 17 is be AUX iajs-1955 87 18 w w NOUN iajs-1955 87 19 - - PUNCT iajs-1955 87 20 closed closed ADJ iajs-1955 87 21 , , PUNCT iajs-1955 87 22 then then ADV iajs-1955 87 23 every every DET iajs-1955 87 24 submodule submodule NOUN iajs-1955 87 25 of of ADP iajs-1955 87 26 m m PROPN iajs-1955 87 27 is be AUX iajs-1955 87 28 a a DET iajs-1955 87 29 direct direct ADJ iajs-1955 87 30 summand summand NOUN iajs-1955 87 31 . . PUNCT iajs-1955 88 1 provided provide VERB iajs-1955 88 2 that that SCONJ iajs-1955 88 3 m m PROPN iajs-1955 88 4 is be AUX iajs-1955 88 5 a a DET iajs-1955 88 6 semi semi ADJ iajs-1955 88 7 simple simple ADJ iajs-1955 88 8 . . PUNCT iajs-1955 89 1 proof proof NOUN iajs-1955 89 2 since since SCONJ iajs-1955 89 3 every every DET iajs-1955 89 4 submodule submodule NOUN iajs-1955 89 5 of of ADP iajs-1955 89 6 m m PROPN iajs-1955 89 7 is be AUX iajs-1955 89 8 w w NOUN iajs-1955 89 9 - - PUNCT iajs-1955 89 10 closed closed ADJ iajs-1955 89 11 , , PUNCT iajs-1955 89 12 then then ADV iajs-1955 89 13 every every DET iajs-1955 89 14 submodule submodule NOUN iajs-1955 89 15 of of ADP iajs-1955 89 16 m m PROPN iajs-1955 89 17 is be AUX iajs-1955 89 18 a a DET iajs-1955 89 19 closed closed ADJ iajs-1955 89 20 . . PUNCT iajs-1955 90 1 hence hence ADV iajs-1955 90 2 by by ADP iajs-1955 90 3 [ [ X iajs-1955 90 4 8 8 NUM iajs-1955 90 5 , , PUNCT iajs-1955 90 6 exc(6 exc(6 ADJ iajs-1955 90 7 - - PUNCT iajs-1955 90 8 c c NOUN iajs-1955 90 9 ) ) PUNCT iajs-1955 90 10 , , PUNCT iajs-1955 90 11 p.139 p.139 VERB iajs-1955 90 12 ] ] PUNCT iajs-1955 90 13 " " PUNCT iajs-1955 90 14 every every DET iajs-1955 90 15 submodule submodule NOUN iajs-1955 90 16 of of ADP iajs-1955 90 17 m m PROPN iajs-1955 90 18 is be AUX iajs-1955 90 19 a a DET iajs-1955 90 20 direct direct ADJ iajs-1955 90 21 summand summand NOUN iajs-1955 90 22 of of ADP iajs-1955 90 23 m m NOUN iajs-1955 90 24 " " PUNCT iajs-1955 90 25 . . PUNCT iajs-1955 91 1 the the DET iajs-1955 91 2 following follow VERB iajs-1955 91 3 corollary corollary NOUN iajs-1955 91 4 is be AUX iajs-1955 91 5 a a DET iajs-1955 91 6 direct direct ADJ iajs-1955 91 7 consequence consequence NOUN iajs-1955 91 8 of of ADP iajs-1955 91 9 proposition(2.19 proposition(2.19 NOUN iajs-1955 91 10 ) ) PUNCT iajs-1955 91 11 . . PUNCT iajs-1955 92 1 corollary corollary ADJ iajs-1955 92 2 ( ( PUNCT iajs-1955 92 3 2.20 2.20 NUM iajs-1955 92 4 ) ) PUNCT iajs-1955 92 5 if if SCONJ iajs-1955 92 6 every every DET iajs-1955 92 7 submodule submodule NOUN iajs-1955 92 8 of of ADP iajs-1955 92 9 a a DET iajs-1955 92 10 module module NOUN iajs-1955 92 11 m m NOUN iajs-1955 92 12 is be AUX iajs-1955 92 13 a a DET iajs-1955 92 14 w w NOUN iajs-1955 92 15 - - PUNCT iajs-1955 92 16 closed closed ADJ iajs-1955 92 17 , , PUNCT iajs-1955 92 18 then then ADV iajs-1955 92 19 m m VERB iajs-1955 92 20 is be AUX iajs-1955 92 21 a a DET iajs-1955 92 22 semi semi ADJ iajs-1955 92 23 - - ADJ iajs-1955 92 24 simple simple ADJ iajs-1955 92 25 . . PUNCT iajs-1955 93 1 proposition(2.21 proposition(2.21 NOUN iajs-1955 93 2 ) ) PUNCT iajs-1955 93 3 if if SCONJ iajs-1955 93 4 e e PROPN iajs-1955 93 5 and and CCONJ iajs-1955 93 6 d d PROPN iajs-1955 93 7 are be AUX iajs-1955 93 8 submodules submodule NOUN iajs-1955 93 9 of of ADP iajs-1955 93 10 a a DET iajs-1955 93 11 module module NOUN iajs-1955 93 12 m m NOUN iajs-1955 93 13 with with ADP iajs-1955 93 14 𝐸 𝐸 PROPN iajs-1955 93 15 𝐷 𝐷 NOUN iajs-1955 93 16 , , PUNCT iajs-1955 93 17 and and CCONJ iajs-1955 93 18 𝐸 𝐸 PROPN iajs-1955 93 19 𝑀 𝑀 PROPN iajs-1955 93 20 , , PUNCT iajs-1955 93 21 then then ADV iajs-1955 93 22 𝐸 𝐸 ADJ iajs-1955 93 23 𝐷. 𝐷. NOUN iajs-1955 93 24 proof proof NOUN iajs-1955 93 25 let let VERB iajs-1955 93 26 𝐹 𝐹 PROPN iajs-1955 93 27 𝐷 𝐷 PROPN iajs-1955 93 28 , , PUNCT iajs-1955 93 29 then then ADV iajs-1955 93 30 𝐹 𝐹 PROPN iajs-1955 93 31 𝑀 𝑀 PROPN iajs-1955 93 32 , , PUNCT iajs-1955 93 33 and and CCONJ iajs-1955 93 34 e e NOUN iajs-1955 93 35 is be AUX iajs-1955 93 36 a a DET iajs-1955 93 37 weak weak ADJ iajs-1955 93 38 essential essential ADJ iajs-1955 93 39 submodule submodule NOUN iajs-1955 93 40 of of ADP iajs-1955 93 41 f. f. PROPN iajs-1955 93 42 but but CCONJ iajs-1955 93 43 𝐸 𝐸 PROPN iajs-1955 93 44 𝑀 𝑀 PROPN iajs-1955 93 45 , , PUNCT iajs-1955 93 46 then then ADV iajs-1955 93 47 e e X iajs-1955 93 48 = = NOUN iajs-1955 93 49 f. f. PROPN iajs-1955 93 50 hence hence ADV iajs-1955 93 51 𝐸 𝐸 PROPN iajs-1955 93 52 𝐷. 𝐷. PROPN iajs-1955 93 53 as as ADP iajs-1955 93 54 a a DET iajs-1955 93 55 direct direct ADJ iajs-1955 93 56 application application NOUN iajs-1955 93 57 of of ADP iajs-1955 93 58 proposition(2.21 proposition(2.21 NOUN iajs-1955 93 59 ) ) PUNCT iajs-1955 93 60 we we PRON iajs-1955 93 61 get get VERB iajs-1955 93 62 the the DET iajs-1955 93 63 following follow VERB iajs-1955 93 64 results result NOUN iajs-1955 93 65 . . PUNCT iajs-1955 94 1 corollary corollary ADJ iajs-1955 94 2 ( ( PUNCT iajs-1955 94 3 2.22 2.22 NUM iajs-1955 94 4 ) ) PUNCT iajs-1955 94 5 if if SCONJ iajs-1955 94 6 e e PROPN iajs-1955 94 7 and and CCONJ iajs-1955 94 8 d d PROPN iajs-1955 94 9 are be AUX iajs-1955 94 10 submodules submodule NOUN iajs-1955 94 11 of of ADP iajs-1955 94 12 a a DET iajs-1955 94 13 module module NOUN iajs-1955 94 14 m m NOUN iajs-1955 94 15 with with ADP iajs-1955 94 16 𝐸 𝐸 PROPN iajs-1955 94 17 ∩ ∩ ADJ iajs-1955 94 18 𝐷 𝐷 NOUN iajs-1955 94 19 is be AUX iajs-1955 94 20 a a DET iajs-1955 94 21 w w NOUN iajs-1955 94 22 - - PUNCT iajs-1955 94 23 closed closed ADJ iajs-1955 94 24 submodule submodule NOUN iajs-1955 94 25 in in ADP iajs-1955 94 26 m m PROPN iajs-1955 94 27 , , PUNCT iajs-1955 94 28 then then ADV iajs-1955 94 29 𝐸 𝐸 PROPN iajs-1955 94 30 ∩ ∩ ADJ iajs-1955 94 31 𝐷 𝐷 NOUN iajs-1955 94 32 is be AUX iajs-1955 94 33 a a DET iajs-1955 94 34 w w NOUN iajs-1955 94 35 - - PUNCT iajs-1955 94 36 closed closed ADJ iajs-1955 94 37 submodule submodule NOUN iajs-1955 94 38 in in ADP iajs-1955 94 39 e e PROPN iajs-1955 94 40 and and CCONJ iajs-1955 94 41 d. d. PROPN iajs-1955 94 42 corollary corollary PROPN iajs-1955 94 43 ( ( PUNCT iajs-1955 94 44 2.23 2.23 NUM iajs-1955 94 45 ) ) PUNCT iajs-1955 94 46 if if SCONJ iajs-1955 94 47 m m NOUN iajs-1955 94 48 is be AUX iajs-1955 94 49 a a DET iajs-1955 94 50 module module NOUN iajs-1955 94 51 , , PUNCT iajs-1955 94 52 and and CCONJ iajs-1955 94 53 e e NOUN iajs-1955 94 54 , , PUNCT iajs-1955 94 55 u u PRON iajs-1955 94 56 are be AUX iajs-1955 94 57 w w NOUN iajs-1955 94 58 - - PUNCT iajs-1955 94 59 closed closed ADJ iajs-1955 94 60 submodules submodule NOUN iajs-1955 94 61 in in ADP iajs-1955 94 62 m m PROPN iajs-1955 94 63 , , PUNCT iajs-1955 94 64 then then ADV iajs-1955 94 65 e e NOUN iajs-1955 94 66 and and CCONJ iajs-1955 94 67 u u NOUN iajs-1955 94 68 are be AUX iajs-1955 94 69 w w NOUN iajs-1955 94 70 - - PUNCT iajs-1955 94 71 closed closed ADJ iajs-1955 94 72 submodules submodule NOUN iajs-1955 94 73 in in ADP iajs-1955 94 74 e e PROPN iajs-1955 94 75 + + CCONJ iajs-1955 94 76 u. u. NOUN iajs-1955 94 77 corollary(2.24 corollary(2.24 NOUN iajs-1955 94 78 ) ) PUNCT iajs-1955 94 79 if if SCONJ iajs-1955 94 80 m m NOUN iajs-1955 94 81 is be AUX iajs-1955 94 82 an an DET iajs-1955 94 83 r r NOUN iajs-1955 94 84 - - PUNCT iajs-1955 94 85 module module NOUN iajs-1955 94 86 , , PUNCT iajs-1955 94 87 and and CCONJ iajs-1955 94 88 e e NOUN iajs-1955 94 89 is be AUX iajs-1955 94 90 a a DET iajs-1955 94 91 w w NOUN iajs-1955 94 92 - - PUNCT iajs-1955 94 93 closed closed ADJ iajs-1955 94 94 submodule submodule NOUN iajs-1955 94 95 in in ADP iajs-1955 94 96 m m PROPN iajs-1955 94 97 , , PUNCT iajs-1955 94 98 then then ADV iajs-1955 94 99 e e PROPN iajs-1955 94 100 is be AUX iajs-1955 94 101 a a DET iajs-1955 94 102 w w NOUN iajs-1955 94 103 - - PUNCT iajs-1955 94 104 closed closed ADJ iajs-1955 94 105 submodule submodule NOUN iajs-1955 94 106 in in ADP iajs-1955 94 107 √𝐸. √𝐸. PROPN iajs-1955 94 108 proof proof NOUN iajs-1955 94 109 since since SCONJ iajs-1955 94 110 𝐸 𝐸 PROPN iajs-1955 94 111 √𝐸 √𝐸 ADP iajs-1955 94 112 𝑀 𝑀 PROPN iajs-1955 94 113 , , PUNCT iajs-1955 94 114 and and CCONJ iajs-1955 94 115 e e NOUN iajs-1955 94 116 is be AUX iajs-1955 94 117 a a DET iajs-1955 94 118 w w NOUN iajs-1955 94 119 - - PUNCT iajs-1955 94 120 closed closed ADJ iajs-1955 94 121 submodule submodule NOUN iajs-1955 94 122 in in ADP iajs-1955 94 123 m m PROPN iajs-1955 94 124 then then ADV iajs-1955 94 125 by by ADP iajs-1955 94 126 proposition(2.21 proposition(2.21 NOUN iajs-1955 94 127 ) ) PUNCT iajs-1955 94 128 , , PUNCT iajs-1955 94 129 e e X iajs-1955 94 130 is be AUX iajs-1955 94 131 a a DET iajs-1955 94 132 w w NOUN iajs-1955 94 133 - - PUNCT iajs-1955 94 134 closed closed ADJ iajs-1955 94 135 submodule submodule NOUN iajs-1955 94 136 in in ADP iajs-1955 94 137 √𝐸. √𝐸. PROPN iajs-1955 94 138 remark remark NOUN iajs-1955 94 139 ( ( PUNCT iajs-1955 94 140 2.25 2.25 NUM iajs-1955 94 141 ) ) PUNCT iajs-1955 94 142 a a DET iajs-1955 94 143 direct direct ADJ iajs-1955 94 144 summand summand NOUN iajs-1955 94 145 of of ADP iajs-1955 94 146 a a DET iajs-1955 94 147 module module NOUN iajs-1955 94 148 m m NOUN iajs-1955 94 149 is be AUX iajs-1955 94 150 not not PART iajs-1955 94 151 necessary necessary ADJ iajs-1955 94 152 w w NOUN iajs-1955 94 153 - - PUNCT iajs-1955 94 154 closed closed ADJ iajs-1955 94 155 submodule submodule NOUN iajs-1955 94 156 in in ADP iajs-1955 94 157 m m PROPN iajs-1955 94 158 , , PUNCT iajs-1955 94 159 as as ADP iajs-1955 94 160 the the DET iajs-1955 94 161 following follow VERB iajs-1955 94 162 example example NOUN iajs-1955 94 163 show show NOUN iajs-1955 94 164 : : PUNCT iajs-1955 94 165 let let VERB iajs-1955 94 166 m=𝑍 m=𝑍 NOUN iajs-1955 94 167 as as ADP iajs-1955 94 168 a a DET iajs-1955 94 169 z z NOUN iajs-1955 94 170 - - PUNCT iajs-1955 94 171 module module NOUN iajs-1955 94 172 , , PUNCT iajs-1955 94 173 where where SCONJ iajs-1955 94 174 𝑍 𝑍 VERB iajs-1955 94 175 〈 〈 NOUN iajs-1955 94 176 3 3 NUM iajs-1955 94 177 〉 〉 NOUN iajs-1955 94 178 ⨁ ⨁ PROPN iajs-1955 94 179 〈 〈 PROPN iajs-1955 94 180 8 8 NUM iajs-1955 94 181 〉 〉 NOUN iajs-1955 94 182 , , PUNCT iajs-1955 94 183 the the DET iajs-1955 94 184 direct direct ADJ iajs-1955 94 185 summand summand NOUN iajs-1955 94 186 〈 〈 PROPN iajs-1955 94 187 3 3 NUM iajs-1955 94 188 〉 〉 NOUN iajs-1955 94 189 is be AUX iajs-1955 94 190 not not PART iajs-1955 94 191 w w NOUN iajs-1955 94 192 - - PUNCT iajs-1955 94 193 closed closed ADJ iajs-1955 94 194 submodule submodule NOUN iajs-1955 94 195 in in ADP iajs-1955 94 196 𝑍 𝑍 PROPN iajs-1955 94 197 . . PUNCT iajs-1955 95 1 since since SCONJ iajs-1955 95 2 〈 〈 NOUN iajs-1955 95 3 3 3 NUM iajs-1955 95 4 〉 〉 NOUN iajs-1955 95 5 is be AUX iajs-1955 95 6 a a DET iajs-1955 95 7 weak weak ADJ iajs-1955 95 8 essential essential ADJ iajs-1955 95 9 in in ADP iajs-1955 95 10 𝑍 𝑍 NOUN iajs-1955 95 11 . . PUNCT iajs-1955 96 1 proposition(2.26 proposition(2.26 X iajs-1955 96 2 ) ) PUNCT iajs-1955 96 3 let let VERB iajs-1955 96 4 𝑋 𝑋 NOUN iajs-1955 96 5 𝑋 𝑋 PROPN iajs-1955 96 6 ⨁ ⨁ PROPN iajs-1955 96 7 𝑋 𝑋 PROPN iajs-1955 96 8 be be VERB iajs-1955 96 9 a a DET iajs-1955 96 10 module module NOUN iajs-1955 96 11 , , PUNCT iajs-1955 96 12 where where SCONJ iajs-1955 96 13 𝑋 𝑋 NOUN iajs-1955 96 14 and and CCONJ iajs-1955 96 15 𝑋 𝑋 PROPN iajs-1955 96 16 are be AUX iajs-1955 96 17 submodules submodule NOUN iajs-1955 96 18 of of ADP iajs-1955 96 19 x x NOUN iajs-1955 96 20 , , PUNCT iajs-1955 96 21 and and CCONJ iajs-1955 96 22 let let VERB iajs-1955 96 23 e e PRON iajs-1955 96 24 be be AUX iajs-1955 96 25 a a DET iajs-1955 96 26 non non ADJ iajs-1955 96 27 zero zero NUM iajs-1955 96 28 w w ADV iajs-1955 96 29 - - PUNCT iajs-1955 96 30 closed closed ADJ iajs-1955 96 31 submodule submodule NOUN iajs-1955 96 32 in in ADP iajs-1955 96 33 𝑋 𝑋 PROPN iajs-1955 96 34 and and CCONJ iajs-1955 96 35 d d PROPN iajs-1955 96 36 is be AUX iajs-1955 96 37 a a DET iajs-1955 96 38 non non ADJ iajs-1955 96 39 zero zero NUM iajs-1955 96 40 w w ADV iajs-1955 96 41 - - PUNCT iajs-1955 96 42 closed closed ADJ iajs-1955 96 43 submodule submodule NOUN iajs-1955 96 44 in in ADP iajs-1955 96 45 𝑋 𝑋 PROPN iajs-1955 96 46 such such ADJ iajs-1955 96 47 that that SCONJ iajs-1955 96 48 ann ann PROPN iajs-1955 96 49 𝑋 𝑋 PROPN iajs-1955 96 50 + + CCONJ iajs-1955 96 51 ann ann PROPN iajs-1955 96 52 𝑋 𝑋 PROPN iajs-1955 96 53 = = SYM iajs-1955 96 54 r r NOUN iajs-1955 96 55 , , PUNCT iajs-1955 96 56 and and CCONJ iajs-1955 96 57 all all DET iajs-1955 96 58 weak weak ADJ iajs-1955 96 59 - - PUNCT iajs-1955 96 60 essential essential ADJ iajs-1955 96 61 extensions extension NOUN iajs-1955 96 62 of of ADP iajs-1955 96 63 𝐸 𝐸 PROPN iajs-1955 96 64 ⨁ ⨁ PROPN iajs-1955 96 65 𝐷 𝐷 PROPN iajs-1955 96 66 are be AUX iajs-1955 96 67 completely completely ADV iajs-1955 96 68 essential essential ADJ iajs-1955 96 69 submodule submodule NOUN iajs-1955 96 70 of of ADP iajs-1955 96 71 𝑋 𝑋 PROPN iajs-1955 96 72 ⨁ ⨁ PROPN iajs-1955 96 73 𝑋 𝑋 PROPN iajs-1955 96 74 . . PUNCT iajs-1955 97 1 then then ADV iajs-1955 97 2 𝐸 𝐸 PROPN iajs-1955 97 3 ⨁ ⨁ PROPN iajs-1955 97 4 𝐷 𝐷 PROPN iajs-1955 97 5 is be AUX iajs-1955 97 6 a a DET iajs-1955 97 7 w w NOUN iajs-1955 97 8 - - PUNCT iajs-1955 97 9 closed closed ADJ iajs-1955 97 10 submodule submodule NOUN iajs-1955 97 11 in in ADP iajs-1955 97 12 𝑋 𝑋 PROPN iajs-1955 97 13 ⨁ ⨁ PROPN iajs-1955 97 14 𝑋 𝑋 PROPN iajs-1955 97 15 . . PUNCT iajs-1955 97 16             SPACE iajs-1955 98 1 170 170 NUM iajs-1955 98 2     SPACE iajs-1955 98 3 | | ADV iajs-1955 98 4     SPACE iajs-1955 98 5 mathmatics mathmatic NOUN iajs-1955 98 6 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 98 7     SPACE iajs-1955 98 8 2018 2018 NUM iajs-1955 98 9 ) ) PUNCT iajs-1955 99 1 عام عام ADP iajs-1955 99 2 2العدد 2العدد NUM iajs-1955 99 3 ( ( PUNCT iajs-1955 99 4 31لمجلد 31لمجلد NUM iajs-1955 99 5 ا ا NOUN iajs-1955 99 6 مجلة مجلة NOUN iajs-1955 99 7 إبن إبن VERB iajs-1955 99 8 الهيثم الهيثم ADJ iajs-1955 99 9 للعلوم للعلوم NOUN iajs-1955 99 10 الصرفة الصرفة NOUN iajs-1955 100 1 و و PRON iajs-1955 100 2 التطبيقية التطبيقية ADJ iajs-1955 100 3 ibn ibn PROPN iajs-1955 100 4 al al PROPN iajs-1955 100 5 - - PUNCT iajs-1955 100 6 haitham haitham PROPN iajs-1955 100 7 jour jour X iajs-1955 100 8 . . PROPN iajs-1955 100 9 for for ADP iajs-1955 100 10 pure pure ADJ iajs-1955 100 11 & & CCONJ iajs-1955 100 12 appl appl PROPN iajs-1955 100 13 . . PUNCT iajs-1955 101 1 sci sci PROPN iajs-1955 101 2 . . PUNCT iajs-1955 101 3 vol vol NOUN iajs-1955 101 4 . . PROPN iajs-1955 102 1 31 31 NUM iajs-1955 102 2 ( ( PUNCT iajs-1955 102 3 2 2 NUM iajs-1955 102 4 ) ) PUNCT iajs-1955 102 5 2018 2018 NUM iajs-1955 102 6 proof proof NOUN iajs-1955 102 7 let let VERB iajs-1955 102 8 𝑆 𝑆 PROPN iajs-1955 102 9 𝑋 𝑋 PROPN iajs-1955 102 10 with with ADP iajs-1955 102 11 𝐸 𝐸 PROPN iajs-1955 102 12 ⨁ ⨁ PROPN iajs-1955 102 13 𝐷 𝐷 PROPN iajs-1955 102 14 " " PUNCT iajs-1955 102 15 is be AUX iajs-1955 102 16 a a DET iajs-1955 102 17 weak weak ADJ iajs-1955 102 18 essential essential ADJ iajs-1955 102 19 submodule submodule NOUN iajs-1955 102 20 in in ADP iajs-1955 102 21 s s NOUN iajs-1955 102 22 " " PUNCT iajs-1955 102 23 . . PUNCT iajs-1955 103 1 since since SCONJ iajs-1955 103 2 s s PROPN iajs-1955 103 3 is be AUX iajs-1955 103 4 a a DET iajs-1955 103 5 submodule submodule NOUN iajs-1955 103 6 of of ADP iajs-1955 103 7 x x X iajs-1955 103 8 and and CCONJ iajs-1955 103 9 ann ann PROPN iajs-1955 103 10 𝑋 𝑋 PROPN iajs-1955 103 11 + + CCONJ iajs-1955 103 12 ann ann PROPN iajs-1955 103 13 𝑋 𝑋 PROPN iajs-1955 103 14 = = PROPN iajs-1955 103 15 r r NOUN iajs-1955 103 16 , , PUNCT iajs-1955 103 17 then then ADV iajs-1955 103 18 by by ADP iajs-1955 103 19 [ [ PUNCT iajs-1955 103 20 9 9 NUM iajs-1955 103 21 , , PUNCT iajs-1955 103 22 prop(4.2 prop(4.2 NOUN iajs-1955 103 23 ) ) PUNCT iajs-1955 103 24 ] ] PUNCT iajs-1955 103 25 , , PUNCT iajs-1955 103 26 𝑆 𝑆 PROPN iajs-1955 103 27 𝑆 𝑆 PROPN iajs-1955 103 28 ⨁ ⨁ PROPN iajs-1955 103 29 𝑆 𝑆 PROPN iajs-1955 103 30 , , PUNCT iajs-1955 103 31 where where SCONJ iajs-1955 103 32 𝑆 𝑆 PROPN iajs-1955 103 33 is be AUX iajs-1955 103 34 a a DET iajs-1955 103 35 submodule submodule NOUN iajs-1955 103 36 of of ADP iajs-1955 103 37 𝑋 𝑋 PROPN iajs-1955 103 38 and and CCONJ iajs-1955 103 39 𝑆 𝑆 PROPN iajs-1955 103 40 is be AUX iajs-1955 103 41 a a DET iajs-1955 103 42 submodule submodule NOUN iajs-1955 103 43 of of ADP iajs-1955 103 44 𝑋 𝑋 PROPN iajs-1955 103 45 . . PUNCT iajs-1955 104 1 thus thus ADV iajs-1955 104 2 𝐸 𝐸 PROPN iajs-1955 104 3 ⨁ ⨁ PROPN iajs-1955 104 4 𝐷 𝐷 PROPN iajs-1955 104 5 is be AUX iajs-1955 104 6 a a DET iajs-1955 104 7 weak weak ADJ iajs-1955 104 8 essential essential ADJ iajs-1955 104 9 submodule submodule NOUN iajs-1955 104 10 in in ADP iajs-1955 104 11 𝑆 𝑆 PROPN iajs-1955 104 12 ⨁ ⨁ PROPN iajs-1955 104 13 𝑆 𝑆 PROPN iajs-1955 104 14 . . PUNCT iajs-1955 105 1 but but CCONJ iajs-1955 105 2 by by ADP iajs-1955 105 3 hybothesis hybothesis NOUN iajs-1955 105 4 s s NOUN iajs-1955 105 5 is be AUX iajs-1955 105 6 a a DET iajs-1955 105 7 completely completely ADV iajs-1955 105 8 essential essential ADJ iajs-1955 105 9 , , PUNCT iajs-1955 105 10 therefore therefore ADV iajs-1955 105 11 𝐸 𝐸 PROPN iajs-1955 105 12 ⨁ ⨁ PROPN iajs-1955 105 13 𝐷 𝐷 PROPN iajs-1955 105 14 is be AUX iajs-1955 105 15 an an DET iajs-1955 105 16 essential essential ADJ iajs-1955 105 17 submodule submodule NOUN iajs-1955 105 18 in in ADP iajs-1955 105 19 𝑆 𝑆 PROPN iajs-1955 105 20 𝑆 𝑆 PROPN iajs-1955 105 21 ⨁ ⨁ PROPN iajs-1955 105 22 𝑆 𝑆 PROPN iajs-1955 105 23 , , PUNCT iajs-1955 105 24 thus thus ADV iajs-1955 105 25 by by ADP iajs-1955 105 26 [ [ X iajs-1955 105 27 10 10 NUM iajs-1955 105 28 , , PUNCT iajs-1955 105 29 prop(5.20 prop(5.20 NUM iajs-1955 105 30 ) ) PUNCT iajs-1955 105 31 ] ] PUNCT iajs-1955 105 32 we we PRON iajs-1955 105 33 are be AUX iajs-1955 105 34 , , PUNCT iajs-1955 105 35 " " PUNCT iajs-1955 105 36 e e NOUN iajs-1955 105 37 is be AUX iajs-1955 105 38 an an DET iajs-1955 105 39 essential essential ADJ iajs-1955 105 40 submodule submodule NOUN iajs-1955 105 41 in in ADP iajs-1955 105 42 𝑆 𝑆 PROPN iajs-1955 105 43 and and CCONJ iajs-1955 105 44 d d PROPN iajs-1955 105 45 is be AUX iajs-1955 105 46 an an DET iajs-1955 105 47 essential essential ADJ iajs-1955 105 48 submodule submodule NOUN iajs-1955 105 49 in in ADP iajs-1955 105 50 𝑆 𝑆 PROPN iajs-1955 105 51 " " PUNCT iajs-1955 105 52 . . PUNCT iajs-1955 106 1 since since SCONJ iajs-1955 106 2 both both DET iajs-1955 106 3 e e NOUN iajs-1955 106 4 and and CCONJ iajs-1955 106 5 d d NOUN iajs-1955 106 6 are be AUX iajs-1955 106 7 w w NOUN iajs-1955 106 8 - - PUNCT iajs-1955 106 9 closed closed ADJ iajs-1955 106 10 , , PUNCT iajs-1955 106 11 it it PRON iajs-1955 106 12 is be AUX iajs-1955 106 13 a a DET iajs-1955 106 14 clear clear ADJ iajs-1955 106 15 that that SCONJ iajs-1955 106 16 e e PROPN iajs-1955 106 17 and and CCONJ iajs-1955 106 18 d d PROPN iajs-1955 106 19 are be AUX iajs-1955 106 20 closed close VERB iajs-1955 106 21 submodules submodule NOUN iajs-1955 106 22 in in ADP iajs-1955 106 23 𝑆 𝑆 PROPN iajs-1955 106 24 and and CCONJ iajs-1955 106 25 𝑆 𝑆 PROPN iajs-1955 106 26 respectively respectively ADV iajs-1955 106 27 . . PUNCT iajs-1955 107 1 then then ADV iajs-1955 107 2 e=𝑆 e=𝑆 PROPN iajs-1955 107 3 and and CCONJ iajs-1955 107 4 d=𝑆 d=𝑆 NUM iajs-1955 107 5 , , PUNCT iajs-1955 107 6 thus thus ADV iajs-1955 107 7 𝐸 𝐸 PROPN iajs-1955 107 8 ⨁ ⨁ PROPN iajs-1955 107 9 𝐷=𝑆 𝐷=𝑆 NOUN iajs-1955 107 10 ⨁ ⨁ PROPN iajs-1955 107 11 𝑆 𝑆 PROPN iajs-1955 107 12 . . PUNCT iajs-1955 108 1 that that PRON iajs-1955 108 2 is be AUX iajs-1955 108 3 𝐸 𝐸 PROPN iajs-1955 108 4 ⨁ ⨁ PROPN iajs-1955 108 5 𝐷 𝐷 PROPN iajs-1955 108 6 is be AUX iajs-1955 108 7 a a DET iajs-1955 108 8 w w NOUN iajs-1955 108 9 - - PUNCT iajs-1955 108 10 closed closed ADJ iajs-1955 108 11 submodule submodule NOUN iajs-1955 108 12 in in ADP iajs-1955 108 13 x. x. PROPN iajs-1955 108 14 proposition(2.27 proposition(2.27 PROPN iajs-1955 108 15 ) ) PUNCT iajs-1955 108 16 let let VERB iajs-1955 108 17 𝑋 𝑋 NOUN iajs-1955 108 18 𝑋 𝑋 PROPN iajs-1955 108 19 ⨁ ⨁ PROPN iajs-1955 108 20 𝑋 𝑋 PROPN iajs-1955 108 21 be be VERB iajs-1955 108 22 a a DET iajs-1955 108 23 module module NOUN iajs-1955 108 24 , , PUNCT iajs-1955 108 25 where where SCONJ iajs-1955 108 26 𝑋 𝑋 NOUN iajs-1955 108 27 and and CCONJ iajs-1955 108 28 𝑋 𝑋 PROPN iajs-1955 108 29 are be AUX iajs-1955 108 30 submodules submodule NOUN iajs-1955 108 31 of of ADP iajs-1955 108 32 x x SYM iajs-1955 108 33 such such ADJ iajs-1955 108 34 that that SCONJ iajs-1955 108 35 ann ann PROPN iajs-1955 108 36 𝑋 𝑋 PROPN iajs-1955 108 37 + + CCONJ iajs-1955 108 38 ann ann PROPN iajs-1955 108 39 𝑋 𝑋 PROPN iajs-1955 108 40 = = PROPN iajs-1955 108 41 r r NOUN iajs-1955 108 42 and and CCONJ iajs-1955 108 43 all all DET iajs-1955 108 44 submodules submodule NOUN iajs-1955 108 45 of of ADP iajs-1955 108 46 x x SYM iajs-1955 108 47 are be AUX iajs-1955 108 48 completely completely ADV iajs-1955 108 49 essential essential ADJ iajs-1955 108 50 submodule submodule NOUN iajs-1955 108 51 of of ADP iajs-1955 108 52 x. x. NOUN iajs-1955 108 53 if if SCONJ iajs-1955 108 54 e e PROPN iajs-1955 108 55 and and CCONJ iajs-1955 108 56 d d PROPN iajs-1955 108 57 are be AUX iajs-1955 108 58 non non X iajs-1955 108 59 zero zero NUM iajs-1955 108 60 submodules submodule NOUN iajs-1955 108 61 of of ADP iajs-1955 108 62 𝑋 𝑋 PROPN iajs-1955 108 63 and and CCONJ iajs-1955 108 64 𝑋 𝑋 PROPN iajs-1955 108 65 respectively respectively ADV iajs-1955 108 66 , , PUNCT iajs-1955 108 67 then then ADV iajs-1955 108 68 𝐸 𝐸 PROPN iajs-1955 108 69 ⨁ ⨁ PROPN iajs-1955 108 70 𝐷 𝐷 PROPN iajs-1955 108 71 is be AUX iajs-1955 108 72 a a DET iajs-1955 108 73 w w NOUN iajs-1955 108 74 - - PUNCT iajs-1955 108 75 closed closed ADJ iajs-1955 108 76 submodule submodule NOUN iajs-1955 108 77 in in ADP iajs-1955 108 78 x x SYM iajs-1955 108 79 if if SCONJ iajs-1955 108 80 and and CCONJ iajs-1955 108 81 only only ADV iajs-1955 108 82 if if SCONJ iajs-1955 108 83 e e NOUN iajs-1955 108 84 is be AUX iajs-1955 108 85 a a DET iajs-1955 108 86 w w NOUN iajs-1955 108 87 - - PUNCT iajs-1955 108 88 closed closed ADJ iajs-1955 108 89 submodule submodule NOUN iajs-1955 108 90 in in ADP iajs-1955 108 91 𝑋 𝑋 PROPN iajs-1955 108 92 and and CCONJ iajs-1955 108 93 d d PROPN iajs-1955 108 94 is be AUX iajs-1955 108 95 a a DET iajs-1955 108 96 w w NOUN iajs-1955 108 97 - - PUNCT iajs-1955 108 98 closed closed ADJ iajs-1955 108 99 submodule submodule NOUN iajs-1955 108 100 in in ADP iajs-1955 108 101 𝑋 𝑋 PROPN iajs-1955 108 102 . . PUNCT iajs-1955 109 1 proof proof NOUN iajs-1955 109 2 ⟸ ⟸ ADJ iajs-1955 109 3 suppose suppose VERB iajs-1955 109 4 that that SCONJ iajs-1955 109 5 𝐸 𝐸 PROPN iajs-1955 109 6 ⨁ ⨁ PROPN iajs-1955 109 7 𝐷 𝐷 PROPN iajs-1955 109 8 " " PUNCT iajs-1955 109 9 is be AUX iajs-1955 109 10 weak weak ADJ iajs-1955 109 11 essential essential ADJ iajs-1955 109 12 submodule submodule NOUN iajs-1955 109 13 of of ADP iajs-1955 109 14 k k NOUN iajs-1955 109 15 " " PUNCT iajs-1955 109 16 , , PUNCT iajs-1955 109 17 " " PUNCT iajs-1955 109 18 where where SCONJ iajs-1955 109 19 k k PROPN iajs-1955 109 20 is be AUX iajs-1955 109 21 a a DET iajs-1955 109 22 submodule submodule NOUN iajs-1955 109 23 of of ADP iajs-1955 109 24 m m NOUN iajs-1955 109 25 " " PUNCT iajs-1955 109 26 . . PUNCT iajs-1955 110 1 hence hence ADV iajs-1955 110 2 by by ADP iajs-1955 110 3 [ [ X iajs-1955 110 4 1 1 NUM iajs-1955 110 5 , , PUNCT iajs-1955 110 6 prop(4.2 prop(4.2 NOUN iajs-1955 110 7 ) ) PUNCT iajs-1955 110 8 ] ] PUNCT iajs-1955 111 1 𝐾 𝐾 PROPN iajs-1955 111 2 𝐾 𝐾 PROPN iajs-1955 111 3 ⨁ ⨁ PROPN iajs-1955 111 4 𝐾 𝐾 PROPN iajs-1955 111 5 where where SCONJ iajs-1955 111 6 𝐾 𝐾 PROPN iajs-1955 111 7 is be AUX iajs-1955 111 8 a a DET iajs-1955 111 9 submodule submodule NOUN iajs-1955 111 10 of of ADP iajs-1955 111 11 𝑋 𝑋 PROPN iajs-1955 111 12 and𝐾 and𝐾 NOUN iajs-1955 111 13 is be AUX iajs-1955 111 14 a a DET iajs-1955 111 15 submodule submodule NOUN iajs-1955 111 16 of of ADP iajs-1955 111 17 𝑋 𝑋 PROPN iajs-1955 111 18 . . PUNCT iajs-1955 112 1 thus thus ADV iajs-1955 112 2 𝐸 𝐸 PROPN iajs-1955 112 3 ⨁ ⨁ PROPN iajs-1955 112 4 𝐷 𝐷 PROPN iajs-1955 112 5 is be AUX iajs-1955 112 6 weak weak ADJ iajs-1955 112 7 essential essential ADJ iajs-1955 112 8 submodule submodule NOUN iajs-1955 112 9 in in ADP iajs-1955 112 10 𝐾 𝐾 PROPN iajs-1955 112 11 ⨁ ⨁ PROPN iajs-1955 112 12 𝐾 𝐾 PROPN iajs-1955 112 13 . . PUNCT iajs-1955 113 1 but but CCONJ iajs-1955 113 2 𝐾 𝐾 PROPN iajs-1955 113 3 ⨁ ⨁ PROPN iajs-1955 113 4 𝐾 𝐾 PROPN iajs-1955 113 5 is be AUX iajs-1955 113 6 a a DET iajs-1955 113 7 completely completely ADV iajs-1955 113 8 " " PUNCT iajs-1955 113 9 essential essential ADJ iajs-1955 113 10 submodule submodule NOUN iajs-1955 113 11 of of ADP iajs-1955 113 12 " " PUNCT iajs-1955 113 13 x x NOUN iajs-1955 113 14 , , PUNCT iajs-1955 113 15 then then ADV iajs-1955 113 16 𝐸 𝐸 PROPN iajs-1955 113 17 ⨁ ⨁ PROPN iajs-1955 113 18 𝐷 𝐷 PROPN iajs-1955 113 19 " " PUNCT iajs-1955 113 20 is be AUX iajs-1955 113 21 an an DET iajs-1955 113 22 essential essential ADJ iajs-1955 113 23 submodule submodule NOUN iajs-1955 113 24 of of ADP iajs-1955 113 25 " " PUNCT iajs-1955 113 26 𝐾 𝐾 PROPN iajs-1955 113 27 ⨁ ⨁ PROPN iajs-1955 113 28 𝐾 𝐾 PROPN iajs-1955 113 29 . . PUNCT iajs-1955 114 1 hence hence ADV iajs-1955 114 2 by by ADP iajs-1955 114 3 [ [ X iajs-1955 114 4 10 10 NUM iajs-1955 114 5 , , PUNCT iajs-1955 114 6 prop(5.20 prop(5.20 NUM iajs-1955 114 7 ) ) PUNCT iajs-1955 114 8 , , PUNCT iajs-1955 114 9 p.15 p.15 AUX iajs-1955 114 10 ] ] X iajs-1955 114 11 we we PRON iajs-1955 114 12 get get VERB iajs-1955 114 13 " " PUNCT iajs-1955 114 14 e e NOUN iajs-1955 114 15 is be AUX iajs-1955 114 16 an an DET iajs-1955 114 17 essential essential ADJ iajs-1955 114 18 submodule submodule NOUN iajs-1955 114 19 in in ADP iajs-1955 114 20 𝐾 𝐾 PROPN iajs-1955 114 21 and and CCONJ iajs-1955 114 22 d d PROPN iajs-1955 114 23 is be AUX iajs-1955 114 24 an an DET iajs-1955 114 25 essential essential ADJ iajs-1955 114 26 submodule submodule NOUN iajs-1955 114 27 in in ADP iajs-1955 114 28 𝐾 𝐾 PROPN iajs-1955 114 29 " " PUNCT iajs-1955 114 30 . . PUNCT iajs-1955 115 1 but but CCONJ iajs-1955 115 2 by by ADP iajs-1955 115 3 [ [ X iajs-1955 115 4 2 2 X iajs-1955 115 5 ] ] PUNCT iajs-1955 115 6 every every DET iajs-1955 115 7 essential essential ADJ iajs-1955 115 8 submodule submodule NOUN iajs-1955 115 9 is be AUX iajs-1955 115 10 a a DET iajs-1955 115 11 weak weak ADJ iajs-1955 115 12 essential essential ADJ iajs-1955 115 13 . . PUNCT iajs-1955 116 1 hence hence ADV iajs-1955 116 2 e e X iajs-1955 116 3 " " PUNCT iajs-1955 116 4 is be AUX iajs-1955 116 5 a a DET iajs-1955 116 6 weak weak ADJ iajs-1955 116 7 essential essential ADJ iajs-1955 116 8 submodule submodule NOUN iajs-1955 116 9 in in ADP iajs-1955 116 10 𝐾 𝐾 PROPN iajs-1955 116 11 " " PUNCT iajs-1955 116 12 and and CCONJ iajs-1955 116 13 d d PROPN iajs-1955 116 14 is be AUX iajs-1955 116 15 a a DET iajs-1955 116 16 weak weak ADJ iajs-1955 116 17 essential essential ADJ iajs-1955 116 18 submodule submodule NOUN iajs-1955 116 19 in in ADP iajs-1955 116 20 𝐾 𝐾 PROPN iajs-1955 116 21 . . PUNCT iajs-1955 117 1 but but CCONJ iajs-1955 117 2 e e NOUN iajs-1955 117 3 and and CCONJ iajs-1955 117 4 d d PROPN iajs-1955 117 5 are be AUX iajs-1955 117 6 w w NOUN iajs-1955 117 7 - - PUNCT iajs-1955 117 8 closed closed ADJ iajs-1955 117 9 submodules submodule NOUN iajs-1955 117 10 of of ADP iajs-1955 117 11 x x NOUN iajs-1955 117 12 , , PUNCT iajs-1955 117 13 then then ADV iajs-1955 117 14 e=𝐾 e=𝐾 NOUN iajs-1955 117 15 and and CCONJ iajs-1955 117 16 d=𝐾 d=𝐾 NOUN iajs-1955 117 17 . . PUNCT iajs-1955 118 1 thus thus ADV iajs-1955 118 2 𝐸 𝐸 PROPN iajs-1955 118 3 ⨁ ⨁ PROPN iajs-1955 118 4 𝐷=𝐾 𝐷=𝐾 PROPN iajs-1955 118 5 ⨁ ⨁ PROPN iajs-1955 118 6 𝐾 𝐾 PROPN iajs-1955 118 7 . . PUNCT iajs-1955 119 1 that that PRON iajs-1955 119 2 is be AUX iajs-1955 119 3 𝐸 𝐸 PROPN iajs-1955 119 4 ⨁ ⨁ PROPN iajs-1955 119 5 𝐷 𝐷 PROPN iajs-1955 119 6 is be AUX iajs-1955 119 7 a a DET iajs-1955 119 8 w w NOUN iajs-1955 119 9 - - PUNCT iajs-1955 119 10 closed closed ADJ iajs-1955 119 11 submodule submodule NOUN iajs-1955 119 12 in in ADP iajs-1955 119 13 x. x. PROPN iajs-1955 119 14 ⟹ ⟹ PROPN iajs-1955 119 15 assume assume VERB iajs-1955 119 16 that that SCONJ iajs-1955 119 17 e e PRON iajs-1955 119 18 " " PUNCT iajs-1955 119 19 is be AUX iajs-1955 119 20 a a DET iajs-1955 119 21 weak weak ADJ iajs-1955 119 22 essential essential ADJ iajs-1955 119 23 submodule submodule NOUN iajs-1955 119 24 in in ADP iajs-1955 119 25 l l NOUN iajs-1955 119 26 " " PUNCT iajs-1955 119 27 where where SCONJ iajs-1955 119 28 l l NOUN iajs-1955 119 29 is be AUX iajs-1955 119 30 a a DET iajs-1955 119 31 submodule submodule NOUN iajs-1955 119 32 of of ADP iajs-1955 119 33 x x PRON iajs-1955 119 34 , , PUNCT iajs-1955 119 35 we we PRON iajs-1955 119 36 have have AUX iajs-1955 119 37 d d NOUN iajs-1955 119 38 is be AUX iajs-1955 119 39 a a DET iajs-1955 119 40 weak weak ADJ iajs-1955 119 41 essential essential ADJ iajs-1955 119 42 submodule submodule NOUN iajs-1955 119 43 in in ADP iajs-1955 119 44 d. d. PROPN iajs-1955 119 45 but but CCONJ iajs-1955 119 46 by by ADP iajs-1955 119 47 hypothesis hypothesis NOUN iajs-1955 119 48 all all DET iajs-1955 119 49 submodules submodule NOUN iajs-1955 119 50 of of ADP iajs-1955 119 51 x x SYM iajs-1955 119 52 are be AUX iajs-1955 119 53 completely completely ADV iajs-1955 119 54 essential essential ADJ iajs-1955 119 55 , , PUNCT iajs-1955 119 56 then then ADV iajs-1955 119 57 e e PROPN iajs-1955 119 58 is be AUX iajs-1955 119 59 an an DET iajs-1955 119 60 essential essential ADJ iajs-1955 119 61 submodule submodule NOUN iajs-1955 119 62 in in ADP iajs-1955 119 63 l l PROPN iajs-1955 119 64 and and CCONJ iajs-1955 119 65 d d PROPN iajs-1955 119 66 is be AUX iajs-1955 119 67 an an DET iajs-1955 119 68 essential essential ADJ iajs-1955 119 69 submodule submodule NOUN iajs-1955 119 70 in in ADP iajs-1955 119 71 d. d. PROPN iajs-1955 119 72 hence hence ADV iajs-1955 119 73 by by ADP iajs-1955 119 74 [ [ X iajs-1955 119 75 10 10 NUM iajs-1955 119 76 , , PUNCT iajs-1955 119 77 prop(5.20 prop(5.20 NUM iajs-1955 119 78 ) ) PUNCT iajs-1955 119 79 , , PUNCT iajs-1955 119 80 p.15 p.15 PROPN iajs-1955 119 81 ] ] PUNCT iajs-1955 119 82 , , PUNCT iajs-1955 119 83 we we PRON iajs-1955 119 84 have have VERB iajs-1955 119 85 . . PUNCT iajs-1955 120 1 𝐸 𝐸 PROPN iajs-1955 120 2 ⨁ ⨁ PROPN iajs-1955 120 3 𝐷 𝐷 PROPN iajs-1955 120 4 is be AUX iajs-1955 120 5 an an DET iajs-1955 120 6 essential essential ADJ iajs-1955 120 7 submodule submodule NOUN iajs-1955 120 8 in in ADP iajs-1955 120 9 𝐿 𝐿 PROPN iajs-1955 120 10 ⨁ ⨁ PROPN iajs-1955 120 11 𝐷 𝐷 PROPN iajs-1955 120 12 , , PUNCT iajs-1955 120 13 which which PRON iajs-1955 120 14 implies imply VERB iajs-1955 120 15 that that SCONJ iajs-1955 120 16 𝐸 𝐸 PROPN iajs-1955 120 17 ⨁ ⨁ PROPN iajs-1955 120 18 𝐷 𝐷 PROPN iajs-1955 120 19 is be AUX iajs-1955 120 20 a a DET iajs-1955 120 21 weak weak ADJ iajs-1955 120 22 essential essential ADJ iajs-1955 120 23 submodule submodule NOUN iajs-1955 120 24 in in ADP iajs-1955 120 25 𝐿 𝐿 PROPN iajs-1955 120 26 ⨁ ⨁ PROPN iajs-1955 120 27 𝐷. 𝐷. PROPN iajs-1955 120 28 hence hence ADV iajs-1955 120 29 𝐸 𝐸 NOUN iajs-1955 120 30 ⨁ ⨁ PROPN iajs-1955 120 31 𝐷= 𝐷= PROPN iajs-1955 120 32 𝐿 𝐿 PROPN iajs-1955 120 33 ⨁ ⨁ PROPN iajs-1955 120 34 𝐷. 𝐷. PROPN iajs-1955 120 35 that that PRON iajs-1955 120 36 is be AUX iajs-1955 120 37 e e NOUN iajs-1955 120 38 = = NOUN iajs-1955 120 39 l l NOUN iajs-1955 120 40 , , PUNCT iajs-1955 120 41 implies imply VERB iajs-1955 120 42 that that SCONJ iajs-1955 120 43 e e NOUN iajs-1955 120 44 is be AUX iajs-1955 120 45 a a DET iajs-1955 120 46 w w NOUN iajs-1955 120 47 - - PUNCT iajs-1955 120 48 closed closed ADJ iajs-1955 120 49 submodule submodule NOUN iajs-1955 120 50 in𝑋 in𝑋 NOUN iajs-1955 120 51 . . PUNCT iajs-1955 121 1 in in ADP iajs-1955 121 2 similar similar ADJ iajs-1955 121 3 way way NOUN iajs-1955 121 4 we we PRON iajs-1955 121 5 can can AUX iajs-1955 121 6 prove prove VERB iajs-1955 121 7 that that SCONJ iajs-1955 121 8 d d NOUN iajs-1955 121 9 is be AUX iajs-1955 121 10 w w NOUN iajs-1955 121 11 - - PUNCT iajs-1955 121 12 closed closed ADJ iajs-1955 121 13 submodule submodule NOUN iajs-1955 121 14 in𝑋 in𝑋 NOUN iajs-1955 121 15 . . PUNCT iajs-1955 122 1 it it PRON iajs-1955 122 2 is be AUX iajs-1955 122 3 well well ADV iajs-1955 122 4 - - PUNCT iajs-1955 122 5 known know VERB iajs-1955 122 6 that that SCONJ iajs-1955 122 7 a a DET iajs-1955 122 8 fully fully ADV iajs-1955 122 9 semi semi ADJ iajs-1955 122 10 - - ADJ iajs-1955 122 11 prime prime ADJ iajs-1955 122 12 module module NOUN iajs-1955 122 13 is be AUX iajs-1955 122 14 a a DET iajs-1955 122 15 completely completely ADV iajs-1955 122 16 essential essential ADJ iajs-1955 122 17 [ [ X iajs-1955 122 18 3 3 NUM iajs-1955 122 19 , , PUNCT iajs-1955 122 20 cor(2.6 cor(2.6 NUM iajs-1955 122 21 ) ) PUNCT iajs-1955 122 22 ] ] PUNCT iajs-1955 122 23 . . PUNCT iajs-1955 123 1 so so ADV iajs-1955 123 2 we we PRON iajs-1955 123 3 have have VERB iajs-1955 123 4 the the DET iajs-1955 123 5 following follow VERB iajs-1955 123 6 result result NOUN iajs-1955 123 7 . . PUNCT iajs-1955 124 1 corollary(2.28 corollary(2.28 NOUN iajs-1955 124 2 ) ) PUNCT iajs-1955 124 3 if if SCONJ iajs-1955 124 4 𝑋 𝑋 PROPN iajs-1955 124 5 𝑋 𝑋 PROPN iajs-1955 124 6 ⨁ ⨁ PROPN iajs-1955 124 7 𝑋 𝑋 PROPN iajs-1955 124 8 is be AUX iajs-1955 124 9 a a DET iajs-1955 124 10 module module NOUN iajs-1955 124 11 , , PUNCT iajs-1955 124 12 where where SCONJ iajs-1955 124 13 𝑋 𝑋 NOUN iajs-1955 124 14 and and CCONJ iajs-1955 124 15 𝑋 𝑋 PROPN iajs-1955 124 16 are be AUX iajs-1955 124 17 submodules submodule NOUN iajs-1955 124 18 of of ADP iajs-1955 124 19 x x PUNCT iajs-1955 124 20 with with ADP iajs-1955 124 21 ann ann PROPN iajs-1955 124 22 𝑋 𝑋 PROPN iajs-1955 124 23 + + CCONJ iajs-1955 124 24 ann ann PROPN iajs-1955 124 25 𝑋 𝑋 PROPN iajs-1955 124 26 = = PROPN iajs-1955 124 27 r r NOUN iajs-1955 124 28 and and CCONJ iajs-1955 124 29 all all DET iajs-1955 124 30 submodules submodule NOUN iajs-1955 124 31 of of ADP iajs-1955 124 32 x x SYM iajs-1955 124 33 are be AUX iajs-1955 124 34 fully fully ADV iajs-1955 124 35 - - PUNCT iajs-1955 124 36 semi semi ADJ iajs-1955 124 37 - - ADJ iajs-1955 124 38 prime prime ADJ iajs-1955 124 39 . . PUNCT iajs-1955 125 1 if if SCONJ iajs-1955 125 2 e e X iajs-1955 125 3 , , PUNCT iajs-1955 125 4 d d X iajs-1955 125 5 are be AUX iajs-1955 125 6 submodules submodule NOUN iajs-1955 125 7 of of ADP iajs-1955 125 8 𝑋 𝑋 NOUN iajs-1955 125 9 and and CCONJ iajs-1955 125 10 𝑋 𝑋 PROPN iajs-1955 125 11 respectively respectively ADV iajs-1955 125 12 , , PUNCT iajs-1955 125 13 then then ADV iajs-1955 125 14 𝐸 𝐸 PROPN iajs-1955 125 15 ⨁ ⨁ PROPN iajs-1955 125 16 𝐷 𝐷 PROPN iajs-1955 125 17 is be AUX iajs-1955 125 18 a a DET iajs-1955 125 19 w w NOUN iajs-1955 125 20 - - PUNCT iajs-1955 125 21 closed closed ADJ iajs-1955 125 22 submodule submodule NOUN iajs-1955 125 23 in in ADP iajs-1955 125 24 x x SYM iajs-1955 125 25 if if SCONJ iajs-1955 125 26 and and CCONJ iajs-1955 125 27 only only ADV iajs-1955 125 28 if if SCONJ iajs-1955 125 29 e e NOUN iajs-1955 125 30 is be AUX iajs-1955 125 31 a a DET iajs-1955 125 32 w w NOUN iajs-1955 125 33 - - PUNCT iajs-1955 125 34 closed closed ADJ iajs-1955 125 35 submodule submodule NOUN iajs-1955 125 36 in in ADP iajs-1955 125 37 𝑋 𝑋 PROPN iajs-1955 125 38 and and CCONJ iajs-1955 125 39 d d PROPN iajs-1955 125 40 is be AUX iajs-1955 125 41 a a DET iajs-1955 125 42 w w NOUN iajs-1955 125 43 - - PUNCT iajs-1955 125 44 closed closed ADJ iajs-1955 125 45 submodule submodule NOUN iajs-1955 125 46 in in ADP iajs-1955 125 47 𝑋 𝑋 PROPN iajs-1955 125 48 . . PUNCT iajs-1955 126 1 the the DET iajs-1955 126 2 following follow VERB iajs-1955 126 3 remark remark NOUN iajs-1955 126 4 shows show VERB iajs-1955 126 5 that that SCONJ iajs-1955 126 6 w w NOUN iajs-1955 126 7 - - PUNCT iajs-1955 126 8 closed closed ADJ iajs-1955 126 9 property property NOUN iajs-1955 126 10 is be AUX iajs-1955 126 11 not not PART iajs-1955 126 12 algebrice algebrice ADJ iajs-1955 126 13 property property NOUN iajs-1955 126 14 . . PUNCT iajs-1955 127 1 remark(2.29 remark(2.29 NOUN iajs-1955 127 2 ) ) PUNCT iajs-1955 127 3 if if SCONJ iajs-1955 127 4 m m NOUN iajs-1955 127 5 is be AUX iajs-1955 127 6 a a DET iajs-1955 127 7 module module NOUN iajs-1955 127 8 , , PUNCT iajs-1955 127 9 and and CCONJ iajs-1955 127 10 x x X iajs-1955 127 11 is be AUX iajs-1955 127 12 a a DET iajs-1955 127 13 w w NOUN iajs-1955 127 14 - - PUNCT iajs-1955 127 15 closed closed ADJ iajs-1955 127 16 submodule submodule NOUN iajs-1955 127 17 of of ADP iajs-1955 127 18 m m PROPN iajs-1955 127 19 , , PUNCT iajs-1955 127 20 and and CCONJ iajs-1955 127 21 y y PROPN iajs-1955 127 22 is be AUX iajs-1955 127 23 asubmodule asubmodule NOUN iajs-1955 127 24 of of ADP iajs-1955 127 25 m m NOUN iajs-1955 127 26 such such ADJ iajs-1955 127 27 that that SCONJ iajs-1955 127 28 x≅y x≅y ADV iajs-1955 127 29 , , PUNCT iajs-1955 127 30 then then ADV iajs-1955 127 31 it it PRON iajs-1955 127 32 is be AUX iajs-1955 127 33 not not PART iajs-1955 127 34 necessary necessary ADJ iajs-1955 127 35 that that SCONJ iajs-1955 127 36 y y PROPN iajs-1955 127 37 is be AUX iajs-1955 127 38 a a DET iajs-1955 127 39 w w NOUN iajs-1955 127 40 - - PUNCT iajs-1955 127 41 closed closed ADJ iajs-1955 127 42 submodule submodule NOUN iajs-1955 127 43 in in ADP iajs-1955 127 44 m m PROPN iajs-1955 127 45 , , PUNCT iajs-1955 127 46 as as ADP iajs-1955 127 47 the the DET iajs-1955 127 48 following following ADJ iajs-1955 127 49 example example NOUN iajs-1955 127 50             SPACE iajs-1955 127 51 171 171 NUM iajs-1955 127 52     SPACE iajs-1955 127 53 | | ADV iajs-1955 127 54     SPACE iajs-1955 127 55 mathmatics mathmatic NOUN iajs-1955 127 56 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 127 57     SPACE iajs-1955 127 58 2018 2018 NUM iajs-1955 127 59 ) ) PUNCT iajs-1955 127 60 عام عام ADP iajs-1955 127 61 2العدد 2العدد NUM iajs-1955 127 62 ( ( PUNCT iajs-1955 127 63 31لمجلد 31لمجلد NUM iajs-1955 127 64 ا ا NOUN iajs-1955 127 65 مجلة مجلة NOUN iajs-1955 127 66 إبن إبن VERB iajs-1955 127 67 الهيثم الهيثم ADJ iajs-1955 127 68 للعلوم للعلوم NOUN iajs-1955 127 69 الصرفة الصرفة NOUN iajs-1955 128 1 و و PRON iajs-1955 128 2 التطبيقية التطبيقية ADJ iajs-1955 128 3 ibn ibn PROPN iajs-1955 128 4 al al PROPN iajs-1955 128 5 - - PUNCT iajs-1955 128 6 haitham haitham PROPN iajs-1955 128 7 jour jour X iajs-1955 128 8 . . PROPN iajs-1955 128 9 for for ADP iajs-1955 128 10 pure pure ADJ iajs-1955 128 11 & & CCONJ iajs-1955 128 12 appl appl PROPN iajs-1955 128 13 . . PUNCT iajs-1955 129 1 sci sci PROPN iajs-1955 129 2 . . PUNCT iajs-1955 129 3 vol vol NOUN iajs-1955 129 4 . . PROPN iajs-1955 130 1 31 31 NUM iajs-1955 130 2 ( ( PUNCT iajs-1955 130 3 2 2 NUM iajs-1955 130 4 ) ) PUNCT iajs-1955 130 5 2018 2018 NUM iajs-1955 130 6 shows show VERB iajs-1955 130 7 : : PUNCT iajs-1955 130 8 the the DET iajs-1955 130 9 z z NOUN iajs-1955 130 10 - - PUNCT iajs-1955 130 11 module module NOUN iajs-1955 130 12 z z NOUN iajs-1955 130 13 is be AUX iajs-1955 130 14 a a DET iajs-1955 130 15 w w NOUN iajs-1955 130 16 - - PUNCT iajs-1955 130 17 closed closed ADJ iajs-1955 130 18 in in ADP iajs-1955 130 19 itself itself PRON iajs-1955 130 20 and and CCONJ iajs-1955 130 21 z z PROPN iajs-1955 130 22 ≅ ≅ PROPN iajs-1955 130 23 3z 3z NUM iajs-1955 130 24 , , PUNCT iajs-1955 130 25 but but CCONJ iajs-1955 130 26 3z 3z NUM iajs-1955 130 27 as as ADP iajs-1955 130 28 a a DET iajs-1955 130 29 z z NOUN iajs-1955 130 30 - - PUNCT iajs-1955 130 31 module module NOUN iajs-1955 130 32 is be AUX iajs-1955 130 33 not not PART iajs-1955 130 34 a a DET iajs-1955 130 35 wclosed wclose VERB iajs-1955 130 36 submodule submodule NOUN iajs-1955 130 37 in in ADP iajs-1955 130 38 z z PROPN iajs-1955 130 39 , , PUNCT iajs-1955 130 40 since since SCONJ iajs-1955 130 41 3z 3z NUM iajs-1955 130 42 " " PUNCT iajs-1955 130 43 is be AUX iajs-1955 130 44 a a DET iajs-1955 130 45 weak weak ADJ iajs-1955 130 46 - - PUNCT iajs-1955 130 47 essential essential ADJ iajs-1955 130 48 submodule submodule NOUN iajs-1955 130 49 of of ADP iajs-1955 130 50 z z PROPN iajs-1955 130 51 " " PUNCT iajs-1955 130 52 . . PUNCT iajs-1955 131 1 we we PRON iajs-1955 131 2 introduce introduce VERB iajs-1955 131 3 the the DET iajs-1955 131 4 following follow VERB iajs-1955 131 5 lemma lemma PROPN iajs-1955 131 6 , , PUNCT iajs-1955 131 7 before before SCONJ iajs-1955 131 8 we we PRON iajs-1955 131 9 give give VERB iajs-1955 131 10 the the DET iajs-1955 131 11 next next ADJ iajs-1955 131 12 proposition proposition NOUN iajs-1955 131 13 . . PUNCT iajs-1955 132 1 lemma(2.30 lemma(2.30 NOUN iajs-1955 132 2 ) ) PUNCT iajs-1955 132 3 let let VERB iajs-1955 132 4 𝑓 𝑓 DET iajs-1955 132 5 ∈ ∈ PROPN iajs-1955 132 6 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1955 132 7 𝑀 𝑀 PROPN iajs-1955 132 8 , , PUNCT iajs-1955 132 9 𝑀 𝑀 PROPN iajs-1955 132 10 be be AUX iajs-1955 132 11 module module NOUN iajs-1955 132 12 an an DET iajs-1955 132 13 epimorphism epimorphism NOUN iajs-1955 132 14 with with ADP iajs-1955 132 15 𝐾𝑒𝑟 𝐾𝑒𝑟 PROPN iajs-1955 132 16 𝑓 𝑓 DET iajs-1955 132 17 𝑆𝑟𝑎𝑑 𝑆𝑟𝑎𝑑 PROPN iajs-1955 132 18 𝑀 𝑀 PROPN iajs-1955 132 19 , , PUNCT iajs-1955 132 20 if if SCONJ iajs-1955 132 21 𝐸 𝐸 PROPN iajs-1955 132 22 𝑀 𝑀 PROPN iajs-1955 132 23 . . PUNCT iajs-1955 133 1 then then ADV iajs-1955 133 2 𝑓 𝑓 DET iajs-1955 133 3 𝐸 𝐸 ADJ iajs-1955 133 4 𝑤𝑒𝑎𝑘 𝑤𝑒𝑎𝑘 NOUN iajs-1955 133 5 𝑀1 𝑀1 ADV iajs-1955 133 6 . . PUNCT iajs-1955 134 1 proof proof NOUN iajs-1955 134 2 assume assume VERB iajs-1955 134 3 that that SCONJ iajs-1955 134 4 𝐸 𝐸 PROPN iajs-1955 134 5 𝑀 𝑀 PROPN iajs-1955 134 6 , , PUNCT iajs-1955 134 7 and and CCONJ iajs-1955 134 8 𝑓 𝑓 DET iajs-1955 134 9 𝐸 𝐸 PROPN iajs-1955 134 10 ∩ ∩ ADJ iajs-1955 134 11 𝑆 𝑆 PROPN iajs-1955 134 12 0 0 NUM iajs-1955 134 13 where where SCONJ iajs-1955 134 14 s s VERB iajs-1955 134 15 is be AUX iajs-1955 134 16 a a DET iajs-1955 134 17 semi semi ADJ iajs-1955 134 18 - - ADJ iajs-1955 134 19 prime prime ADJ iajs-1955 134 20 submodule submodule NOUN iajs-1955 134 21 of of ADP iajs-1955 134 22 𝑀 𝑀 PROPN iajs-1955 134 23 . . PUNCT iajs-1955 135 1 but but CCONJ iajs-1955 135 2 𝐾𝑒𝑟 𝐾𝑒𝑟 PROPN iajs-1955 135 3 𝑓 𝑓 DET iajs-1955 135 4 𝑆𝑟𝑎𝑑 𝑆𝑟𝑎𝑑 PROPN iajs-1955 135 5 𝑀 𝑀 PROPN iajs-1955 135 6 𝑆 𝑆 PROPN iajs-1955 135 7 for for ADP iajs-1955 135 8 all all DET iajs-1955 135 9 semi semi ADJ iajs-1955 135 10 - - ADJ iajs-1955 135 11 prime prime ADJ iajs-1955 135 12 submodule submodule NOUN iajs-1955 135 13 s s PROPN iajs-1955 135 14 of of ADP iajs-1955 135 15 𝑀 𝑀 PROPN iajs-1955 135 16 , , PUNCT iajs-1955 135 17 hence hence ADV iajs-1955 135 18 by by ADP iajs-1955 135 19 [ [ X iajs-1955 135 20 5 5 NUM iajs-1955 135 21 , , PUNCT iajs-1955 135 22 prop(2.1)(a prop(2.1)(a PROPN iajs-1955 135 23 ) ) PUNCT iajs-1955 135 24 ] ] PUNCT iajs-1955 136 1 𝑓 𝑓 DET iajs-1955 136 2 𝑆 𝑆 PROPN iajs-1955 136 3 is be AUX iajs-1955 136 4 a a DET iajs-1955 136 5 semi semi ADJ iajs-1955 136 6 - - ADJ iajs-1955 136 7 prime prime ADJ iajs-1955 136 8 submodule submodule NOUN iajs-1955 136 9 of of ADP iajs-1955 136 10 𝑀 𝑀 PROPN iajs-1955 136 11 . . PUNCT iajs-1955 137 1 that that PRON iajs-1955 137 2 is be AUX iajs-1955 137 3 𝐸 𝐸 PROPN iajs-1955 137 4 ∩ ∩ NOUN iajs-1955 137 5 𝑓 𝑓 DET iajs-1955 137 6 𝑆 𝑆 PROPN iajs-1955 137 7 0 0 NUM iajs-1955 137 8 , , PUNCT iajs-1955 137 9 but but CCONJ iajs-1955 137 10 e e X iajs-1955 137 11 " " PUNCT iajs-1955 137 12 is be AUX iajs-1955 137 13 a a DET iajs-1955 137 14 weak weak ADJ iajs-1955 137 15 essential essential ADJ iajs-1955 137 16 submodule submodule NOUN iajs-1955 137 17 of of ADP iajs-1955 137 18 𝑀 𝑀 PROPN iajs-1955 137 19 " " PUNCT iajs-1955 137 20 , , PUNCT iajs-1955 137 21 then then ADV iajs-1955 137 22 𝑓 𝑓 DET iajs-1955 137 23 𝑆 𝑆 PROPN iajs-1955 137 24 0 0 NUM iajs-1955 137 25 . . PUNCT iajs-1955 138 1 implies imply VERB iajs-1955 138 2 that that SCONJ iajs-1955 138 3 𝑆 𝑆 PROPN iajs-1955 138 4 𝐾𝑒𝑟𝑓 𝐾𝑒𝑟𝑓 PROPN iajs-1955 138 5 𝑓 𝑓 DET iajs-1955 138 6 𝐸 𝐸 PROPN iajs-1955 138 7 , , PUNCT iajs-1955 138 8 and and CCONJ iajs-1955 138 9 hence hence ADV iajs-1955 138 10 𝑓 𝑓 DET iajs-1955 138 11 𝐸 𝐸 PROPN iajs-1955 138 12 ∩ ∩ ADJ iajs-1955 138 13 𝑆 𝑆 PROPN iajs-1955 138 14 0 0 NUM iajs-1955 138 15 implies imply VERB iajs-1955 138 16 that that SCONJ iajs-1955 138 17 𝑆 𝑆 PROPN iajs-1955 138 18 0 0 NUM iajs-1955 138 19 . . PUNCT iajs-1955 139 1 then then ADV iajs-1955 139 2 𝑓 𝑓 DET iajs-1955 139 3 𝐸 𝐸 ADJ iajs-1955 139 4 𝑤𝑒𝑎𝑘 𝑤𝑒𝑎𝑘 NOUN iajs-1955 139 5 𝑀1 𝑀1 PROPN iajs-1955 139 6 . . PUNCT iajs-1955 140 1 proposition(2.31 proposition(2.31 NOUN iajs-1955 140 2 ) ) PUNCT iajs-1955 140 3 let let VERB iajs-1955 140 4 𝑔 𝑔 NOUN iajs-1955 140 5 : : PUNCT iajs-1955 140 6 𝑀 𝑀 PROPN iajs-1955 140 7 → → SYM iajs-1955 140 8 𝑀 𝑀 PROPN iajs-1955 140 9 be be VERB iajs-1955 140 10 a a DET iajs-1955 140 11 module module NOUN iajs-1955 140 12 epimorphism epimorphism NOUN iajs-1955 140 13 , , PUNCT iajs-1955 140 14 and and CCONJ iajs-1955 140 15 let let VERB iajs-1955 140 16 e e PRON iajs-1955 140 17 be be AUX iajs-1955 140 18 a a DET iajs-1955 140 19 submodule submodule NOUN iajs-1955 140 20 of of ADP iajs-1955 140 21 𝑀 𝑀 PROPN iajs-1955 140 22 such such ADJ iajs-1955 140 23 that that DET iajs-1955 140 24 ker ker PROPN iajs-1955 140 25 𝑔 𝑔 PROPN iajs-1955 140 26 𝑆𝑟𝑎𝑑 𝑆𝑟𝑎𝑑 PROPN iajs-1955 140 27 𝑀 𝑀 PROPN iajs-1955 140 28 ∩ ∩ NOUN iajs-1955 140 29 𝐸.if 𝐸.if SCONJ iajs-1955 140 30 e e NOUN iajs-1955 140 31 is be AUX iajs-1955 140 32 a a DET iajs-1955 140 33 w w NOUN iajs-1955 140 34 - - PUNCT iajs-1955 140 35 closed close VERB iajs-1955 140 36 submodule submodule NOUN iajs-1955 140 37 in𝑀 in𝑀 ADV iajs-1955 140 38 then then ADV iajs-1955 140 39 𝑔 𝑔 PROPN iajs-1955 140 40 𝐸 𝐸 PROPN iajs-1955 140 41 is be AUX iajs-1955 140 42 a a DET iajs-1955 140 43 w w NOUN iajs-1955 140 44 - - PUNCT iajs-1955 140 45 closed closed ADJ iajs-1955 140 46 submodule submodule NOUN iajs-1955 140 47 in in ADP iajs-1955 140 48 𝑀 𝑀 PROPN iajs-1955 140 49 . . PUNCT iajs-1955 141 1 proof proof NOUN iajs-1955 141 2 suppose suppose VERB iajs-1955 141 3 that that SCONJ iajs-1955 141 4 e e PROPN iajs-1955 141 5 is be AUX iajs-1955 141 6 a a DET iajs-1955 141 7 w w NOUN iajs-1955 141 8 - - PUNCT iajs-1955 141 9 closed closed ADJ iajs-1955 141 10 submodule submodule NOUN iajs-1955 141 11 in in ADP iajs-1955 141 12 𝑀 𝑀 PROPN iajs-1955 141 13 , , PUNCT iajs-1955 141 14 and and CCONJ iajs-1955 141 15 let let VERB iajs-1955 141 16 𝑔 𝑔 PROPN iajs-1955 141 17 𝐸 𝐸 PROPN iajs-1955 141 18 " " PUNCT iajs-1955 141 19 is be AUX iajs-1955 141 20 a a DET iajs-1955 141 21 weak weak ADJ iajs-1955 141 22 essential essential ADJ iajs-1955 141 23 submodule submodule NOUN iajs-1955 141 24 of of ADP iajs-1955 141 25 l l PROPN iajs-1955 141 26 " " PUNCT iajs-1955 141 27 , , PUNCT iajs-1955 141 28 where where SCONJ iajs-1955 141 29 l l NOUN iajs-1955 141 30 is be AUX iajs-1955 141 31 a a DET iajs-1955 141 32 submodule submodule NOUN iajs-1955 141 33 of of ADP iajs-1955 141 34 𝑀 𝑀 PROPN iajs-1955 141 35 . . PUNCT iajs-1955 142 1 since since SCONJ iajs-1955 142 2 ker ker PROPN iajs-1955 142 3 𝑔 𝑔 PROPN iajs-1955 142 4 𝑆𝑟𝑎𝑑 𝑆𝑟𝑎𝑑 PROPN iajs-1955 142 5 𝑀 𝑀 PROPN iajs-1955 142 6 ∩ ∩ NOUN iajs-1955 142 7 𝐸. 𝐸. VERB iajs-1955 142 8 hence hence ADV iajs-1955 142 9 by by ADP iajs-1955 142 10 lemma(2.30 lemma(2.30 NOUN iajs-1955 142 11 ) ) PUNCT iajs-1955 142 12 , , PUNCT iajs-1955 142 13 we we PRON iajs-1955 142 14 get get VERB iajs-1955 142 15 𝑔 𝑔 PROPN iajs-1955 142 16 𝑔 𝑔 PROPN iajs-1955 142 17 𝐸 𝐸 PROPN iajs-1955 142 18 is be AUX iajs-1955 142 19 a a DET iajs-1955 142 20 weak weak ADJ iajs-1955 142 21 essential essential ADJ iajs-1955 142 22 submodule submodule NOUN iajs-1955 142 23 in in ADP iajs-1955 142 24 𝑔 𝑔 PROPN iajs-1955 142 25 𝐿 𝐿 PROPN iajs-1955 142 26 , , PUNCT iajs-1955 142 27 where where SCONJ iajs-1955 142 28 𝑔 𝑔 PROPN iajs-1955 142 29 𝐿 𝐿 PROPN iajs-1955 142 30 is be AUX iajs-1955 142 31 a a DET iajs-1955 142 32 submodule submodule NOUN iajs-1955 142 33 of of ADP iajs-1955 142 34 𝑀 𝑀 PROPN iajs-1955 142 35 , , PUNCT iajs-1955 142 36 but but CCONJ iajs-1955 142 37 𝐾𝑒𝑟𝑔 𝐾𝑒𝑟𝑔 PROPN iajs-1955 142 38 𝐸 𝐸 PROPN iajs-1955 142 39 , , PUNCT iajs-1955 142 40 then then ADV iajs-1955 142 41 𝑔 𝑔 PROPN iajs-1955 142 42 𝑔 𝑔 PROPN iajs-1955 142 43 𝐸 𝐸 PROPN iajs-1955 142 44 𝐸 𝐸 PROPN iajs-1955 142 45 , , PUNCT iajs-1955 142 46 i.e i.e PROPN iajs-1955 142 47 e e NOUN iajs-1955 142 48 is be AUX iajs-1955 142 49 a a DET iajs-1955 142 50 weak weak ADJ iajs-1955 142 51 essential essential ADJ iajs-1955 142 52 in in ADP iajs-1955 142 53 𝑔 𝑔 PROPN iajs-1955 142 54 𝐿 𝐿 PROPN iajs-1955 142 55 . . PUNCT iajs-1955 143 1 but but CCONJ iajs-1955 143 2 e e NOUN iajs-1955 143 3 is be AUX iajs-1955 143 4 a a DET iajs-1955 143 5 wclosed wclosed ADJ iajs-1955 143 6 submodule submodule NOUN iajs-1955 143 7 in in ADP iajs-1955 143 8 𝑀 𝑀 PROPN iajs-1955 143 9 , , PUNCT iajs-1955 143 10 then then ADV iajs-1955 143 11 e=𝑔 e=𝑔 PROPN iajs-1955 143 12 𝐿 𝐿 PROPN iajs-1955 143 13 , , PUNCT iajs-1955 143 14 and and CCONJ iajs-1955 143 15 since since SCONJ iajs-1955 143 16 𝑔 𝑔 PROPN iajs-1955 143 17 𝑖𝑠 𝑖𝑠 PROPN iajs-1955 143 18 an an DET iajs-1955 143 19 epimorphism epimorphism NOUN iajs-1955 144 1 so so ADV iajs-1955 144 2 , , PUNCT iajs-1955 144 3 𝑔 𝑔 PROPN iajs-1955 144 4 𝐸 𝐸 PROPN iajs-1955 144 5 𝐿. 𝐿. VERB iajs-1955 144 6 hence hence ADV iajs-1955 144 7 𝑔 𝑔 PROPN iajs-1955 144 8 𝐸 𝐸 PROPN iajs-1955 144 9 is be AUX iajs-1955 144 10 a a DET iajs-1955 144 11 w w NOUN iajs-1955 144 12 - - PUNCT iajs-1955 144 13 closed closed ADJ iajs-1955 144 14 submodule submodule NOUN iajs-1955 144 15 in in ADP iajs-1955 144 16 𝑀 𝑀 PROPN iajs-1955 144 17 . . PUNCT iajs-1955 145 1 as as ADP iajs-1955 145 2 a a DET iajs-1955 145 3 direct direct ADJ iajs-1955 145 4 consequence consequence NOUN iajs-1955 145 5 of of ADP iajs-1955 145 6 proposition(2.31 proposition(2.31 NOUN iajs-1955 145 7 ) ) PUNCT iajs-1955 145 8 we we PRON iajs-1955 145 9 get get VERB iajs-1955 145 10 the the DET iajs-1955 145 11 following follow VERB iajs-1955 145 12 corollary corollary NOUN iajs-1955 145 13 . . PUNCT iajs-1955 146 1 corollary(2.32 corollary(2.32 PUNCT iajs-1955 146 2 ) ) PUNCT iajs-1955 146 3 : : PUNCT iajs-1955 146 4 if if SCONJ iajs-1955 146 5 e e PROPN iajs-1955 146 6 and and CCONJ iajs-1955 146 7 d d PROPN iajs-1955 146 8 are be AUX iajs-1955 146 9 submodules submodule NOUN iajs-1955 146 10 of of ADP iajs-1955 146 11 a a DET iajs-1955 146 12 module module NOUN iajs-1955 146 13 m m NOUN iajs-1955 146 14 with with ADP iajs-1955 146 15 𝐸 𝐸 PROPN iajs-1955 146 16 𝑠𝑟𝑎𝑑 𝑠𝑟𝑎𝑑 NOUN iajs-1955 146 17 𝑀 𝑀 PROPN iajs-1955 146 18 ∩ ∩ NOUN iajs-1955 146 19 𝐷. 𝐷. VERB iajs-1955 146 20 if if SCONJ iajs-1955 146 21 d d NOUN iajs-1955 146 22 is be AUX iajs-1955 146 23 a a DET iajs-1955 146 24 w w NOUN iajs-1955 146 25 - - PUNCT iajs-1955 146 26 closed closed ADJ iajs-1955 146 27 submodule submodule NOUN iajs-1955 146 28 in in ADP iajs-1955 146 29 m m PROPN iajs-1955 146 30 , , PUNCT iajs-1955 146 31 then then ADV iajs-1955 146 32 is be AUX iajs-1955 146 33 a a DET iajs-1955 146 34 w w NOUN iajs-1955 146 35 - - PUNCT iajs-1955 146 36 closed closed ADJ iajs-1955 146 37 submodule submodule NOUN iajs-1955 146 38 in in ADP iajs-1955 146 39 . . PUNCT iajs-1955 147 1 the the DET iajs-1955 147 2 following follow VERB iajs-1955 147 3 proposition proposition NOUN iajs-1955 147 4 gives give VERB iajs-1955 147 5 a a DET iajs-1955 147 6 relation relation NOUN iajs-1955 147 7 between between ADP iajs-1955 147 8 y y PROPN iajs-1955 147 9 - - PUNCT iajs-1955 147 10 closed close VERB iajs-1955 147 11 submodule submodule NOUN iajs-1955 147 12 and and CCONJ iajs-1955 147 13 w w NOUN iajs-1955 147 14 - - PUNCT iajs-1955 147 15 closed closed ADJ iajs-1955 147 16 submodule submodule NOUN iajs-1955 147 17 in in ADP iajs-1955 147 18 the the DET iajs-1955 147 19 class class NOUN iajs-1955 147 20 of of ADP iajs-1955 147 21 a a DET iajs-1955 147 22 fully fully ADV iajs-1955 147 23 semi semi ADJ iajs-1955 147 24 - - ADJ iajs-1955 147 25 prime prime ADJ iajs-1955 147 26 module module NOUN iajs-1955 147 27 . . PUNCT iajs-1955 148 1 proposition proposition NOUN iajs-1955 148 2 ( ( PUNCT iajs-1955 148 3 2.33 2.33 NUM iajs-1955 148 4 ) ) PUNCT iajs-1955 148 5 let let VERB iajs-1955 148 6 m m PRON iajs-1955 148 7 be be AUX iajs-1955 148 8 a a DET iajs-1955 148 9 fully fully ADV iajs-1955 148 10 semi semi ADJ iajs-1955 148 11 - - ADJ iajs-1955 148 12 prime prime ADJ iajs-1955 148 13 module module NOUN iajs-1955 148 14 . . PUNCT iajs-1955 149 1 then then ADV iajs-1955 149 2 every every DET iajs-1955 149 3 non non PROPN iajs-1955 149 4 zero zero NUM iajs-1955 149 5 y y ADJ iajs-1955 149 6 - - PUNCT iajs-1955 149 7 closed close VERB iajs-1955 149 8 submodule submodule NOUN iajs-1955 149 9 is be AUX iajs-1955 149 10 a a DET iajs-1955 149 11 wclosed wclosed ADJ iajs-1955 149 12 submodule submodule NOUN iajs-1955 149 13 . . PUNCT iajs-1955 150 1 proof proof NOUN iajs-1955 150 2 let let VERB iajs-1955 150 3 e e PRON iajs-1955 150 4 be be AUX iajs-1955 150 5 a a DET iajs-1955 150 6 non non ADJ iajs-1955 150 7 zero zero NUM iajs-1955 150 8 y y ADV iajs-1955 150 9 - - PUNCT iajs-1955 150 10 closed close VERB iajs-1955 150 11 submodule submodule NOUN iajs-1955 150 12 in in ADP iajs-1955 150 13 m m PROPN iajs-1955 150 14 , , PUNCT iajs-1955 150 15 then then ADV iajs-1955 150 16 by by ADP iajs-1955 150 17 [ [ PUNCT iajs-1955 150 18 11 11 NUM iajs-1955 150 19 ] ] PUNCT iajs-1955 150 20 , , PUNCT iajs-1955 150 21 every every DET iajs-1955 150 22 y y PROPN iajs-1955 150 23 - - PUNCT iajs-1955 150 24 closed close VERB iajs-1955 150 25 submodule submodule NOUN iajs-1955 150 26 is be AUX iajs-1955 150 27 a a DET iajs-1955 150 28 closed closed ADJ iajs-1955 150 29 . . PUNCT iajs-1955 151 1 hence hence ADV iajs-1955 151 2 e e PROPN iajs-1955 151 3 is be AUX iajs-1955 151 4 a a DET iajs-1955 151 5 closed closed ADJ iajs-1955 151 6 , , PUNCT iajs-1955 151 7 then then ADV iajs-1955 151 8 by by ADP iajs-1955 151 9 proposition(2.14 proposition(2.14 NOUN iajs-1955 151 10 ) ) PUNCT iajs-1955 151 11 , , PUNCT iajs-1955 151 12 e e X iajs-1955 151 13 is be AUX iajs-1955 151 14 a a DET iajs-1955 151 15 w w NOUN iajs-1955 151 16 - - PUNCT iajs-1955 151 17 closed closed ADJ iajs-1955 151 18 submodule submodule NOUN iajs-1955 151 19 in in ADP iajs-1955 151 20 m. m. NOUN iajs-1955 151 21 " " PUNCT iajs-1955 151 22 the the DET iajs-1955 151 23 following follow VERB iajs-1955 151 24 proposition proposition NOUN iajs-1955 151 25 shows show VERB iajs-1955 151 26 that that SCONJ iajs-1955 151 27 in in ADP iajs-1955 151 28 the the DET iajs-1955 151 29 class class NOUN iajs-1955 151 30 of of ADP iajs-1955 151 31 non non ADJ iajs-1955 151 32 - - ADJ iajs-1955 151 33 singular singular ADJ iajs-1955 151 34 modules module NOUN iajs-1955 151 35 " " PUNCT iajs-1955 151 36 , , PUNCT iajs-1955 151 37 the the DET iajs-1955 151 38 class class NOUN iajs-1955 151 39 of of ADP iajs-1955 151 40 wclosed wclose VERB iajs-1955 151 41 submodules submodule NOUN iajs-1955 151 42 is be AUX iajs-1955 151 43 contained contain VERB iajs-1955 151 44 in in ADP iajs-1955 151 45 the the DET iajs-1955 151 46 class class NOUN iajs-1955 151 47 of of ADP iajs-1955 151 48 y y PROPN iajs-1955 151 49 - - PUNCT iajs-1955 151 50 closed close VERB iajs-1955 151 51 submodules submodule NOUN iajs-1955 151 52 . . PUNCT iajs-1955 152 1 proposition proposition NOUN iajs-1955 152 2 ( ( PUNCT iajs-1955 152 3 2.34 2.34 NUM iajs-1955 152 4 ) ) PUNCT iajs-1955 152 5 if if SCONJ iajs-1955 152 6 m m NOUN iajs-1955 152 7 is be AUX iajs-1955 152 8 a a DET iajs-1955 152 9 non non ADJ iajs-1955 152 10 singular singular NOUN iajs-1955 152 11 module module NOUN iajs-1955 152 12 and and CCONJ iajs-1955 152 13 e e NOUN iajs-1955 152 14 is be AUX iajs-1955 152 15 a a DET iajs-1955 152 16 w w NOUN iajs-1955 152 17 - - PUNCT iajs-1955 152 18 closed closed ADJ iajs-1955 152 19 submodule submodule NOUN iajs-1955 152 20 of of ADP iajs-1955 152 21 m m PROPN iajs-1955 152 22 , , PUNCT iajs-1955 152 23 then then ADV iajs-1955 152 24 e e PROPN iajs-1955 152 25 is be AUX iajs-1955 152 26 a a DET iajs-1955 152 27 y y ADV iajs-1955 152 28 - - PUNCT iajs-1955 152 29 closed close VERB iajs-1955 152 30 submodule submodule NOUN iajs-1955 152 31 of of ADP iajs-1955 152 32 m. m. NOUN iajs-1955 152 33             SPACE iajs-1955 152 34 172 172 NUM iajs-1955 152 35     SPACE iajs-1955 152 36 | | ADV iajs-1955 152 37     SPACE iajs-1955 152 38 mathmatics mathmatic NOUN iajs-1955 152 39 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 152 40     SPACE iajs-1955 152 41 2018 2018 NUM iajs-1955 152 42 ) ) PUNCT iajs-1955 152 43 عام عام ADP iajs-1955 152 44 2العدد 2العدد NUM iajs-1955 152 45 ( ( PUNCT iajs-1955 152 46 31لمجلد 31لمجلد NUM iajs-1955 152 47 ا ا NOUN iajs-1955 152 48 مجلة مجلة NOUN iajs-1955 152 49 إبن إبن VERB iajs-1955 152 50 الهيثم الهيثم ADJ iajs-1955 152 51 للعلوم للعلوم NOUN iajs-1955 152 52 الصرفة الصرفة NOUN iajs-1955 153 1 و و PRON iajs-1955 153 2 التطبيقية التطبيقية ADJ iajs-1955 153 3 ibn ibn PROPN iajs-1955 153 4 al al PROPN iajs-1955 153 5 - - PUNCT iajs-1955 153 6 haitham haitham PROPN iajs-1955 153 7 jour jour X iajs-1955 153 8 . . PROPN iajs-1955 153 9 for for ADP iajs-1955 153 10 pure pure ADJ iajs-1955 153 11 & & CCONJ iajs-1955 153 12 appl appl PROPN iajs-1955 153 13 . . PUNCT iajs-1955 154 1 sci sci PROPN iajs-1955 154 2 . . PUNCT iajs-1955 154 3 vol vol NOUN iajs-1955 154 4 . . PROPN iajs-1955 155 1 31 31 NUM iajs-1955 155 2 ( ( PUNCT iajs-1955 155 3 2 2 NUM iajs-1955 155 4 ) ) PUNCT iajs-1955 155 5 2018 2018 NUM iajs-1955 155 6 proof proof NOUN iajs-1955 155 7 let let VERB iajs-1955 155 8 e e PRON iajs-1955 155 9 be be AUX iajs-1955 155 10 a a DET iajs-1955 155 11 w w NOUN iajs-1955 155 12 - - PUNCT iajs-1955 155 13 closed closed ADJ iajs-1955 155 14 submodule submodule NOUN iajs-1955 155 15 in in ADP iajs-1955 155 16 m m PROPN iajs-1955 155 17 then then ADV iajs-1955 155 18 e e NOUN iajs-1955 155 19 " " PUNCT iajs-1955 155 20 is be AUX iajs-1955 155 21 a a DET iajs-1955 155 22 closed closed ADJ iajs-1955 155 23 submodule submodule NOUN iajs-1955 155 24 in in ADP iajs-1955 155 25 m m PROPN iajs-1955 155 26 " " PUNCT iajs-1955 155 27 , , PUNCT iajs-1955 155 28 but but CCONJ iajs-1955 155 29 m m NOUN iajs-1955 155 30 is be AUX iajs-1955 155 31 a a DET iajs-1955 155 32 non non ADJ iajs-1955 155 33 - - ADJ iajs-1955 155 34 singular singular ADJ iajs-1955 155 35 r r NOUN iajs-1955 155 36 - - PUNCT iajs-1955 155 37 module module NOUN iajs-1955 155 38 , , PUNCT iajs-1955 155 39 then then ADV iajs-1955 155 40 by by ADP iajs-1955 155 41 [ [ PUNCT iajs-1955 155 42 11 11 NUM iajs-1955 155 43 , , PUNCT iajs-1955 155 44 prop(2.1)(2 prop(2.1)(2 NOUN iajs-1955 155 45 ) ) PUNCT iajs-1955 155 46 ] ] PUNCT iajs-1955 156 1 e e X iajs-1955 156 2 is be AUX iajs-1955 156 3 a a DET iajs-1955 156 4 y y ADV iajs-1955 156 5 - - PUNCT iajs-1955 156 6 closed close VERB iajs-1955 156 7 submodule submodule NOUN iajs-1955 156 8 in in ADP iajs-1955 156 9 m. m. NOUN iajs-1955 156 10 the the DET iajs-1955 156 11 following follow VERB iajs-1955 156 12 proposition proposition NOUN iajs-1955 156 13 shows show VERB iajs-1955 156 14 that that SCONJ iajs-1955 156 15 in in ADP iajs-1955 156 16 the the DET iajs-1955 156 17 class class NOUN iajs-1955 156 18 of of ADP iajs-1955 156 19 non non ADJ iajs-1955 156 20 - - ADJ iajs-1955 156 21 singular singular ADJ iajs-1955 156 22 and and CCONJ iajs-1955 156 23 fully fully ADV iajs-1955 156 24 semi semi ADJ iajs-1955 156 25 - - ADJ iajs-1955 156 26 prime prime ADJ iajs-1955 156 27 rmodule rmodule NOUN iajs-1955 156 28 , , PUNCT iajs-1955 156 29 w w NOUN iajs-1955 156 30 - - PUNCT iajs-1955 156 31 closed closed ADJ iajs-1955 156 32 submodule submodule NOUN iajs-1955 156 33 , , PUNCT iajs-1955 156 34 y y NOUN iajs-1955 156 35 - - PUNCT iajs-1955 156 36 closed close VERB iajs-1955 156 37 submodule submodule NOUN iajs-1955 156 38 and and CCONJ iajs-1955 156 39 closed closed ADJ iajs-1955 156 40 submodule submodule NOUN iajs-1955 156 41 are be AUX iajs-1955 156 42 equivalent equivalent ADJ iajs-1955 156 43 : : PUNCT iajs-1955 156 44 proposition proposition NOUN iajs-1955 156 45 ( ( PUNCT iajs-1955 156 46 2.35 2.35 NUM iajs-1955 156 47 ) ) PUNCT iajs-1955 156 48 let let VERB iajs-1955 156 49 m m PRON iajs-1955 156 50 be be AUX iajs-1955 156 51 a a DET iajs-1955 156 52 fully fully ADV iajs-1955 156 53 semi semi ADJ iajs-1955 156 54 - - ADJ iajs-1955 156 55 prime prime ADJ iajs-1955 156 56 and and CCONJ iajs-1955 156 57 non non ADJ iajs-1955 156 58 - - ADJ iajs-1955 156 59 singular singular ADJ iajs-1955 156 60 module module NOUN iajs-1955 156 61 , , PUNCT iajs-1955 156 62 " " PUNCT iajs-1955 156 63 and and CCONJ iajs-1955 156 64 e e PRON iajs-1955 156 65 be be AUX iajs-1955 156 66 a a DET iajs-1955 156 67 non non ADJ iajs-1955 156 68 zero zero NUM iajs-1955 156 69 submodule submodule NOUN iajs-1955 156 70 of of ADP iajs-1955 156 71 m. m. NOUN iajs-1955 156 72 then then ADV iajs-1955 156 73 the the DET iajs-1955 156 74 following follow VERB iajs-1955 156 75 statements statement NOUN iajs-1955 156 76 are be AUX iajs-1955 156 77 equivalent equivalent ADJ iajs-1955 156 78 " " PUNCT iajs-1955 156 79 : : PUNCT iajs-1955 156 80 1e 1e PROPN iajs-1955 156 81 is be AUX iajs-1955 156 82 a a DET iajs-1955 156 83 y y NOUN iajs-1955 156 84 - - PUNCT iajs-1955 156 85 closed close VERB iajs-1955 156 86 submodule submodule NOUN iajs-1955 156 87 . . PUNCT iajs-1955 157 1 2e 2e PROPN iajs-1955 157 2 is be AUX iajs-1955 157 3 a a DET iajs-1955 157 4 closed closed ADJ iajs-1955 157 5 submodule submodule NOUN iajs-1955 157 6 . . PUNCT iajs-1955 158 1 3e 3e X iajs-1955 158 2 is be AUX iajs-1955 158 3 a a DET iajs-1955 158 4 w w NOUN iajs-1955 158 5 - - PUNCT iajs-1955 158 6 closed closed ADJ iajs-1955 158 7 submodule submodule NOUN iajs-1955 158 8 . . PUNCT iajs-1955 159 1 proof proof NOUN iajs-1955 159 2 𝟏 𝟏 NUM iajs-1955 159 3 ⟹ ⟹ PROPN iajs-1955 159 4 𝟐 𝟐 NUM iajs-1955 159 5 follows follow VERB iajs-1955 159 6 by by ADP iajs-1955 159 7 [ [ X iajs-1955 159 8 11 11 NUM iajs-1955 159 9 ] ] PUNCT iajs-1955 159 10 . . PUNCT iajs-1955 160 1 𝟐 𝟐 NUM iajs-1955 160 2 ⟹ ⟹ NUM iajs-1955 160 3 𝟑 𝟑 NUM iajs-1955 160 4 follows follow VERB iajs-1955 160 5 by by ADP iajs-1955 160 6 proposition(2.14 proposition(2.14 NOUN iajs-1955 160 7 ) ) PUNCT iajs-1955 160 8 . . PUNCT iajs-1955 161 1 𝟑 𝟑 NUM iajs-1955 161 2 ⟹ ⟹ NUM iajs-1955 161 3 𝟏 𝟏 PROPN iajs-1955 161 4 follows follow VERB iajs-1955 161 5 by by ADP iajs-1955 161 6 proposition(2.34 proposition(2.34 NOUN iajs-1955 161 7 ) ) PUNCT iajs-1955 161 8 . . PUNCT iajs-1955 162 1 3 3 X iajs-1955 162 2 . . X iajs-1955 162 3 w w X iajs-1955 162 4 - - PUNCT iajs-1955 162 5 closed close VERB iajs-1955 162 6 submodule submodule NOUN iajs-1955 162 7 in in ADP iajs-1955 162 8 multiplication multiplication NOUN iajs-1955 162 9 modules module NOUN iajs-1955 162 10 in in ADP iajs-1955 162 11 this this DET iajs-1955 162 12 section section NOUN iajs-1955 162 13 , , PUNCT iajs-1955 162 14 we we PRON iajs-1955 162 15 establishe establishe VERB iajs-1955 162 16 some some DET iajs-1955 162 17 relationships relationship NOUN iajs-1955 162 18 between between ADP iajs-1955 162 19 w w NOUN iajs-1955 162 20 - - PUNCT iajs-1955 162 21 closed closed ADJ iajs-1955 162 22 submodule submodule NOUN iajs-1955 162 23 and and CCONJ iajs-1955 162 24 multiplication multiplication NOUN iajs-1955 162 25 modules module NOUN iajs-1955 162 26 . . PUNCT iajs-1955 163 1 " " PUNCT iajs-1955 163 2 first first ADV iajs-1955 163 3 we we PRON iajs-1955 163 4 introduce introduce VERB iajs-1955 163 5 the the DET iajs-1955 163 6 following follow VERB iajs-1955 163 7 definition definition NOUN iajs-1955 163 8 " " PUNCT iajs-1955 163 9 . . PUNCT iajs-1955 164 1 definition(3.1 definition(3.1 ADJ iajs-1955 164 2 ) ) PUNCT iajs-1955 164 3 a a DET iajs-1955 164 4 non non ADJ iajs-1955 164 5 - - ADJ iajs-1955 164 6 zero zero NUM iajs-1955 164 7 semi semi ADJ iajs-1955 164 8 - - ADJ iajs-1955 164 9 prime prime ADJ iajs-1955 164 10 submodule submodule NOUN iajs-1955 164 11 e e PROPN iajs-1955 164 12 of of ADP iajs-1955 164 13 a a DET iajs-1955 164 14 module module NOUN iajs-1955 164 15 m m VERB iajs-1955 164 16 is be AUX iajs-1955 164 17 called call VERB iajs-1955 164 18 minimal minimal ADJ iajs-1955 164 19 semi semi ADJ iajs-1955 164 20 - - ADJ iajs-1955 164 21 prime prime ADJ iajs-1955 164 22 submodule submodule NOUN iajs-1955 164 23 of of ADP iajs-1955 164 24 m m PROPN iajs-1955 164 25 , , PUNCT iajs-1955 164 26 if if SCONJ iajs-1955 164 27 whenever whenever SCONJ iajs-1955 164 28 s s VERB iajs-1955 164 29 " " PUNCT iajs-1955 164 30 is be AUX iajs-1955 164 31 a a DET iajs-1955 164 32 non non ADJ iajs-1955 164 33 zero zero NUM iajs-1955 164 34 semi semi ADJ iajs-1955 164 35 - - ADJ iajs-1955 164 36 prime prime ADJ iajs-1955 164 37 submodule submodule NOUN iajs-1955 164 38 of of ADP iajs-1955 164 39 m m PROPN iajs-1955 164 40 such such ADJ iajs-1955 164 41 that that SCONJ iajs-1955 164 42 " " PUNCT iajs-1955 164 43 𝑆 𝑆 PROPN iajs-1955 164 44 𝐸 𝐸 PROPN iajs-1955 164 45 , , PUNCT iajs-1955 164 46 then then ADV iajs-1955 164 47 s s PROPN iajs-1955 164 48 = = PROPN iajs-1955 164 49 e. e. PROPN iajs-1955 164 50 that that PRON iajs-1955 164 51 is be AUX iajs-1955 164 52 by by ADP iajs-1955 164 53 minimal minimal ADJ iajs-1955 164 54 semi semi ADJ iajs-1955 164 55 - - ADJ iajs-1955 164 56 prime prime ADJ iajs-1955 164 57 submodule submodule NOUN iajs-1955 164 58 e e PROPN iajs-1955 164 59 of of ADP iajs-1955 164 60 m m VERB iajs-1955 164 61 we we PRON iajs-1955 164 62 mean mean VERB iajs-1955 164 63 a a DET iajs-1955 164 64 semi semi ADJ iajs-1955 164 65 - - ADJ iajs-1955 164 66 prime prime ADJ iajs-1955 164 67 submodule submodule NOUN iajs-1955 164 68 which which PRON iajs-1955 164 69 is be AUX iajs-1955 164 70 a a DET iajs-1955 164 71 minimal minimal ADJ iajs-1955 164 72 in in ADP iajs-1955 164 73 the the DET iajs-1955 164 74 collection collection NOUN iajs-1955 164 75 of of ADP iajs-1955 164 76 semi semi ADJ iajs-1955 164 77 - - ADJ iajs-1955 164 78 prime prime ADJ iajs-1955 164 79 submodules submodule NOUN iajs-1955 164 80 of of ADP iajs-1955 164 81 m. m. NOUN iajs-1955 164 82 if if SCONJ iajs-1955 164 83 a a PRON iajs-1955 164 84 is be AUX iajs-1955 164 85 a a DET iajs-1955 164 86 proper proper ADJ iajs-1955 164 87 ideal ideal NOUN iajs-1955 164 88 of of ADP iajs-1955 164 89 r r NOUN iajs-1955 164 90 , , PUNCT iajs-1955 164 91 then then ADV iajs-1955 164 92 a a DET iajs-1955 164 93 semi semi ADJ iajs-1955 164 94 - - ADJ iajs-1955 164 95 prime prime ADJ iajs-1955 164 96 ideal ideal NOUN iajs-1955 164 97 b b PROPN iajs-1955 164 98 is be AUX iajs-1955 164 99 called call VERB iajs-1955 164 100 a a DET iajs-1955 164 101 minimal minimal ADJ iajs-1955 164 102 semi semi ADJ iajs-1955 164 103 - - ADJ iajs-1955 164 104 prime prime ADJ iajs-1955 164 105 ideal ideal NOUN iajs-1955 164 106 of of ADP iajs-1955 164 107 a a DET iajs-1955 164 108 provided provide VERB iajs-1955 164 109 that that SCONJ iajs-1955 164 110 𝐴 𝐴 PROPN iajs-1955 164 111 𝐵 𝐵 PROPN iajs-1955 164 112 and and CCONJ iajs-1955 164 113 is be AUX iajs-1955 164 114 minimal minimal ADJ iajs-1955 164 115 semi semi ADJ iajs-1955 164 116 - - ADJ iajs-1955 164 117 prime prime ADJ iajs-1955 164 118 ideal ideal NOUN iajs-1955 164 119 of of ADP iajs-1955 164 120 a a DET iajs-1955 164 121 ring ring NOUN iajs-1955 164 122 . . PUNCT iajs-1955 165 1 remark(3.2 remark(3.2 NOUN iajs-1955 165 2 ) ) PUNCT iajs-1955 166 1 in in ADP iajs-1955 166 2 multiplication multiplication NOUN iajs-1955 166 3 module module NOUN iajs-1955 166 4 since since SCONJ iajs-1955 166 5 𝑎𝑛𝑛 𝑎𝑛𝑛 ADP iajs-1955 166 6 𝑀 𝑀 PROPN iajs-1955 166 7 𝑅 𝑅 PROPN iajs-1955 166 8 it it PRON iajs-1955 166 9 follows follow VERB iajs-1955 166 10 that that SCONJ iajs-1955 166 11 by by ADP iajs-1955 166 12 [ [ X iajs-1955 166 13 12 12 NUM iajs-1955 166 14 , , PUNCT iajs-1955 166 15 th(2.5 th(2.5 NOUN iajs-1955 166 16 ) ) PUNCT iajs-1955 166 17 ] ] PUNCT iajs-1955 166 18 , , PUNCT iajs-1955 166 19 there there PRON iajs-1955 166 20 exists exist VERB iajs-1955 166 21 a a DET iajs-1955 166 22 minimal minimal ADJ iajs-1955 166 23 ideal ideal NOUN iajs-1955 166 24 p p NOUN iajs-1955 166 25 of of ADP iajs-1955 166 26 r r NOUN iajs-1955 166 27 such such ADJ iajs-1955 166 28 that that PRON iajs-1955 166 29 𝑎𝑛𝑛 𝑎𝑛𝑛 NOUN iajs-1955 166 30 𝑀 𝑀 PROPN iajs-1955 166 31 𝑃 𝑃 PROPN iajs-1955 166 32 , , PUNCT iajs-1955 166 33 𝑎𝑛𝑑 𝑎𝑛𝑑 PROPN iajs-1955 166 34 𝑀 𝑀 PROPN iajs-1955 166 35 𝑃𝑀. 𝑃𝑀. NOUN iajs-1955 167 1 but but CCONJ iajs-1955 167 2 by by ADP iajs-1955 167 3 [ [ PUNCT iajs-1955 167 4 13 13 NUM iajs-1955 167 5 , , PUNCT iajs-1955 167 6 prop(2.5 prop(2.5 ADJ iajs-1955 167 7 ) ) PUNCT iajs-1955 167 8 , , PUNCT iajs-1955 167 9 p.36 p.36 PROPN iajs-1955 167 10 ] ] X iajs-1955 167 11 pm pm NOUN iajs-1955 167 12 is be AUX iajs-1955 167 13 a a DET iajs-1955 167 14 semi semi ADJ iajs-1955 167 15 - - ADJ iajs-1955 167 16 prime prime ADJ iajs-1955 167 17 submodule submodule NOUN iajs-1955 167 18 of of ADP iajs-1955 167 19 m. m. NOUN iajs-1955 167 20 then then ADV iajs-1955 167 21 from from ADP iajs-1955 167 22 definition(3.1 definition(3.1 ADJ iajs-1955 167 23 ) ) PUNCT iajs-1955 167 24 we we PRON iajs-1955 167 25 get get VERB iajs-1955 167 26 the the DET iajs-1955 167 27 following follow VERB iajs-1955 167 28 facts fact NOUN iajs-1955 167 29 : : PUNCT iajs-1955 167 30 ( ( PUNCT iajs-1955 167 31 a a X iajs-1955 167 32 ) ) PUNCT iajs-1955 167 33 e e NOUN iajs-1955 167 34 is be AUX iajs-1955 167 35 a a DET iajs-1955 167 36 minimal minimal ADJ iajs-1955 167 37 semi semi ADJ iajs-1955 167 38 - - ADJ iajs-1955 167 39 prime prime ADJ iajs-1955 167 40 submodule submodule NOUN iajs-1955 167 41 of of ADP iajs-1955 167 42 m m PROPN iajs-1955 167 43 if if SCONJ iajs-1955 167 44 and and CCONJ iajs-1955 167 45 only only ADV iajs-1955 167 46 if if SCONJ iajs-1955 167 47 there there PRON iajs-1955 167 48 exists exist VERB iajs-1955 167 49 a a DET iajs-1955 167 50 minimal minimal ADJ iajs-1955 167 51 semiprime semiprime NOUN iajs-1955 167 52 ideal ideal NOUN iajs-1955 167 53 a a X iajs-1955 167 54 , , PUNCT iajs-1955 167 55 with with ADP iajs-1955 167 56 𝑎𝑛𝑛 𝑎𝑛𝑛 NOUN iajs-1955 167 57 𝑀 𝑀 PROPN iajs-1955 167 58 𝐴 𝐴 PROPN iajs-1955 167 59 such such ADJ iajs-1955 167 60 that that SCONJ iajs-1955 167 61 𝐸 𝐸 PROPN iajs-1955 167 62 𝐴𝑀 𝐴𝑀 PROPN iajs-1955 167 63 𝑀. 𝑀. PROPN iajs-1955 167 64 ( ( PUNCT iajs-1955 167 65 b b NOUN iajs-1955 167 66 ) ) PUNCT iajs-1955 167 67 eveery eveery NOUN iajs-1955 167 68 semi semi ADJ iajs-1955 167 69 - - ADJ iajs-1955 167 70 prime prime ADJ iajs-1955 167 71 submodule submodule NOUN iajs-1955 167 72 of of ADP iajs-1955 167 73 m m PROPN iajs-1955 167 74 contains contain VERB iajs-1955 167 75 a a DET iajs-1955 167 76 minimal minimal ADJ iajs-1955 167 77 semi semi ADJ iajs-1955 167 78 - - ADJ iajs-1955 167 79 prime prime ADJ iajs-1955 167 80 submodule submodule NOUN iajs-1955 167 81 . . PUNCT iajs-1955 167 82             SPACE iajs-1955 168 1 173 173 NUM iajs-1955 168 2     SPACE iajs-1955 168 3 | | ADV iajs-1955 168 4     SPACE iajs-1955 168 5 mathmatics mathmatic NOUN iajs-1955 168 6 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 168 7     SPACE iajs-1955 168 8 2018 2018 NUM iajs-1955 168 9 ) ) PUNCT iajs-1955 169 1 عام عام ADP iajs-1955 169 2 2العدد 2العدد NUM iajs-1955 169 3 ( ( PUNCT iajs-1955 169 4 31لمجلد 31لمجلد NUM iajs-1955 169 5 ا ا NOUN iajs-1955 169 6 مجلة مجلة NOUN iajs-1955 169 7 إبن إبن VERB iajs-1955 169 8 الهيثم الهيثم ADJ iajs-1955 169 9 للعلوم للعلوم NOUN iajs-1955 169 10 الصرفة الصرفة NOUN iajs-1955 170 1 و و PRON iajs-1955 170 2 التطبيقية التطبيقية ADJ iajs-1955 170 3 ibn ibn PROPN iajs-1955 170 4 al al PROPN iajs-1955 170 5 - - PUNCT iajs-1955 170 6 haitham haitham PROPN iajs-1955 170 7 jour jour X iajs-1955 170 8 . . PROPN iajs-1955 170 9 for for ADP iajs-1955 170 10 pure pure ADJ iajs-1955 170 11 & & CCONJ iajs-1955 170 12 appl appl PROPN iajs-1955 170 13 . . PUNCT iajs-1955 171 1 sci sci PROPN iajs-1955 171 2 . . PUNCT iajs-1955 171 3 vol vol NOUN iajs-1955 171 4 . . PROPN iajs-1955 172 1 31 31 NUM iajs-1955 172 2 ( ( PUNCT iajs-1955 172 3 2 2 NUM iajs-1955 172 4 ) ) SYM iajs-1955 172 5 2018 2018 NUM iajs-1955 172 6 lemma(3.3 lemma(3.3 NOUN iajs-1955 172 7 ) ) PUNCT iajs-1955 172 8 if if SCONJ iajs-1955 172 9 m m NOUN iajs-1955 172 10 is be AUX iajs-1955 172 11 a a DET iajs-1955 172 12 faithful faithful ADJ iajs-1955 172 13 and and CCONJ iajs-1955 172 14 multiplication multiplication NOUN iajs-1955 172 15 module module NOUN iajs-1955 172 16 , , PUNCT iajs-1955 172 17 and and CCONJ iajs-1955 172 18 e e NOUN iajs-1955 172 19 be be AUX iajs-1955 172 20 a a DET iajs-1955 172 21 non non ADJ iajs-1955 172 22 zero zero NUM iajs-1955 172 23 semi semi ADJ iajs-1955 172 24 - - ADJ iajs-1955 172 25 prime prime ADJ iajs-1955 172 26 submodule submodule NOUN iajs-1955 172 27 of of ADP iajs-1955 172 28 m. m. NOUN iajs-1955 172 29 if if SCONJ iajs-1955 172 30 e e PRON iajs-1955 172 31 is be AUX iajs-1955 172 32 not not PART iajs-1955 172 33 minimal minimal ADJ iajs-1955 172 34 semi semi ADJ iajs-1955 172 35 - - ADJ iajs-1955 172 36 prime prime ADJ iajs-1955 172 37 , , PUNCT iajs-1955 172 38 then then ADV iajs-1955 172 39 e e NOUN iajs-1955 172 40 " " PUNCT iajs-1955 172 41 is be AUX iajs-1955 172 42 a a DET iajs-1955 172 43 weak weak ADJ iajs-1955 172 44 - - PUNCT iajs-1955 172 45 essential essential ADJ iajs-1955 172 46 submodule submodule NOUN iajs-1955 172 47 of of ADP iajs-1955 172 48 m m NOUN iajs-1955 172 49 " " PUNCT iajs-1955 172 50 . . PUNCT iajs-1955 173 1 proof proof NOUN iajs-1955 173 2 since since SCONJ iajs-1955 173 3 m m PROPN iajs-1955 173 4 is be AUX iajs-1955 173 5 a a DET iajs-1955 173 6 multiplication multiplication NOUN iajs-1955 173 7 , , PUNCT iajs-1955 173 8 and and CCONJ iajs-1955 173 9 e e NOUN iajs-1955 173 10 is be AUX iajs-1955 173 11 a a DET iajs-1955 173 12 semi semi ADJ iajs-1955 173 13 - - ADJ iajs-1955 173 14 prime prime ADJ iajs-1955 173 15 submodule submodule NOUN iajs-1955 173 16 of of ADP iajs-1955 173 17 m m PROPN iajs-1955 173 18 , , PUNCT iajs-1955 173 19 then then ADV iajs-1955 173 20 by by ADP iajs-1955 173 21 [ [ X iajs-1955 173 22 13,prop(2.5 13,prop(2.5 NUM iajs-1955 173 23 ) ) PUNCT iajs-1955 173 24 , , PUNCT iajs-1955 173 25 p.36 p.36 PROPN iajs-1955 173 26 ] ] X iajs-1955 173 27 ∃ ∃ PROPN iajs-1955 173 28 a a DET iajs-1955 173 29 " " PUNCT iajs-1955 173 30 semi semi ADJ iajs-1955 173 31 - - ADJ iajs-1955 173 32 prime prime ADJ iajs-1955 173 33 ideal ideal NOUN iajs-1955 173 34 k k PROPN iajs-1955 173 35 of of ADP iajs-1955 173 36 r r NOUN iajs-1955 173 37 " " PUNCT iajs-1955 173 38 with with ADP iajs-1955 173 39 0 0 NUM iajs-1955 173 40 𝑎𝑛𝑛𝑀 𝑎𝑛𝑛𝑀 NOUN iajs-1955 173 41 𝐾 𝐾 PROPN iajs-1955 173 42 and and CCONJ iajs-1955 173 43 e e NOUN iajs-1955 173 44 = = NOUN iajs-1955 173 45 km km NOUN iajs-1955 173 46 . . PUNCT iajs-1955 174 1 " " PUNCT iajs-1955 174 2 let let VERB iajs-1955 174 3 s s PRON iajs-1955 174 4 be be AUX iajs-1955 174 5 a a DET iajs-1955 174 6 non non ADJ iajs-1955 174 7 - - ADJ iajs-1955 174 8 zero zero NUM iajs-1955 174 9 semi semi ADJ iajs-1955 174 10 - - ADJ iajs-1955 174 11 prime prime ADJ iajs-1955 174 12 submodule submodule NOUN iajs-1955 174 13 of of ADP iajs-1955 174 14 m m PROPN iajs-1955 174 15 " " PUNCT iajs-1955 174 16 such such ADJ iajs-1955 174 17 that that SCONJ iajs-1955 174 18 𝐸 𝐸 PROPN iajs-1955 174 19 ∩ ∩ NOUN iajs-1955 174 20 𝑆 𝑆 PROPN iajs-1955 174 21 0 0 NUM iajs-1955 174 22 .but .but PUNCT iajs-1955 175 1 e e NOUN iajs-1955 175 2 is be AUX iajs-1955 175 3 not not PART iajs-1955 175 4 minimal minimal ADJ iajs-1955 175 5 semi semi ADJ iajs-1955 175 6 - - ADJ iajs-1955 175 7 prime prime ADJ iajs-1955 175 8 , , PUNCT iajs-1955 175 9 then then ADV iajs-1955 175 10 by by ADP iajs-1955 175 11 remark(3.2)(b remark(3.2)(b PROPN iajs-1955 175 12 ) ) PUNCT iajs-1955 175 13 every every DET iajs-1955 175 14 semi semi ADJ iajs-1955 175 15 - - ADJ iajs-1955 175 16 prime prime ADJ iajs-1955 175 17 submodule submodule NOUN iajs-1955 175 18 of of ADP iajs-1955 175 19 m m PROPN iajs-1955 175 20 contain contain VERB iajs-1955 175 21 a a DET iajs-1955 175 22 minimal minimal ADJ iajs-1955 175 23 semi semi ADJ iajs-1955 175 24 - - ADJ iajs-1955 175 25 prime prime ADJ iajs-1955 175 26 submodule submodule NOUN iajs-1955 175 27 say say VERB iajs-1955 175 28 𝐸 𝐸 PROPN iajs-1955 175 29 𝐸. 𝐸. PROPN iajs-1955 175 30 hence hence ADV iajs-1955 175 31 by by ADP iajs-1955 175 32 remark(3.2)(a remark(3.2)(a PROPN iajs-1955 175 33 ) ) PUNCT iajs-1955 175 34 , , PUNCT iajs-1955 175 35 there there PRON iajs-1955 175 36 exists exist VERB iajs-1955 175 37 a a DET iajs-1955 175 38 minimal minimal ADJ iajs-1955 175 39 semi semi ADJ iajs-1955 175 40 - - ADJ iajs-1955 175 41 prime prime ADJ iajs-1955 175 42 ideal ideal NOUN iajs-1955 175 43 𝐾 𝐾 PROPN iajs-1955 175 44 of of ADP iajs-1955 175 45 r r NOUN iajs-1955 175 46 such such ADJ iajs-1955 176 1 that that PRON iajs-1955 176 2 𝑎𝑛𝑛 𝑎𝑛𝑛 NOUN iajs-1955 176 3 𝑀 𝑀 PROPN iajs-1955 176 4 𝐾 𝐾 PROPN iajs-1955 176 5 and and CCONJ iajs-1955 176 6 𝐸 𝐸 PROPN iajs-1955 176 7 𝐾 𝐾 PROPN iajs-1955 176 8 𝑀 𝑀 PROPN iajs-1955 176 9 𝑀 𝑀 PROPN iajs-1955 176 10 , , PUNCT iajs-1955 176 11 𝐾 𝐾 PROPN iajs-1955 176 12 ∩ ∩ ADJ iajs-1955 176 13 𝑆 𝑆 PROPN iajs-1955 176 14 : : PUNCT iajs-1955 176 15 𝑀 𝑀 PROPN iajs-1955 176 16 𝑀 𝑀 PROPN iajs-1955 176 17 𝐾𝑀 𝐾𝑀 PROPN iajs-1955 176 18 ∩ ∩ ADJ iajs-1955 176 19 𝑆 𝑆 PROPN iajs-1955 176 20 : : PUNCT iajs-1955 176 21 𝑀 𝑀 PROPN iajs-1955 176 22 𝑀 𝑀 PROPN iajs-1955 176 23 𝐸 𝐸 PROPN iajs-1955 176 24 ∩ ∩ NOUN iajs-1955 176 25 𝑆 𝑆 PROPN iajs-1955 176 26 0 0 NUM iajs-1955 176 27 0 0 NUM iajs-1955 176 28 . . PUNCT iajs-1955 177 1 but but CCONJ iajs-1955 177 2 m m PROPN iajs-1955 177 3 is be AUX iajs-1955 177 4 faithful faithful ADJ iajs-1955 177 5 , , PUNCT iajs-1955 177 6 then then ADV iajs-1955 177 7 𝐾 𝐾 PROPN iajs-1955 177 8 ∩ ∩ ADJ iajs-1955 177 9 𝑆 𝑆 PROPN iajs-1955 177 10 : : PUNCT iajs-1955 177 11 𝑀 𝑀 PROPN iajs-1955 177 12 0 0 NUM iajs-1955 177 13 , , PUNCT iajs-1955 177 14 which which PRON iajs-1955 177 15 implies imply VERB iajs-1955 177 16 that that SCONJ iajs-1955 177 17 𝐾 𝐾 PROPN iajs-1955 177 18 ∩ ∩ ADJ iajs-1955 177 19 𝑆 𝑆 PROPN iajs-1955 177 20 : : PUNCT iajs-1955 177 21 𝑀 𝑀 PROPN iajs-1955 177 22 𝐾 𝐾 PROPN iajs-1955 177 23 , , PUNCT iajs-1955 177 24 that that PRON iajs-1955 177 25 is be AUX iajs-1955 177 26 either either CCONJ iajs-1955 177 27 𝐾 𝐾 PROPN iajs-1955 177 28 𝐾 𝐾 PROPN iajs-1955 177 29 𝑜𝑟 𝑜𝑟 ADP iajs-1955 177 30 𝑆 𝑆 PROPN iajs-1955 177 31 : : PUNCT iajs-1955 177 32 𝑀 𝑀 PROPN iajs-1955 177 33 𝐾 𝐾 PROPN iajs-1955 177 34 . . PUNCT iajs-1955 178 1 if if SCONJ iajs-1955 178 2 𝐾 𝐾 PROPN iajs-1955 178 3 𝐾 𝐾 PROPN iajs-1955 178 4 , , PUNCT iajs-1955 178 5 then then ADV iajs-1955 178 6 𝐾𝑀 𝐾𝑀 PROPN iajs-1955 178 7 𝐾 𝐾 PROPN iajs-1955 178 8 𝑀 𝑀 PROPN iajs-1955 178 9 , , PUNCT iajs-1955 178 10 implies imply VERB iajs-1955 178 11 that that SCONJ iajs-1955 178 12 𝐸 𝐸 PROPN iajs-1955 178 13 𝐸 𝐸 PROPN iajs-1955 178 14 which which PRON iajs-1955 178 15 is be AUX iajs-1955 178 16 a a DET iajs-1955 178 17 contradiction contradiction NOUN iajs-1955 178 18 . . PUNCT iajs-1955 179 1 thus thus ADV iajs-1955 179 2 , , PUNCT iajs-1955 179 3 𝑆 𝑆 PROPN iajs-1955 179 4 : : PUNCT iajs-1955 179 5 𝑀 𝑀 PROPN iajs-1955 179 6 𝐾 𝐾 PROPN iajs-1955 179 7 . . PUNCT iajs-1955 180 1 that that PRON iajs-1955 180 2 is be AUX iajs-1955 180 3 𝑆 𝑆 PROPN iajs-1955 180 4 : : PUNCT iajs-1955 180 5 𝑀 𝑀 PROPN iajs-1955 180 6 𝑀 𝑀 PROPN iajs-1955 180 7 𝐾 𝐾 PROPN iajs-1955 180 8 𝑀 𝑀 PROPN iajs-1955 180 9 , , PUNCT iajs-1955 180 10 implies imply VERB iajs-1955 180 11 that that SCONJ iajs-1955 180 12 𝑆 𝑆 PROPN iajs-1955 180 13 𝐸 𝐸 PROPN iajs-1955 180 14 𝐸 𝐸 PROPN iajs-1955 180 15 which which PRON iajs-1955 180 16 is be AUX iajs-1955 180 17 contradict contradict VERB iajs-1955 180 18 the the DET iajs-1955 180 19 minimality minimality NOUN iajs-1955 180 20 of of ADP iajs-1955 180 21 𝐸 𝐸 PROPN iajs-1955 180 22 . . PUNCT iajs-1955 181 1 thus thus ADV iajs-1955 181 2 𝐸 𝐸 ADJ iajs-1955 181 3 ∩ ∩ NOUN iajs-1955 181 4 𝑆 𝑆 PROPN iajs-1955 181 5 0 0 NUM iajs-1955 181 6 is be AUX iajs-1955 181 7 not not PART iajs-1955 181 8 true true ADJ iajs-1955 181 9 . . PUNCT iajs-1955 182 1 thus thus ADV iajs-1955 182 2 𝐸 𝐸 ADJ iajs-1955 182 3 ∩ ∩ ADJ iajs-1955 182 4 𝑆 𝑆 PROPN iajs-1955 182 5 0 0 NUM iajs-1955 182 6 , , PUNCT iajs-1955 182 7 which which PRON iajs-1955 182 8 implies imply VERB iajs-1955 182 9 that that SCONJ iajs-1955 182 10 e e NOUN iajs-1955 182 11 is be AUX iajs-1955 182 12 a a DET iajs-1955 182 13 weak weak ADJ iajs-1955 182 14 essential essential ADJ iajs-1955 182 15 submodule submodule NOUN iajs-1955 182 16 of of ADP iajs-1955 182 17 m. m. NOUN iajs-1955 182 18 proposition proposition NOUN iajs-1955 182 19 ( ( PUNCT iajs-1955 182 20 3.4 3.4 NUM iajs-1955 182 21 ) ) PUNCT iajs-1955 182 22 if if SCONJ iajs-1955 182 23 m m NOUN iajs-1955 182 24 is be AUX iajs-1955 182 25 a a DET iajs-1955 182 26 faithful faithful ADJ iajs-1955 182 27 and and CCONJ iajs-1955 182 28 multiplication multiplication NOUN iajs-1955 182 29 module module NOUN iajs-1955 182 30 , , PUNCT iajs-1955 182 31 and and CCONJ iajs-1955 182 32 e e NOUN iajs-1955 182 33 be be AUX iajs-1955 182 34 a a DET iajs-1955 182 35 non non ADJ iajs-1955 182 36 - - ADJ iajs-1955 182 37 zero zero NUM iajs-1955 182 38 semi semi ADJ iajs-1955 182 39 - - ADJ iajs-1955 182 40 prime prime ADJ iajs-1955 182 41 submodule submodule NOUN iajs-1955 182 42 and and CCONJ iajs-1955 182 43 w w NOUN iajs-1955 182 44 - - PUNCT iajs-1955 182 45 closed closed ADJ iajs-1955 182 46 submodule submodule NOUN iajs-1955 182 47 of of ADP iajs-1955 182 48 m m PROPN iajs-1955 182 49 , , PUNCT iajs-1955 182 50 then then ADV iajs-1955 182 51 e e PROPN iajs-1955 182 52 is be AUX iajs-1955 182 53 a a DET iajs-1955 182 54 minimal minimal ADJ iajs-1955 182 55 semi semi ADJ iajs-1955 182 56 - - ADJ iajs-1955 182 57 prime prime ADJ iajs-1955 182 58 submodule submodule NOUN iajs-1955 182 59 of of ADP iajs-1955 182 60 m. m. NOUN iajs-1955 182 61 proof proof NOUN iajs-1955 182 62 suppose suppose VERB iajs-1955 182 63 that that SCONJ iajs-1955 182 64 e e PROPN iajs-1955 182 65 is be AUX iajs-1955 182 66 not not PART iajs-1955 182 67 minimal minimal ADJ iajs-1955 182 68 semi semi ADJ iajs-1955 182 69 - - ADJ iajs-1955 182 70 prime prime ADJ iajs-1955 182 71 submodule submodule NOUN iajs-1955 182 72 of of ADP iajs-1955 182 73 m m PROPN iajs-1955 182 74 , , PUNCT iajs-1955 182 75 then then ADV iajs-1955 182 76 by by ADP iajs-1955 182 77 lemma(3.3 lemma(3.3 NOUN iajs-1955 182 78 ) ) PUNCT iajs-1955 182 79 , , PUNCT iajs-1955 182 80 e e X iajs-1955 182 81 " " PUNCT iajs-1955 182 82 is be AUX iajs-1955 182 83 a a DET iajs-1955 182 84 weak weak ADJ iajs-1955 182 85 essential essential ADJ iajs-1955 182 86 submodule submodule NOUN iajs-1955 182 87 of of ADP iajs-1955 182 88 m m NOUN iajs-1955 182 89 " " PUNCT iajs-1955 182 90 . . PUNCT iajs-1955 183 1 but but CCONJ iajs-1955 183 2 e e NOUN iajs-1955 183 3 is be AUX iajs-1955 183 4 a a DET iajs-1955 183 5 w w NOUN iajs-1955 183 6 - - PUNCT iajs-1955 183 7 closed closed ADJ iajs-1955 183 8 submodule submodule NOUN iajs-1955 183 9 in in ADP iajs-1955 183 10 m m PROPN iajs-1955 183 11 , , PUNCT iajs-1955 183 12 then then ADV iajs-1955 183 13 e e X iajs-1955 183 14 = = NOUN iajs-1955 183 15 m. m. NOUN iajs-1955 183 16 on on ADP iajs-1955 183 17 the the DET iajs-1955 183 18 other other ADJ iajs-1955 183 19 hand hand NOUN iajs-1955 183 20 e e NOUN iajs-1955 183 21 is be AUX iajs-1955 183 22 a a DET iajs-1955 183 23 semi semi ADJ iajs-1955 183 24 - - ADJ iajs-1955 183 25 prime prime ADJ iajs-1955 183 26 submodule submodule NOUN iajs-1955 183 27 of of ADP iajs-1955 183 28 m m PROPN iajs-1955 183 29 , , PUNCT iajs-1955 183 30 that that SCONJ iajs-1955 183 31 e e NOUN iajs-1955 183 32 must must AUX iajs-1955 183 33 be be AUX iajs-1955 183 34 a a DET iajs-1955 183 35 proper proper ADJ iajs-1955 183 36 submodule submodule NOUN iajs-1955 183 37 of of ADP iajs-1955 183 38 m m PROPN iajs-1955 183 39 , , PUNCT iajs-1955 183 40 so so SCONJ iajs-1955 183 41 we we PRON iajs-1955 183 42 get get VERB iajs-1955 183 43 contradiction contradiction NOUN iajs-1955 183 44 . . PUNCT iajs-1955 184 1 hence hence ADV iajs-1955 184 2 e e PROPN iajs-1955 184 3 must must AUX iajs-1955 184 4 be be AUX iajs-1955 184 5 a a DET iajs-1955 184 6 minimal minimal ADJ iajs-1955 184 7 " " PUNCT iajs-1955 184 8 semi semi ADJ iajs-1955 184 9 - - ADJ iajs-1955 184 10 prime prime ADJ iajs-1955 184 11 submodule submodule NOUN iajs-1955 184 12 of of ADP iajs-1955 184 13 m m NOUN iajs-1955 184 14 " " PUNCT iajs-1955 184 15 . . PUNCT iajs-1955 185 1 proposition proposition NOUN iajs-1955 185 2 ( ( PUNCT iajs-1955 185 3 3.5 3.5 NUM iajs-1955 185 4 ) ) PUNCT iajs-1955 185 5 let let VERB iajs-1955 185 6 m m PRON iajs-1955 185 7 be be AUX iajs-1955 185 8 a a DET iajs-1955 185 9 non non ADJ iajs-1955 185 10 zero zero NUM iajs-1955 185 11 multiplication multiplication NOUN iajs-1955 185 12 module module NOUN iajs-1955 185 13 with with ADP iajs-1955 185 14 only only ADV iajs-1955 185 15 one one NUM iajs-1955 185 16 non non ADJ iajs-1955 185 17 zero zero NUM iajs-1955 185 18 maximal maximal ADJ iajs-1955 185 19 submodule submodule PROPN iajs-1955 185 20 e. e. PROPN iajs-1955 185 21 then then ADV iajs-1955 185 22 e e X iajs-1955 185 23 can can AUX iajs-1955 185 24 not not PART iajs-1955 185 25 be be AUX iajs-1955 185 26 w w NOUN iajs-1955 185 27 - - PUNCT iajs-1955 185 28 closed closed ADJ iajs-1955 185 29 submodule submodule NOUN iajs-1955 185 30 in in ADP iajs-1955 185 31 m. m. NOUN iajs-1955 185 32 proof proof NOUN iajs-1955 185 33 assume assume VERB iajs-1955 185 34 that that SCONJ iajs-1955 185 35 e e PRON iajs-1955 185 36 is be AUX iajs-1955 185 37 a a DET iajs-1955 185 38 w w NOUN iajs-1955 185 39 - - PUNCT iajs-1955 185 40 closed closed ADJ iajs-1955 185 41 submodule submodule NOUN iajs-1955 185 42 in in ADP iajs-1955 185 43 m m PROPN iajs-1955 185 44 , , PUNCT iajs-1955 185 45 then then ADV iajs-1955 185 46 by by ADP iajs-1955 185 47 [ [ X iajs-1955 185 48 3 3 NUM iajs-1955 185 49 , , PUNCT iajs-1955 185 50 prop(2.20 prop(2.20 NUM iajs-1955 185 51 ) ) PUNCT iajs-1955 185 52 ] ] PUNCT iajs-1955 186 1 e e X iajs-1955 186 2 " " PUNCT iajs-1955 186 3 is be AUX iajs-1955 186 4 a a DET iajs-1955 186 5 weak weak ADJ iajs-1955 186 6 essential essential ADJ iajs-1955 186 7 submodule submodule NOUN iajs-1955 186 8 of of ADP iajs-1955 186 9 m m NOUN iajs-1955 186 10 " " PUNCT iajs-1955 186 11 . . PUNCT iajs-1955 187 1 hence hence ADV iajs-1955 187 2 e e X iajs-1955 187 3 = = NOUN iajs-1955 187 4 m. m. NOUN iajs-1955 187 5 " " PUNCT iajs-1955 187 6 but but CCONJ iajs-1955 187 7 this this PRON iajs-1955 187 8 contradict contradict VERB iajs-1955 187 9 the the DET iajs-1955 187 10 maximality maximality NOUN iajs-1955 187 11 of of ADP iajs-1955 187 12 e e NOUN iajs-1955 187 13 " " PUNCT iajs-1955 187 14 . . PUNCT iajs-1955 188 1 therefore therefore ADV iajs-1955 188 2 e e PROPN iajs-1955 188 3 is be AUX iajs-1955 188 4 not not PART iajs-1955 188 5 w w NOUN iajs-1955 188 6 - - PUNCT iajs-1955 188 7 closed closed ADJ iajs-1955 188 8 submodule submodule NOUN iajs-1955 188 9 in in ADP iajs-1955 188 10 m. m. NOUN iajs-1955 188 11 " " PUNCT iajs-1955 188 12 recal recal ADJ iajs-1955 188 13 that that SCONJ iajs-1955 188 14 for for ADP iajs-1955 188 15 any any DET iajs-1955 188 16 module module NOUN iajs-1955 188 17 m m NOUN iajs-1955 188 18 and and CCONJ iajs-1955 188 19 any any DET iajs-1955 188 20 ideals ideal NOUN iajs-1955 188 21 i i PRON iajs-1955 188 22 and and CCONJ iajs-1955 188 23 j j PROPN iajs-1955 188 24 of of ADP iajs-1955 188 25 r r NOUN iajs-1955 188 26 if if SCONJ iajs-1955 188 27 i i PRON iajs-1955 188 28 is be AUX iajs-1955 188 29 a a DET iajs-1955 188 30 semi semi ADJ iajs-1955 188 31 - - ADJ iajs-1955 188 32 prime prime ADJ iajs-1955 188 33 ideal ideal NOUN iajs-1955 188 34 of of ADP iajs-1955 188 35 j j PROPN iajs-1955 189 1 then then ADV iajs-1955 189 2 i i PRON iajs-1955 189 3 m m VERB iajs-1955 189 4 is be AUX iajs-1955 189 5 a a DET iajs-1955 189 6 semi semi ADJ iajs-1955 189 7 - - ADJ iajs-1955 189 8 prime prime ADJ iajs-1955 189 9 submodule submodule NOUN iajs-1955 189 10 of of ADP iajs-1955 189 11 jm jm PROPN iajs-1955 189 12 this this PRON iajs-1955 189 13 is be AUX iajs-1955 189 14 called call VERB iajs-1955 189 15 condition condition NOUN iajs-1955 189 16 ∗ ∗ NOUN iajs-1955 189 17 in in ADP iajs-1955 189 18 [ [ X iajs-1955 189 19 3 3 NUM iajs-1955 189 20 ] ] X iajs-1955 189 21 " " PUNCT iajs-1955 189 22 . . PUNCT iajs-1955 190 1 proposition(3.6 proposition(3.6 VERB iajs-1955 190 2 ) ) PUNCT iajs-1955 190 3 let let VERB iajs-1955 190 4 m m PRON iajs-1955 190 5 be be AUX iajs-1955 190 6 a a DET iajs-1955 190 7 faithful faithful ADJ iajs-1955 190 8 and and CCONJ iajs-1955 190 9 multiplication multiplication NOUN iajs-1955 190 10 module module NOUN iajs-1955 190 11 such such ADJ iajs-1955 190 12 that that SCONJ iajs-1955 190 13 m m PROPN iajs-1955 190 14 satisfies satisfy VERB iajs-1955 190 15 condition condition NOUN iajs-1955 190 16 ∗ ∗ NOUN iajs-1955 190 17 , , PUNCT iajs-1955 190 18 if if SCONJ iajs-1955 190 19 l l NOUN iajs-1955 190 20 is be AUX iajs-1955 190 21 a a DET iajs-1955 190 22 w w NOUN iajs-1955 190 23 - - PUNCT iajs-1955 190 24 closed closed ADJ iajs-1955 190 25 ideal ideal NOUN iajs-1955 190 26 in in ADP iajs-1955 190 27 k k PROPN iajs-1955 190 28 then then ADV iajs-1955 190 29 lm lm INTJ iajs-1955 190 30 is be AUX iajs-1955 190 31 a a DET iajs-1955 190 32 w w NOUN iajs-1955 190 33 - - PUNCT iajs-1955 190 34 closed closed ADJ iajs-1955 190 35 submodule submodule NOUN iajs-1955 190 36 in in ADP iajs-1955 190 37 km km PROPN iajs-1955 190 38 . . PUNCT iajs-1955 191 1 proof proof NOUN iajs-1955 191 2 suppose suppose VERB iajs-1955 191 3 that that SCONJ iajs-1955 191 4 l l NOUN iajs-1955 191 5 is be AUX iajs-1955 191 6 a a DET iajs-1955 191 7 w w NOUN iajs-1955 191 8 - - PUNCT iajs-1955 191 9 closed closed ADJ iajs-1955 191 10 ideal ideal NOUN iajs-1955 191 11 in in ADP iajs-1955 191 12 k k PROPN iajs-1955 191 13 , , PUNCT iajs-1955 191 14 and and CCONJ iajs-1955 191 15 lm lm INTJ iajs-1955 191 16 is be AUX iajs-1955 191 17 a a DET iajs-1955 191 18 weak weak ADJ iajs-1955 191 19 essential essential ADJ iajs-1955 191 20 submodule submodule NOUN iajs-1955 191 21 of of ADP iajs-1955 191 22 t t PROPN iajs-1955 191 23 where where SCONJ iajs-1955 191 24 t t PROPN iajs-1955 191 25 is be AUX iajs-1955 191 26 a a DET iajs-1955 191 27 submodule submodule NOUN iajs-1955 191 28 of of ADP iajs-1955 191 29 km km PROPN iajs-1955 191 30 , , PUNCT iajs-1955 191 31 we we PRON iajs-1955 191 32 have have VERB iajs-1955 191 33 to to PART iajs-1955 191 34 show show VERB iajs-1955 191 35 that that SCONJ iajs-1955 191 36 lm lm PROPN iajs-1955 191 37 = = NOUN iajs-1955 191 38 t. t. NOUN iajs-1955 191 39 since since SCONJ iajs-1955 191 40 m m PROPN iajs-1955 191 41 is be AUX iajs-1955 191 42 a a DET iajs-1955 191 43 multiplication multiplication NOUN iajs-1955 191 44 module module NOUN iajs-1955 191 45 , , PUNCT iajs-1955 191 46 then then ADV iajs-1955 191 47 t t PROPN iajs-1955 191 48 = = NOUN iajs-1955 191 49 pm pm NOUN iajs-1955 191 50 for for ADP iajs-1955 191 51 some some DET iajs-1955 191 52 ideal ideal NOUN iajs-1955 191 53 p p NOUN iajs-1955 191 54 of of ADP iajs-1955 191 55 r r NOUN iajs-1955 191 56 with with ADP iajs-1955 191 57 𝑃 𝑃 NOUN iajs-1955 191 58 𝐾. 𝐾. PROPN iajs-1955 191 59 that that PRON iajs-1955 191 60 is be AUX iajs-1955 191 61 lm lm INTJ iajs-1955 191 62 " " PUNCT iajs-1955 191 63 is be AUX iajs-1955 191 64 a a DET iajs-1955 191 65 weak weak ADJ iajs-1955 191 66 essential essential ADJ iajs-1955 191 67 submodule submodule NOUN iajs-1955 191 68 of of ADP iajs-1955 191 69 pm pm NOUN iajs-1955 191 70 " " PUNCT iajs-1955 191 71 , , PUNCT iajs-1955 191 72 and and CCONJ iajs-1955 191 73 since since SCONJ iajs-1955 191 74 m m PROPN iajs-1955 191 75 is be AUX iajs-1955 191 76 faithful faithful ADJ iajs-1955 191 77 and and CCONJ iajs-1955 191 78 satisfies satisfy VERB iajs-1955 191 79 condition condition NOUN iajs-1955 191 80 ∗ ∗ NOUN iajs-1955 191 81 then then ADV iajs-1955 191 82 by by ADP iajs-1955 191 83 [ [ X iajs-1955 191 84 3,prop(2.17 3,prop(2.17 PROPN iajs-1955 191 85 ) ) PUNCT iajs-1955 191 86 ] ] PUNCT iajs-1955 191 87 , , PUNCT iajs-1955 191 88 we we PRON iajs-1955 191 89 have have AUX iajs-1955 191 90 l l NOUN iajs-1955 191 91 is be AUX iajs-1955 191 92 a a DET iajs-1955 191 93 weak weak ADJ iajs-1955 191 94             SPACE iajs-1955 191 95 174 174 NUM iajs-1955 191 96     SPACE iajs-1955 191 97 | | ADV iajs-1955 191 98     SPACE iajs-1955 191 99 mathmatics mathmatic NOUN iajs-1955 191 100 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 191 101     SPACE iajs-1955 191 102 2018 2018 NUM iajs-1955 191 103 ) ) PUNCT iajs-1955 191 104 عام عام ADP iajs-1955 191 105 2العدد 2العدد NUM iajs-1955 191 106 ( ( PUNCT iajs-1955 191 107 31لمجلد 31لمجلد NUM iajs-1955 191 108 ا ا NOUN iajs-1955 191 109 مجلة مجلة NOUN iajs-1955 191 110 إبن إبن VERB iajs-1955 191 111 الهيثم الهيثم ADJ iajs-1955 191 112 للعلوم للعلوم NOUN iajs-1955 191 113 الصرفة الصرفة NOUN iajs-1955 192 1 و و PRON iajs-1955 192 2 التطبيقية التطبيقية ADJ iajs-1955 192 3 ibn ibn PROPN iajs-1955 192 4 al al PROPN iajs-1955 192 5 - - PUNCT iajs-1955 192 6 haitham haitham PROPN iajs-1955 192 7 jour jour X iajs-1955 192 8 . . PROPN iajs-1955 192 9 for for ADP iajs-1955 192 10 pure pure ADJ iajs-1955 192 11 & & CCONJ iajs-1955 192 12 appl appl PROPN iajs-1955 192 13 . . PUNCT iajs-1955 193 1 sci sci PROPN iajs-1955 193 2 . . PUNCT iajs-1955 193 3 vol vol NOUN iajs-1955 193 4 . . PROPN iajs-1955 194 1 31 31 NUM iajs-1955 194 2 ( ( PUNCT iajs-1955 194 3 2 2 NUM iajs-1955 194 4 ) ) PUNCT iajs-1955 194 5 2018 2018 NUM iajs-1955 194 6 essential essential ADJ iajs-1955 194 7 ideal ideal NOUN iajs-1955 194 8 in in ADP iajs-1955 194 9 p p NOUN iajs-1955 194 10 and and CCONJ iajs-1955 194 11 𝑃 𝑃 NOUN iajs-1955 194 12 𝐾. 𝐾. PROPN iajs-1955 194 13 but but CCONJ iajs-1955 194 14 l l NOUN iajs-1955 194 15 is be AUX iajs-1955 194 16 a a DET iajs-1955 194 17 w w NOUN iajs-1955 194 18 - - PUNCT iajs-1955 194 19 closed closed ADJ iajs-1955 194 20 ideal ideal NOUN iajs-1955 194 21 in in ADP iajs-1955 194 22 k k PROPN iajs-1955 194 23 , , PUNCT iajs-1955 194 24 then then ADV iajs-1955 194 25 l l NOUN iajs-1955 194 26 = = NOUN iajs-1955 194 27 p. p. NOUN iajs-1955 194 28 that that PRON iajs-1955 194 29 is be AUX iajs-1955 194 30 lm lm NOUN iajs-1955 194 31 = = PRON iajs-1955 194 32 pm pm NOUN iajs-1955 194 33 = = NOUN iajs-1955 194 34 t. t. NOUN iajs-1955 194 35 hence hence ADV iajs-1955 194 36 lm lm PROPN iajs-1955 194 37 is be AUX iajs-1955 194 38 a a DET iajs-1955 194 39 w w NOUN iajs-1955 194 40 - - PUNCT iajs-1955 194 41 closed closed ADJ iajs-1955 194 42 submodule submodule NOUN iajs-1955 194 43 in in ADP iajs-1955 194 44 km km PROPN iajs-1955 194 45 . . PUNCT iajs-1955 195 1 the the DET iajs-1955 195 2 following follow VERB iajs-1955 195 3 proposition proposition NOUN iajs-1955 195 4 gives give VERB iajs-1955 195 5 the the DET iajs-1955 195 6 converse converse NOUN iajs-1955 195 7 of of ADP iajs-1955 195 8 proposition(3.6 proposition(3.6 NOUN iajs-1955 195 9 ) ) PUNCT iajs-1955 195 10 . . PUNCT iajs-1955 196 1 proposition proposition NOUN iajs-1955 196 2 ( ( PUNCT iajs-1955 196 3 3.7 3.7 NUM iajs-1955 196 4 ) ) PUNCT iajs-1955 196 5 if if SCONJ iajs-1955 196 6 m m NOUN iajs-1955 196 7 is be AUX iajs-1955 196 8 a a DET iajs-1955 196 9 finitely finitely ADV iajs-1955 196 10 generated generate VERB iajs-1955 196 11 , , PUNCT iajs-1955 196 12 faithful faithful ADJ iajs-1955 196 13 and and CCONJ iajs-1955 196 14 multiplication multiplication NOUN iajs-1955 196 15 module module NOUN iajs-1955 196 16 , , PUNCT iajs-1955 196 17 and and CCONJ iajs-1955 196 18 lm lm INTJ iajs-1955 196 19 is be AUX iajs-1955 196 20 a a DET iajs-1955 196 21 w w NOUN iajs-1955 196 22 - - PUNCT iajs-1955 196 23 closed closed ADJ iajs-1955 196 24 submodule submodule NOUN iajs-1955 196 25 in in ADP iajs-1955 196 26 km km PROPN iajs-1955 196 27 , , PUNCT iajs-1955 196 28 then then ADV iajs-1955 196 29 l l NOUN iajs-1955 196 30 is be AUX iajs-1955 196 31 a a DET iajs-1955 196 32 w w NOUN iajs-1955 196 33 - - PUNCT iajs-1955 196 34 closed closed ADJ iajs-1955 196 35 ideal ideal NOUN iajs-1955 196 36 in in ADP iajs-1955 196 37 k. k. PROPN iajs-1955 196 38 proof proof NOUN iajs-1955 196 39 suppose suppose VERB iajs-1955 196 40 that that SCONJ iajs-1955 196 41 lm lm PROPN iajs-1955 196 42 is be AUX iajs-1955 196 43 a a DET iajs-1955 196 44 w w NOUN iajs-1955 196 45 - - PUNCT iajs-1955 196 46 closed closed ADJ iajs-1955 196 47 submodules submodule NOUN iajs-1955 196 48 in in ADP iajs-1955 196 49 km km PROPN iajs-1955 196 50 , , PUNCT iajs-1955 196 51 where where SCONJ iajs-1955 196 52 l l NOUN iajs-1955 196 53 and and CCONJ iajs-1955 196 54 k k PROPN iajs-1955 196 55 are be AUX iajs-1955 196 56 ideals ideal NOUN iajs-1955 196 57 in in ADP iajs-1955 196 58 r r NOUN iajs-1955 196 59 , , PUNCT iajs-1955 196 60 and and CCONJ iajs-1955 196 61 let let VERB iajs-1955 196 62 l l NOUN iajs-1955 196 63 is be AUX iajs-1955 196 64 a a DET iajs-1955 196 65 weak weak ADJ iajs-1955 196 66 essential essential ADJ iajs-1955 196 67 ideal ideal NOUN iajs-1955 196 68 in in ADP iajs-1955 196 69 u u PRON iajs-1955 196 70 where where SCONJ iajs-1955 196 71 u u NOUN iajs-1955 196 72 is be AUX iajs-1955 196 73 an an DET iajs-1955 196 74 ideal ideal NOUN iajs-1955 196 75 of of ADP iajs-1955 196 76 k. k. PROPN iajs-1955 196 77 " " PUNCT iajs-1955 196 78 since since SCONJ iajs-1955 196 79 m m PROPN iajs-1955 196 80 is be AUX iajs-1955 196 81 finitely finitely ADV iajs-1955 196 82 generated generate VERB iajs-1955 196 83 faithful faithful ADJ iajs-1955 196 84 and and CCONJ iajs-1955 196 85 multiplication multiplication NOUN iajs-1955 196 86 " " PUNCT iajs-1955 196 87 , , PUNCT iajs-1955 196 88 then then ADV iajs-1955 196 89 by by ADP iajs-1955 196 90 [ [ X iajs-1955 196 91 3 3 NUM iajs-1955 196 92 , , PUNCT iajs-1955 196 93 prop(2.18 prop(2.18 NUM iajs-1955 196 94 ) ) PUNCT iajs-1955 196 95 ] ] PUNCT iajs-1955 197 1 we we PRON iajs-1955 197 2 have have AUX iajs-1955 197 3 lm lm INTJ iajs-1955 197 4 is be AUX iajs-1955 197 5 a a DET iajs-1955 197 6 weak weak ADJ iajs-1955 197 7 essential essential ADJ iajs-1955 197 8 in in ADP iajs-1955 197 9 um um INTJ iajs-1955 197 10 which which PRON iajs-1955 197 11 is be AUX iajs-1955 197 12 a a DET iajs-1955 197 13 submodule submodule NOUN iajs-1955 197 14 of of ADP iajs-1955 197 15 km km PROPN iajs-1955 197 16 . . PUNCT iajs-1955 198 1 but but CCONJ iajs-1955 198 2 lm lm INTJ iajs-1955 198 3 is be AUX iajs-1955 198 4 a a DET iajs-1955 198 5 w w NOUN iajs-1955 198 6 - - PUNCT iajs-1955 198 7 closed closed ADJ iajs-1955 198 8 submodule submodule NOUN iajs-1955 198 9 in in ADP iajs-1955 198 10 km km PROPN iajs-1955 198 11 , , PUNCT iajs-1955 198 12 then then ADV iajs-1955 198 13 lm lm NOUN iajs-1955 198 14 = = NOUN iajs-1955 198 15 um um INTJ iajs-1955 198 16 . . PUNCT iajs-1955 199 1 hence hence ADV iajs-1955 199 2 by by ADP iajs-1955 199 3 [ [ X iajs-1955 199 4 12 12 NUM iajs-1955 199 5 , , PUNCT iajs-1955 199 6 th,(3.1 th,(3.1 PROPN iajs-1955 199 7 ) ) PUNCT iajs-1955 199 8 ] ] PUNCT iajs-1955 199 9 , , PUNCT iajs-1955 199 10 l l NOUN iajs-1955 199 11 = = NOUN iajs-1955 199 12 u. u. PROPN iajs-1955 199 13 then then ADV iajs-1955 199 14 l l NOUN iajs-1955 199 15 is be AUX iajs-1955 199 16 a a DET iajs-1955 199 17 w w NOUN iajs-1955 199 18 - - PUNCT iajs-1955 199 19 closed closed ADJ iajs-1955 199 20 ideal ideal NOUN iajs-1955 199 21 in in ADP iajs-1955 199 22 k. k. PROPN iajs-1955 199 23 from from ADP iajs-1955 199 24 proposition proposition NOUN iajs-1955 199 25 ( ( PUNCT iajs-1955 199 26 3.6 3.6 NUM iajs-1955 199 27 ) ) PUNCT iajs-1955 199 28 and and CCONJ iajs-1955 199 29 proposition(3.7 proposition(3.7 NOUN iajs-1955 199 30 ) ) PUNCT iajs-1955 199 31 we we PRON iajs-1955 199 32 get get VERB iajs-1955 199 33 the the DET iajs-1955 199 34 following follow VERB iajs-1955 199 35 corollary corollary NOUN iajs-1955 199 36 . . PUNCT iajs-1955 200 1 corollary(3.8 corollary(3.8 NOUN iajs-1955 200 2 ) ) PUNCT iajs-1955 201 1 " " PUNCT iajs-1955 201 2 if if SCONJ iajs-1955 201 3 m m NOUN iajs-1955 201 4 is be AUX iajs-1955 201 5 a a DET iajs-1955 201 6 finitely finitely ADV iajs-1955 201 7 generated generate VERB iajs-1955 201 8 faithful faithful ADJ iajs-1955 201 9 and and CCONJ iajs-1955 201 10 multiplication multiplication NOUN iajs-1955 201 11 module module NOUN iajs-1955 201 12 which which PRON iajs-1955 201 13 satisfies satisfy VERB iajs-1955 201 14 condition condition NOUN iajs-1955 201 15 ∗ ∗ NOUN iajs-1955 201 16 " " PUNCT iajs-1955 201 17 , , PUNCT iajs-1955 201 18 then then ADV iajs-1955 201 19 l l NOUN iajs-1955 201 20 is be AUX iajs-1955 201 21 a a DET iajs-1955 201 22 w w NOUN iajs-1955 201 23 - - PUNCT iajs-1955 201 24 closed closed ADJ iajs-1955 201 25 ideal ideal NOUN iajs-1955 201 26 in in ADP iajs-1955 201 27 k k PROPN iajs-1955 201 28 if if SCONJ iajs-1955 202 1 and and CCONJ iajs-1955 202 2 only only ADV iajs-1955 202 3 if if SCONJ iajs-1955 202 4 lm lm INTJ iajs-1955 202 5 is be AUX iajs-1955 202 6 a a DET iajs-1955 202 7 w w NOUN iajs-1955 202 8 - - PUNCT iajs-1955 202 9 closed closed ADJ iajs-1955 202 10 submodule submodule NOUN iajs-1955 202 11 in in ADP iajs-1955 202 12 km km PROPN iajs-1955 202 13 . . PUNCT iajs-1955 202 14 theorem(3.9 theorem(3.9 PUNCT iajs-1955 202 15 ) ) PUNCT iajs-1955 202 16 if if SCONJ iajs-1955 202 17 m m NOUN iajs-1955 202 18 is be AUX iajs-1955 202 19 a a DET iajs-1955 202 20 finitely finitely ADV iajs-1955 202 21 generated generate VERB iajs-1955 202 22 faithful faithful ADJ iajs-1955 202 23 and and CCONJ iajs-1955 202 24 multiplication multiplication NOUN iajs-1955 202 25 module module NOUN iajs-1955 202 26 , , PUNCT iajs-1955 202 27 and and CCONJ iajs-1955 202 28 let let VERB iajs-1955 202 29 e e PRON iajs-1955 202 30 be be AUX iajs-1955 202 31 a a DET iajs-1955 202 32 submodule submodule NOUN iajs-1955 202 33 of of ADP iajs-1955 202 34 m m PROPN iajs-1955 202 35 , , PUNCT iajs-1955 202 36 such such ADJ iajs-1955 202 37 that that SCONJ iajs-1955 202 38 m m PROPN iajs-1955 202 39 satisfies satisfy VERB iajs-1955 202 40 condition condition NOUN iajs-1955 202 41 ∗ ∗ NOUN iajs-1955 202 42 , , PUNCT iajs-1955 202 43 " " PUNCT iajs-1955 202 44 then then ADV iajs-1955 202 45 the the DET iajs-1955 202 46 following following ADJ iajs-1955 202 47 statements statement NOUN iajs-1955 202 48 are be AUX iajs-1955 202 49 equivalent equivalent ADJ iajs-1955 202 50 " " PUNCT iajs-1955 202 51 : : PUNCT iajs-1955 202 52 1e 1e PROPN iajs-1955 202 53 is be AUX iajs-1955 202 54 a a DET iajs-1955 202 55 w w NOUN iajs-1955 202 56 - - PUNCT iajs-1955 202 57 closed closed ADJ iajs-1955 202 58 submodule submodule NOUN iajs-1955 202 59 in in ADP iajs-1955 202 60 m. m. NOUN iajs-1955 202 61 2𝐸 2𝐸 NOUN iajs-1955 202 62 : : PUNCT iajs-1955 202 63 𝑀 𝑀 PROPN iajs-1955 202 64 is be AUX iajs-1955 202 65 a a DET iajs-1955 202 66 w w NOUN iajs-1955 202 67 - - PUNCT iajs-1955 202 68 closed closed ADJ iajs-1955 202 69 ideal ideal NOUN iajs-1955 202 70 in in ADP iajs-1955 202 71 r. r. PROPN iajs-1955 202 72 3e 3e X iajs-1955 202 73 = = PRON iajs-1955 202 74 pm pm NOUN iajs-1955 202 75 for for ADP iajs-1955 202 76 some some DET iajs-1955 202 77 w w NOUN iajs-1955 202 78 - - PUNCT iajs-1955 202 79 closed closed ADJ iajs-1955 202 80 ideal ideal NOUN iajs-1955 202 81 p p NOUN iajs-1955 202 82 in in ADP iajs-1955 202 83 r. r. PROPN iajs-1955 202 84 proof proof NOUN iajs-1955 202 85 1 1 NUM iajs-1955 202 86 ⟹ ⟹ NUM iajs-1955 202 87 2 2 NUM iajs-1955 202 88 suppose suppose VERB iajs-1955 202 89 that that SCONJ iajs-1955 202 90 e e PROPN iajs-1955 202 91 is be AUX iajs-1955 202 92 a a DET iajs-1955 202 93 w w NOUN iajs-1955 202 94 - - PUNCT iajs-1955 202 95 closed closed ADJ iajs-1955 202 96 submodule submodule NOUN iajs-1955 202 97 in in ADP iajs-1955 202 98 m. m. NOUN iajs-1955 202 99 since since SCONJ iajs-1955 202 100 m m PROPN iajs-1955 202 101 is be AUX iajs-1955 202 102 a a DET iajs-1955 202 103 multiplication multiplication NOUN iajs-1955 202 104 , , PUNCT iajs-1955 202 105 then then ADV iajs-1955 202 106 by by ADP iajs-1955 202 107 [ [ X iajs-1955 202 108 7 7 X iajs-1955 202 109 ] ] X iajs-1955 202 110 𝐸 𝐸 PROPN iajs-1955 202 111 𝐸 𝐸 PROPN iajs-1955 202 112 : : PUNCT iajs-1955 202 113 𝑀 𝑀 PROPN iajs-1955 202 114 𝑀. 𝑀. PROPN iajs-1955 202 115 put put VERB iajs-1955 202 116 𝐸 𝐸 NOUN iajs-1955 202 117 : : PUNCT iajs-1955 202 118 𝑀 𝑀 PROPN iajs-1955 202 119 𝑃 𝑃 PROPN iajs-1955 202 120 , , PUNCT iajs-1955 202 121 then then ADV iajs-1955 202 122 we we PRON iajs-1955 202 123 have have VERB iajs-1955 202 124 pm pm NOUN iajs-1955 202 125 = = NOUN iajs-1955 202 126 e e NOUN iajs-1955 202 127 is be AUX iajs-1955 202 128 a a DET iajs-1955 202 129 w w NOUN iajs-1955 202 130 - - PUNCT iajs-1955 202 131 closed closed ADJ iajs-1955 202 132 submodule submodule NOUN iajs-1955 202 133 in in ADP iajs-1955 202 134 m. m. NOUN iajs-1955 202 135 hence hence ADV iajs-1955 202 136 by by ADP iajs-1955 202 137 cor(3.8 cor(3.8 NUM iajs-1955 202 138 ) ) PUNCT iajs-1955 202 139 , , PUNCT iajs-1955 202 140 p p PROPN iajs-1955 202 141 is be AUX iajs-1955 202 142 a a DET iajs-1955 202 143 w w NOUN iajs-1955 202 144 - - PUNCT iajs-1955 202 145 closed closed ADJ iajs-1955 202 146 ideal ideal NOUN iajs-1955 202 147 in in ADP iajs-1955 202 148 r. r. PROPN iajs-1955 202 149 that that PRON iajs-1955 202 150 is be AUX iajs-1955 202 151 𝐸 𝐸 PROPN iajs-1955 202 152 : : PUNCT iajs-1955 202 153 𝑀 𝑀 PROPN iajs-1955 202 154 is be AUX iajs-1955 202 155 a a DET iajs-1955 202 156 w w NOUN iajs-1955 202 157 - - PUNCT iajs-1955 202 158 closed closed ADJ iajs-1955 202 159 ideal ideal NOUN iajs-1955 202 160 in in ADP iajs-1955 202 161 r. r. PROPN iajs-1955 202 162 2 2 NUM iajs-1955 202 163 ⟹ ⟹ NUM iajs-1955 202 164 3 3 NUM iajs-1955 202 165 : : PUNCT iajs-1955 202 166 suppose suppose VERB iajs-1955 202 167 that that SCONJ iajs-1955 202 168 𝐸 𝐸 PROPN iajs-1955 202 169 : : PUNCT iajs-1955 202 170 𝑀 𝑀 PROPN iajs-1955 202 171 is be AUX iajs-1955 202 172 a a DET iajs-1955 202 173 w w NOUN iajs-1955 202 174 - - PUNCT iajs-1955 202 175 closed closed ADJ iajs-1955 202 176 ideal ideal NOUN iajs-1955 202 177 in in ADP iajs-1955 202 178 r. r. PROPN iajs-1955 202 179 then then ADV iajs-1955 202 180 𝐸 𝐸 PROPN iajs-1955 202 181 𝐸 𝐸 PROPN iajs-1955 202 182 : : PUNCT iajs-1955 202 183 𝑀 𝑀 PROPN iajs-1955 202 184 𝑀 𝑀 PROPN iajs-1955 202 185 since since SCONJ iajs-1955 202 186 m m PROPN iajs-1955 202 187 is be AUX iajs-1955 202 188 multiplication multiplication NOUN iajs-1955 202 189 , , PUNCT iajs-1955 202 190 i.e i.e PROPN iajs-1955 202 191 e e NOUN iajs-1955 202 192 = = NOUN iajs-1955 202 193 pm pm NOUN iajs-1955 202 194 where where SCONJ iajs-1955 202 195 𝑃 𝑃 NOUN iajs-1955 202 196 𝐸 𝐸 PROPN iajs-1955 202 197 : : PUNCT iajs-1955 202 198 𝑀 𝑀 PROPN iajs-1955 202 199 is be AUX iajs-1955 202 200 a a DET iajs-1955 202 201 w w NOUN iajs-1955 202 202 - - PUNCT iajs-1955 202 203 closed closed ADJ iajs-1955 202 204 ideal ideal NOUN iajs-1955 202 205 in in ADP iajs-1955 202 206 r. r. PROPN iajs-1955 202 207 3 3 NUM iajs-1955 202 208 ⟹ ⟹ NUM iajs-1955 202 209 1 1 NUM iajs-1955 202 210 : : PUNCT iajs-1955 202 211 suppose suppose VERB iajs-1955 202 212 that that SCONJ iajs-1955 202 213 e e NOUN iajs-1955 202 214 = = NOUN iajs-1955 202 215 pm pm NOUN iajs-1955 202 216 for for ADP iajs-1955 202 217 some some DET iajs-1955 202 218 w w NOUN iajs-1955 202 219 - - PUNCT iajs-1955 202 220 closed closed ADJ iajs-1955 202 221 submodule submodule NOUN iajs-1955 202 222 p p PROPN iajs-1955 202 223 in in ADP iajs-1955 202 224 r. r. PROPN iajs-1955 202 225 then then ADV iajs-1955 202 226 by by ADP iajs-1955 202 227 cor(3.8 cor(3.8 NUM iajs-1955 202 228 ) ) PUNCT iajs-1955 202 229 , , PUNCT iajs-1955 202 230 pm pm NOUN iajs-1955 202 231 = = NOUN iajs-1955 202 232 e e NOUN iajs-1955 202 233 is be AUX iajs-1955 202 234 a a DET iajs-1955 202 235 w w NOUN iajs-1955 202 236 - - PUNCT iajs-1955 202 237 closed closed ADJ iajs-1955 202 238 submodule submodule NOUN iajs-1955 202 239 in in ADP iajs-1955 202 240 rm rm PROPN iajs-1955 202 241 = = NOUN iajs-1955 202 242 m. m. NOUN iajs-1955 202 243 4chain 4chain NUM iajs-1955 202 244 conditions condition NOUN iajs-1955 202 245 on on ADP iajs-1955 202 246 w w ADV iajs-1955 202 247 - - PUNCT iajs-1955 202 248 closed closed ADJ iajs-1955 202 249 submodules submodule NOUN iajs-1955 202 250 we we PRON iajs-1955 202 251 start start VERB iajs-1955 202 252 this this DET iajs-1955 202 253 section section NOUN iajs-1955 202 254 by by ADP iajs-1955 202 255 introducing introduce VERB iajs-1955 202 256 the the DET iajs-1955 202 257 definitions definition NOUN iajs-1955 202 258 of of ADP iajs-1955 202 259 a a DET iajs-1955 202 260 modules module NOUN iajs-1955 202 261 that that PRON iajs-1955 202 262 have have AUX iajs-1955 202 263 ascending ascend VERB iajs-1955 202 264 ( ( PUNCT iajs-1955 202 265 descending descend VERB iajs-1955 202 266 ) ) PUNCT iajs-1955 202 267 chain chain NOUN iajs-1955 202 268 condition condition NOUN iajs-1955 202 269 on on ADP iajs-1955 202 270 w w ADJ iajs-1955 202 271 - - PUNCT iajs-1955 202 272 closed closed ADJ iajs-1955 202 273 submodules submodule NOUN iajs-1955 202 274 . . PUNCT iajs-1955 203 1 definition(4.1 definition(4.1 NOUN iajs-1955 203 2 ) ) PUNCT iajs-1955 203 3 a a DET iajs-1955 203 4 module module NOUN iajs-1955 203 5 m m NOUN iajs-1955 203 6 is be AUX iajs-1955 203 7 said say VERB iajs-1955 203 8 to to PART iajs-1955 203 9 have have VERB iajs-1955 203 10 the the DET iajs-1955 203 11 ascending ascend VERB iajs-1955 203 12 chain chain NOUN iajs-1955 203 13 condition condition NOUN iajs-1955 203 14 on on ADP iajs-1955 203 15 w w NOUN iajs-1955 203 16 - - PUNCT iajs-1955 203 17 closed closed ADJ iajs-1955 203 18             SPACE iajs-1955 203 19 175 175 NUM iajs-1955 203 20     SPACE iajs-1955 203 21 | | ADV iajs-1955 203 22     SPACE iajs-1955 203 23 mathmatics mathmatic NOUN iajs-1955 203 24 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 203 25     SPACE iajs-1955 203 26 2018 2018 NUM iajs-1955 203 27 ) ) PUNCT iajs-1955 203 28 عام عام ADP iajs-1955 203 29 2العدد 2العدد NUM iajs-1955 203 30 ( ( PUNCT iajs-1955 203 31 31لمجلد 31لمجلد NUM iajs-1955 203 32 ا ا NOUN iajs-1955 203 33 مجلة مجلة NOUN iajs-1955 203 34 إبن إبن VERB iajs-1955 203 35 الهيثم الهيثم ADJ iajs-1955 203 36 للعلوم للعلوم NOUN iajs-1955 203 37 الصرفة الصرفة NOUN iajs-1955 204 1 و و PRON iajs-1955 204 2 التطبيقية التطبيقية ADJ iajs-1955 204 3 ibn ibn PROPN iajs-1955 204 4 al al PROPN iajs-1955 204 5 - - PUNCT iajs-1955 204 6 haitham haitham PROPN iajs-1955 204 7 jour jour X iajs-1955 204 8 . . PROPN iajs-1955 204 9 for for ADP iajs-1955 204 10 pure pure ADJ iajs-1955 204 11 & & CCONJ iajs-1955 204 12 appl appl PROPN iajs-1955 204 13 . . PUNCT iajs-1955 205 1 sci sci PROPN iajs-1955 205 2 . . PUNCT iajs-1955 205 3 vol vol NOUN iajs-1955 205 4 . . PROPN iajs-1955 206 1 31 31 NUM iajs-1955 206 2 ( ( PUNCT iajs-1955 206 3 2 2 NUM iajs-1955 206 4 ) ) PUNCT iajs-1955 206 5 2018 2018 NUM iajs-1955 206 6 submodule submodule NOUN iajs-1955 206 7 ( ( PUNCT iajs-1955 206 8 briefly briefly NOUN iajs-1955 206 9 acc acc PROPN iajs-1955 206 10 on on ADP iajs-1955 206 11 w w NOUN iajs-1955 206 12 - - PUNCT iajs-1955 206 13 closed closed ADJ iajs-1955 206 14 submodules submodule NOUN iajs-1955 206 15 ) ) PUNCT iajs-1955 206 16 , , PUNCT iajs-1955 206 17 if if SCONJ iajs-1955 206 18 every every DET iajs-1955 206 19 ascending ascend VERB iajs-1955 206 20 chain chain NOUN iajs-1955 206 21 𝐸 𝐸 PROPN iajs-1955 206 22 ⊆ ⊆ NUM iajs-1955 206 23 𝐸 𝐸 PROPN iajs-1955 206 24 ⊆ ⊆ NUM iajs-1955 206 25 . . PUNCT iajs-1955 206 26 . . PUNCT iajs-1955 206 27 . . PUNCT iajs-1955 207 1 of of ADP iajs-1955 207 2 w w NOUN iajs-1955 207 3 - - PUNCT iajs-1955 207 4 closed closed ADJ iajs-1955 207 5 submodule submodule NOUN iajs-1955 207 6 in in ADP iajs-1955 207 7 m m PROPN iajs-1955 207 8 is be AUX iajs-1955 207 9 finite finite ADJ iajs-1955 207 10 . . PUNCT iajs-1955 208 1 that that PRON iajs-1955 208 2 is be AUX iajs-1955 208 3 ∃ ∃ PROPN iajs-1955 208 4 𝑚 𝑚 X iajs-1955 208 5 ∈ ∈ PROPN iajs-1955 208 6 𝑍 𝑍 VERB iajs-1955 208 7 such such ADJ iajs-1955 208 8 that that SCONJ iajs-1955 208 9 𝐸 𝐸 PROPN iajs-1955 208 10 𝐸 𝐸 PROPN iajs-1955 208 11 for for ADP iajs-1955 208 12 all all DET iajs-1955 208 13 𝑛 𝑛 DET iajs-1955 208 14 𝑚. 𝑚. ADJ iajs-1955 208 15 definition(4.2 definition(4.2 NOUN iajs-1955 208 16 ) ) PUNCT iajs-1955 208 17 a a DET iajs-1955 208 18 module module NOUN iajs-1955 208 19 m m NOUN iajs-1955 208 20 is be AUX iajs-1955 208 21 said say VERB iajs-1955 208 22 to to PART iajs-1955 208 23 have have VERB iajs-1955 208 24 the the DET iajs-1955 208 25 descending descend VERB iajs-1955 208 26 chain chain NOUN iajs-1955 208 27 condition condition NOUN iajs-1955 208 28 on on ADP iajs-1955 208 29 w w NOUN iajs-1955 208 30 - - PUNCT iajs-1955 208 31 closed close VERB iajs-1955 208 32 submodule submodule NOUN iajs-1955 208 33 ( ( PUNCT iajs-1955 208 34 briefly briefly NOUN iajs-1955 208 35 dcc dcc PROPN iajs-1955 208 36 on on ADP iajs-1955 208 37 w w NOUN iajs-1955 208 38 - - PUNCT iajs-1955 208 39 closed closed ADJ iajs-1955 208 40 submodules submodule NOUN iajs-1955 208 41 ) ) PUNCT iajs-1955 208 42 , , PUNCT iajs-1955 208 43 if if SCONJ iajs-1955 208 44 every every DET iajs-1955 208 45 descending descend VERB iajs-1955 208 46 chain chain NOUN iajs-1955 208 47 𝐸 𝐸 PROPN iajs-1955 208 48 ⊇ ⊇ NOUN iajs-1955 208 49 𝐸 𝐸 PROPN iajs-1955 208 50 ⊇ ⊇ NOUN iajs-1955 208 51 . . PUNCT iajs-1955 208 52 . . PUNCT iajs-1955 209 1 . . PUNCT iajs-1955 210 1 of of ADP iajs-1955 210 2 w w NOUN iajs-1955 210 3 - - PUNCT iajs-1955 210 4 closed closed ADJ iajs-1955 210 5 submodule submodule NOUN iajs-1955 210 6 in in ADP iajs-1955 210 7 m m PROPN iajs-1955 210 8 is be AUX iajs-1955 210 9 finite finite ADJ iajs-1955 210 10 . . PUNCT iajs-1955 211 1 that that PRON iajs-1955 211 2 is be AUX iajs-1955 211 3 ∃ ∃ PROPN iajs-1955 211 4 𝑚 𝑚 X iajs-1955 211 5 ∈ ∈ PROPN iajs-1955 211 6 𝑍 𝑍 VERB iajs-1955 211 7 such such ADJ iajs-1955 211 8 that that SCONJ iajs-1955 211 9 𝐸 𝐸 PROPN iajs-1955 211 10 𝐸 𝐸 PROPN iajs-1955 211 11 for for ADP iajs-1955 211 12 all all DET iajs-1955 211 13 𝑛 𝑛 DET iajs-1955 211 14 𝑚. 𝑚. ADJ iajs-1955 211 15 remarks remark NOUN iajs-1955 211 16 ( ( PUNCT iajs-1955 211 17 4.3 4.3 NUM iajs-1955 211 18 ) ) PUNCT iajs-1955 211 19 1𝑍𝑝 1𝑍𝑝 NUM iajs-1955 211 20 as as ADP iajs-1955 211 21 a a DET iajs-1955 211 22 z z NOUN iajs-1955 211 23 - - PUNCT iajs-1955 211 24 module module NOUN iajs-1955 211 25 satisfies satisfie NOUN iajs-1955 211 26 dcc dcc PROPN iajs-1955 211 27 on on ADP iajs-1955 211 28 w w NOUN iajs-1955 211 29 - - PUNCT iajs-1955 211 30 closed closed ADJ iajs-1955 211 31 submodules submodule NOUN iajs-1955 211 32 , , PUNCT iajs-1955 211 33 but but CCONJ iajs-1955 212 1 𝑍𝑝 𝑍𝑝 NOUN iajs-1955 212 2 as as ADP iajs-1955 212 3 a a DET iajs-1955 212 4 z z NOUN iajs-1955 212 5 - - PUNCT iajs-1955 212 6 modules module NOUN iajs-1955 212 7 does do AUX iajs-1955 212 8 not not PART iajs-1955 212 9 satisfies satisfy VERB iajs-1955 212 10 acc acc PROPN iajs-1955 212 11 on on ADP iajs-1955 212 12 w w NOUN iajs-1955 212 13 - - PUNCT iajs-1955 212 14 closed closed ADJ iajs-1955 212 15 submodules submodule NOUN iajs-1955 212 16 because because SCONJ iajs-1955 212 17 𝑍𝑝 𝑍𝑝 NOUN iajs-1955 212 18 is be AUX iajs-1955 212 19 an an DET iajs-1955 212 20 artinian artinian NOUN iajs-1955 212 21 but but CCONJ iajs-1955 212 22 not not PART iajs-1955 212 23 noetherian noetherian ADJ iajs-1955 212 24 . . PUNCT iajs-1955 213 1 2z 2z NUM iajs-1955 213 2 as as ADP iajs-1955 213 3 z z NOUN iajs-1955 213 4 - - PUNCT iajs-1955 213 5 module module NOUN iajs-1955 213 6 satisfies satisfie NOUN iajs-1955 213 7 ( ( PUNCT iajs-1955 213 8 acc acc PROPN iajs-1955 213 9 ) ) PUNCT iajs-1955 213 10 on on ADP iajs-1955 213 11 w w ADJ iajs-1955 213 12 - - PUNCT iajs-1955 213 13 closed closed ADJ iajs-1955 213 14 submodules submodule NOUN iajs-1955 213 15 , , PUNCT iajs-1955 213 16 but but CCONJ iajs-1955 213 17 does do AUX iajs-1955 213 18 not not PART iajs-1955 213 19 satisfies satisfie NOUN iajs-1955 213 20 ( ( PUNCT iajs-1955 213 21 dcc dcc PROPN iajs-1955 213 22 ) ) PUNCT iajs-1955 213 23 on on ADP iajs-1955 213 24 wclosed wclose VERB iajs-1955 213 25 submodules submodule NOUN iajs-1955 214 1 because because SCONJ iajs-1955 214 2 z z PROPN iajs-1955 214 3 as as ADP iajs-1955 214 4 a a DET iajs-1955 214 5 z z NOUN iajs-1955 214 6 - - PUNCT iajs-1955 214 7 module module NOUN iajs-1955 214 8 is be AUX iajs-1955 214 9 a a DET iajs-1955 214 10 noetherian noetherian ADJ iajs-1955 214 11 but but CCONJ iajs-1955 214 12 not not PART iajs-1955 214 13 artinian artinian ADJ iajs-1955 214 14 . . PUNCT iajs-1955 215 1 proposition proposition NOUN iajs-1955 215 2 ( ( PUNCT iajs-1955 215 3 4.4 4.4 NUM iajs-1955 215 4 ) ) PUNCT iajs-1955 215 5 if if SCONJ iajs-1955 215 6 m m NOUN iajs-1955 215 7 is be AUX iajs-1955 215 8 a a DET iajs-1955 215 9 module module NOUN iajs-1955 215 10 and and CCONJ iajs-1955 215 11 satisfies satisfie NOUN iajs-1955 215 12 ( ( PUNCT iajs-1955 215 13 dcc dcc PROPN iajs-1955 215 14 ) ) PUNCT iajs-1955 215 15 on on ADP iajs-1955 215 16 closed closed ADJ iajs-1955 215 17 submodules submodule NOUN iajs-1955 215 18 , , PUNCT iajs-1955 215 19 then then ADV iajs-1955 215 20 m m VERB iajs-1955 215 21 satisfies satisfie NOUN iajs-1955 215 22 ( ( PUNCT iajs-1955 215 23 dcc dcc PROPN iajs-1955 215 24 ) ) PUNCT iajs-1955 215 25 on on ADP iajs-1955 215 26 w w ADJ iajs-1955 215 27 - - PUNCT iajs-1955 215 28 closed closed ADJ iajs-1955 215 29 submodules submodule NOUN iajs-1955 215 30 . . PUNCT iajs-1955 216 1 proof proof NOUN iajs-1955 216 2 let let VERB iajs-1955 216 3 𝐸 𝐸 PROPN iajs-1955 216 4 ⊇ ⊇ NOUN iajs-1955 216 5 𝐸 𝐸 PROPN iajs-1955 216 6 ⊇ ⊇ NOUN iajs-1955 216 7 . . PUNCT iajs-1955 216 8 . . PUNCT iajs-1955 216 9 . . PUNCT iajs-1955 217 1 " " PUNCT iajs-1955 217 2 be be AUX iajs-1955 217 3 a a DET iajs-1955 217 4 descending descend VERB iajs-1955 217 5 chain chain NOUN iajs-1955 217 6 " " PUNCT iajs-1955 217 7 of of ADP iajs-1955 217 8 w w NOUN iajs-1955 217 9 - - PUNCT iajs-1955 217 10 closed closed ADJ iajs-1955 217 11 submodules submodule NOUN iajs-1955 217 12 of of ADP iajs-1955 217 13 m. m. NOUN iajs-1955 217 14 but but CCONJ iajs-1955 217 15 by by ADP iajs-1955 217 16 remark(2.2 remark(2.2 NOUN iajs-1955 217 17 ) ) PUNCT iajs-1955 217 18 every every DET iajs-1955 217 19 w w NOUN iajs-1955 217 20 - - PUNCT iajs-1955 217 21 closed closed ADJ iajs-1955 217 22 submodule submodule NOUN iajs-1955 217 23 is be AUX iajs-1955 217 24 closed closed ADJ iajs-1955 217 25 , , PUNCT iajs-1955 217 26 then then ADV iajs-1955 217 27 𝐸 𝐸 PROPN iajs-1955 217 28 is be AUX iajs-1955 217 29 a a DET iajs-1955 217 30 closed closed ADJ iajs-1955 217 31 submodule submodule NOUN iajs-1955 217 32 for for ADP iajs-1955 217 33 each each DET iajs-1955 217 34 i=1,2 i=1,2 NOUN iajs-1955 217 35 , , PUNCT iajs-1955 217 36 . . PUNCT iajs-1955 217 37 . . PUNCT iajs-1955 217 38 . . PUNCT iajs-1955 218 1 . . PUNCT iajs-1955 219 1 since since SCONJ iajs-1955 219 2 m m PRON iajs-1955 219 3 satisfies satisfie NOUN iajs-1955 219 4 ( ( PUNCT iajs-1955 219 5 dcc dcc PROPN iajs-1955 219 6 ) ) PUNCT iajs-1955 219 7 on on ADP iajs-1955 219 8 closed closed ADJ iajs-1955 219 9 submodule submodule NOUN iajs-1955 219 10 , , PUNCT iajs-1955 219 11 then then ADV iajs-1955 219 12 ∃ ∃ PROPN iajs-1955 219 13 𝑚 𝑚 X iajs-1955 219 14 ∈ ∈ PROPN iajs-1955 219 15 𝑍 𝑍 VERB iajs-1955 219 16 such such ADJ iajs-1955 219 17 that that SCONJ iajs-1955 219 18 𝐸 𝐸 PROPN iajs-1955 219 19 𝐸 𝐸 PROPN iajs-1955 219 20 for for ADP iajs-1955 219 21 each each DET iajs-1955 219 22 𝑛 𝑛 DET iajs-1955 219 23 𝑚. 𝑚. NOUN iajs-1955 219 24 thus thus ADV iajs-1955 219 25 , , PUNCT iajs-1955 219 26 m m VERB iajs-1955 219 27 satisfies satisfie NOUN iajs-1955 219 28 ( ( PUNCT iajs-1955 219 29 dcc dcc PROPN iajs-1955 219 30 ) ) PUNCT iajs-1955 219 31 on on ADP iajs-1955 219 32 w w ADJ iajs-1955 219 33 - - PUNCT iajs-1955 219 34 closed closed ADJ iajs-1955 219 35 submodules submodule NOUN iajs-1955 219 36 . . PUNCT iajs-1955 220 1 the the DET iajs-1955 220 2 proof proof NOUN iajs-1955 220 3 of of ADP iajs-1955 220 4 the the DET iajs-1955 220 5 following follow VERB iajs-1955 220 6 proposition proposition NOUN iajs-1955 220 7 is be AUX iajs-1955 220 8 similar similar ADJ iajs-1955 220 9 to to ADP iajs-1955 220 10 the the DET iajs-1955 220 11 proof proof NOUN iajs-1955 220 12 of of ADP iajs-1955 220 13 proposition proposition NOUN iajs-1955 220 14 ( ( PUNCT iajs-1955 220 15 4.4 4.4 NUM iajs-1955 220 16 ) ) PUNCT iajs-1955 220 17 and and CCONJ iajs-1955 220 18 hence hence ADV iajs-1955 220 19 is be AUX iajs-1955 220 20 omited omit VERB iajs-1955 220 21 . . PUNCT iajs-1955 221 1 proposition proposition NOUN iajs-1955 221 2 ( ( PUNCT iajs-1955 221 3 4.5 4.5 NUM iajs-1955 221 4 ) ) PUNCT iajs-1955 221 5 if if SCONJ iajs-1955 221 6 m m NOUN iajs-1955 221 7 is be AUX iajs-1955 221 8 a a DET iajs-1955 221 9 module module NOUN iajs-1955 221 10 and and CCONJ iajs-1955 221 11 satisfies satisfie NOUN iajs-1955 221 12 ( ( PUNCT iajs-1955 221 13 acc acc PROPN iajs-1955 221 14 ) ) PUNCT iajs-1955 221 15 on on ADP iajs-1955 221 16 closed closed ADJ iajs-1955 221 17 submodules submodule NOUN iajs-1955 221 18 , , PUNCT iajs-1955 221 19 then then ADV iajs-1955 221 20 m m VERB iajs-1955 221 21 satisfies satisfie NOUN iajs-1955 221 22 ( ( PUNCT iajs-1955 221 23 acc acc PROPN iajs-1955 221 24 ) ) PUNCT iajs-1955 221 25 on on ADP iajs-1955 221 26 wclosed wclose VERB iajs-1955 221 27 submodules submodule NOUN iajs-1955 221 28 . . PUNCT iajs-1955 222 1 since since SCONJ iajs-1955 222 2 w w NOUN iajs-1955 222 3 - - PUNCT iajs-1955 222 4 closed closed ADJ iajs-1955 222 5 submodules submodule NOUN iajs-1955 222 6 and and CCONJ iajs-1955 222 7 closed closed ADJ iajs-1955 222 8 submodules submodule NOUN iajs-1955 222 9 are be AUX iajs-1955 222 10 equivalent equivalent ADJ iajs-1955 222 11 in in ADP iajs-1955 222 12 the the DET iajs-1955 222 13 class class NOUN iajs-1955 222 14 of of ADP iajs-1955 222 15 fully fully ADV iajs-1955 222 16 semiprime semiprime NOUN iajs-1955 222 17 modules module NOUN iajs-1955 222 18 by by ADP iajs-1955 222 19 proposition proposition NOUN iajs-1955 222 20 ( ( PUNCT iajs-1955 222 21 2.14 2.14 NUM iajs-1955 222 22 ) ) PUNCT iajs-1955 222 23 , , PUNCT iajs-1955 222 24 " " PUNCT iajs-1955 222 25 we we PRON iajs-1955 222 26 get get VERB iajs-1955 222 27 the the DET iajs-1955 222 28 following follow VERB iajs-1955 222 29 results result NOUN iajs-1955 222 30 " " PUNCT iajs-1955 222 31 . . PUNCT iajs-1955 223 1 proposition proposition NOUN iajs-1955 223 2 ( ( PUNCT iajs-1955 223 3 4.6 4.6 NUM iajs-1955 223 4 ) ) PUNCT iajs-1955 223 5 if if SCONJ iajs-1955 223 6 m m NOUN iajs-1955 223 7 is be AUX iajs-1955 223 8 a a DET iajs-1955 223 9 fully fully ADV iajs-1955 223 10 semi semi ADJ iajs-1955 223 11 - - ADJ iajs-1955 223 12 prime prime ADJ iajs-1955 223 13 module module NOUN iajs-1955 223 14 , , PUNCT iajs-1955 223 15 then then ADV iajs-1955 223 16 m m VERB iajs-1955 223 17 satisfies satisfie NOUN iajs-1955 223 18 ( ( PUNCT iajs-1955 223 19 acc acc PROPN iajs-1955 223 20 ) ) PUNCT iajs-1955 223 21 on on ADP iajs-1955 223 22 w w ADJ iajs-1955 223 23 - - PUNCT iajs-1955 223 24 closed closed ADJ iajs-1955 223 25 submodules submodule NOUN iajs-1955 223 26 if if SCONJ iajs-1955 223 27 and and CCONJ iajs-1955 223 28 only only ADV iajs-1955 223 29 if if SCONJ iajs-1955 223 30 m m NOUN iajs-1955 223 31 satisfies satisfie NOUN iajs-1955 223 32 ( ( PUNCT iajs-1955 223 33 acc acc PROPN iajs-1955 223 34 ) ) PUNCT iajs-1955 223 35 on on ADP iajs-1955 223 36 closed closed ADJ iajs-1955 223 37 submodules submodule NOUN iajs-1955 223 38 . . PUNCT iajs-1955 224 1 proof proof NOUN iajs-1955 224 2 ⟹ ⟹ ADV iajs-1955 224 3 let let VERB iajs-1955 224 4 𝐸 𝐸 PRON iajs-1955 224 5 ⊆ ⊆ NUM iajs-1955 224 6 𝐸 𝐸 PROPN iajs-1955 224 7 ⊆ ⊆ NUM iajs-1955 224 8 . . PUNCT iajs-1955 224 9 . . PUNCT iajs-1955 224 10 . . PUNCT iajs-1955 225 1 " " PUNCT iajs-1955 225 2 be be AUX iajs-1955 225 3 ascending ascend VERB iajs-1955 225 4 chain chain NOUN iajs-1955 225 5 of of ADP iajs-1955 225 6 closed closed ADJ iajs-1955 225 7 submodules submodule NOUN iajs-1955 225 8 " " PUNCT iajs-1955 225 9 . . PUNCT iajs-1955 226 1 then then ADV iajs-1955 226 2 by by ADP iajs-1955 226 3 prop(2.14 prop(2.14 NOUN iajs-1955 226 4 ) ) PUNCT iajs-1955 226 5 , , PUNCT iajs-1955 226 6 𝐸 𝐸 PROPN iajs-1955 226 7 is be AUX iajs-1955 226 8 a a DET iajs-1955 226 9 w w NOUN iajs-1955 226 10 - - PUNCT iajs-1955 226 11 closed closed ADJ iajs-1955 226 12 submodule submodule NOUN iajs-1955 226 13 for for ADP iajs-1955 226 14 each each DET iajs-1955 226 15 i=1,2 i=1,2 NOUN iajs-1955 226 16 , , PUNCT iajs-1955 226 17 . . PUNCT iajs-1955 226 18 . . PUNCT iajs-1955 226 19 . . PUNCT iajs-1955 227 1 . . PUNCT iajs-1955 228 1 but but CCONJ iajs-1955 228 2 m m VERB iajs-1955 228 3 satisfies satisfie NOUN iajs-1955 228 4 ( ( PUNCT iajs-1955 228 5 acc acc PROPN iajs-1955 228 6 ) ) PUNCT iajs-1955 228 7 on on ADP iajs-1955 228 8 wclosed wclose VERB iajs-1955 228 9 submodules submodule NOUN iajs-1955 228 10 , , PUNCT iajs-1955 228 11 so so ADV iajs-1955 228 12 ∃ ∃ PROPN iajs-1955 228 13 𝑚 𝑚 X iajs-1955 228 14 ∈ ∈ PROPN iajs-1955 228 15 𝑍 𝑍 VERB iajs-1955 228 16 such such ADJ iajs-1955 228 17 that that SCONJ iajs-1955 228 18 𝐸 𝐸 PROPN iajs-1955 228 19 𝐸 𝐸 PROPN iajs-1955 228 20 for for ADP iajs-1955 228 21 all all DET iajs-1955 228 22 𝑛 𝑛 DET iajs-1955 228 23 𝑚. 𝑚. NOUN iajs-1955 228 24 thus thus ADV iajs-1955 228 25 m m VERB iajs-1955 228 26 satisfies satisfie NOUN iajs-1955 228 27 ( ( PUNCT iajs-1955 228 28 acc acc PROPN iajs-1955 228 29 ) ) PUNCT iajs-1955 228 30 on on ADP iajs-1955 228 31 closed closed ADJ iajs-1955 228 32 submodules submodule NOUN iajs-1955 228 33 . . PUNCT iajs-1955 229 1 ⟸ ⟸ ADJ iajs-1955 229 2 by by ADP iajs-1955 229 3 proposition proposition NOUN iajs-1955 229 4 ( ( PUNCT iajs-1955 229 5 4.5 4.5 NUM iajs-1955 229 6 ) ) PUNCT iajs-1955 229 7 . . PUNCT iajs-1955 230 1 the the DET iajs-1955 230 2 proof proof NOUN iajs-1955 230 3 of of ADP iajs-1955 230 4 the the DET iajs-1955 230 5 following follow VERB iajs-1955 230 6 proposition proposition NOUN iajs-1955 230 7 is be AUX iajs-1955 230 8 similar similar ADJ iajs-1955 230 9 to to ADP iajs-1955 230 10 proof proof NOUN iajs-1955 230 11 of of ADP iajs-1955 230 12 proposition proposition NOUN iajs-1955 230 13 ( ( PUNCT iajs-1955 230 14 4.6 4.6 NUM iajs-1955 230 15 ) ) PUNCT iajs-1955 230 16 . . PUNCT iajs-1955 231 1 proposition proposition NOUN iajs-1955 231 2 ( ( PUNCT iajs-1955 231 3 4.7 4.7 NUM iajs-1955 231 4 ) ) PUNCT iajs-1955 231 5 let let VERB iajs-1955 231 6 m m PRON iajs-1955 231 7 be be AUX iajs-1955 231 8 a a DET iajs-1955 231 9 fully fully ADV iajs-1955 231 10 semi semi ADJ iajs-1955 231 11 - - ADJ iajs-1955 231 12 prime prime ADJ iajs-1955 231 13 module module NOUN iajs-1955 231 14 . . PUNCT iajs-1955 232 1 " " PUNCT iajs-1955 232 2 then then ADV iajs-1955 232 3 m m VERB iajs-1955 232 4 satisfies satisfie NOUN iajs-1955 232 5 ( ( PUNCT iajs-1955 232 6 dcc dcc PROPN iajs-1955 232 7 ) ) PUNCT iajs-1955 232 8 on on ADP iajs-1955 232 9 closed closed ADJ iajs-1955 232 10 submodules submodule NOUN iajs-1955 232 11 if if SCONJ iajs-1955 232 12 and and CCONJ iajs-1955 232 13 only only ADV iajs-1955 232 14 if if SCONJ iajs-1955 232 15 m m NOUN iajs-1955 232 16 satisfies satisfie NOUN iajs-1955 232 17 ( ( PUNCT iajs-1955 232 18 dcc dcc PROPN iajs-1955 232 19 ) ) PUNCT iajs-1955 232 20 " " PUNCT iajs-1955 232 21 on on ADP iajs-1955 232 22 w w NOUN iajs-1955 232 23 - - PUNCT iajs-1955 232 24 closed closed ADJ iajs-1955 232 25 submodules submodule NOUN iajs-1955 232 26 . . PUNCT iajs-1955 232 27             SPACE iajs-1955 233 1 176 176 NUM iajs-1955 233 2     SPACE iajs-1955 233 3 | | ADV iajs-1955 233 4     SPACE iajs-1955 233 5 mathmatics mathmatic NOUN iajs-1955 233 6 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 233 7     SPACE iajs-1955 233 8 2018 2018 NUM iajs-1955 233 9 ) ) PUNCT iajs-1955 233 10 عام عام ADP iajs-1955 234 1 2العدد 2العدد NUM iajs-1955 234 2 ( ( PUNCT iajs-1955 234 3 31لمجلد 31لمجلد NUM iajs-1955 234 4 ا ا NOUN iajs-1955 234 5 مجلة مجلة NOUN iajs-1955 234 6 إبن إبن VERB iajs-1955 234 7 الهيثم الهيثم ADJ iajs-1955 234 8 للعلوم للعلوم NOUN iajs-1955 234 9 الصرفة الصرفة NOUN iajs-1955 235 1 و و PRON iajs-1955 235 2 التطبيقية التطبيقية ADJ iajs-1955 235 3 ibn ibn PROPN iajs-1955 235 4 al al PROPN iajs-1955 235 5 - - PUNCT iajs-1955 235 6 haitham haitham PROPN iajs-1955 235 7 jour jour X iajs-1955 235 8 . . PROPN iajs-1955 235 9 for for ADP iajs-1955 235 10 pure pure ADJ iajs-1955 235 11 & & CCONJ iajs-1955 235 12 appl appl PROPN iajs-1955 235 13 . . PUNCT iajs-1955 236 1 sci sci PROPN iajs-1955 236 2 . . PUNCT iajs-1955 236 3 vol vol NOUN iajs-1955 236 4 . . PROPN iajs-1955 237 1 31 31 NUM iajs-1955 237 2 ( ( PUNCT iajs-1955 237 3 2 2 NUM iajs-1955 237 4 ) ) PUNCT iajs-1955 237 5 2018 2018 NUM iajs-1955 237 6 proposition(4.8 proposition(4.8 NOUN iajs-1955 237 7 ) ) PUNCT iajs-1955 237 8 if if SCONJ iajs-1955 237 9 m m NOUN iajs-1955 237 10 is be AUX iajs-1955 237 11 a a DET iajs-1955 237 12 module module NOUN iajs-1955 237 13 , , PUNCT iajs-1955 237 14 and and CCONJ iajs-1955 237 15 𝐸 𝐸 PROPN iajs-1955 237 16 ⊆ ⊆ NUM iajs-1955 237 17 𝐸 𝐸 PROPN iajs-1955 237 18 ⊆ ⊆ NUM iajs-1955 237 19 . . PUNCT iajs-1955 237 20 . . PUNCT iajs-1955 237 21 . . PUNCT iajs-1955 238 1 " " PUNCT iajs-1955 238 2 be be AUX iajs-1955 238 3 ascending ascend VERB iajs-1955 238 4 chain chain NOUN iajs-1955 238 5 of of ADP iajs-1955 238 6 submodules submodule NOUN iajs-1955 238 7 such such ADJ iajs-1955 238 8 that that SCONJ iajs-1955 238 9 " " PUNCT iajs-1955 238 10 each each DET iajs-1955 238 11 weak weak ADJ iajs-1955 238 12 essential essential ADJ iajs-1955 238 13 extension extension NOUN iajs-1955 238 14 of of ADP iajs-1955 238 15 𝐸 𝐸 PROPN iajs-1955 238 16 is be AUX iajs-1955 238 17 a a DET iajs-1955 238 18 completely completely ADV iajs-1955 238 19 essential essential ADJ iajs-1955 238 20 for for ADP iajs-1955 238 21 each each DET iajs-1955 238 22 i=1,2 i=1,2 NOUN iajs-1955 238 23 , , PUNCT iajs-1955 238 24 . . PUNCT iajs-1955 238 25 . . PUNCT iajs-1955 239 1 . . PUNCT iajs-1955 240 1 , , PUNCT iajs-1955 240 2 then then ADV iajs-1955 240 3 m m VERB iajs-1955 240 4 satisfies satisfie NOUN iajs-1955 240 5 ( ( PUNCT iajs-1955 240 6 acc acc PROPN iajs-1955 240 7 ) ) PUNCT iajs-1955 240 8 on on ADP iajs-1955 240 9 w w ADJ iajs-1955 240 10 - - PUNCT iajs-1955 240 11 closed closed ADJ iajs-1955 240 12 submodules submodule NOUN iajs-1955 240 13 if if SCONJ iajs-1955 240 14 and and CCONJ iajs-1955 240 15 only only ADV iajs-1955 240 16 if if SCONJ iajs-1955 240 17 m m NOUN iajs-1955 240 18 satisfies satisfie NOUN iajs-1955 240 19 ( ( PUNCT iajs-1955 240 20 acc acc PROPN iajs-1955 240 21 ) ) PUNCT iajs-1955 240 22 on on ADP iajs-1955 240 23 closed closed ADJ iajs-1955 240 24 submodules submodule NOUN iajs-1955 240 25 . . PUNCT iajs-1955 241 1 proof proof NOUN iajs-1955 241 2 ⟹ ⟹ ADV iajs-1955 241 3 let let VERB iajs-1955 241 4 𝐸 𝐸 PRON iajs-1955 241 5 ⊆ ⊆ NUM iajs-1955 241 6 𝐸 𝐸 PROPN iajs-1955 241 7 ⊆ ⊆ NUM iajs-1955 241 8 . . PUNCT iajs-1955 241 9 . . PUNCT iajs-1955 241 10 . . PUNCT iajs-1955 242 1 " " PUNCT iajs-1955 242 2 be be AUX iajs-1955 242 3 ascending ascend VERB iajs-1955 242 4 chain chain NOUN iajs-1955 242 5 " " PUNCT iajs-1955 242 6 of of ADP iajs-1955 242 7 closed closed ADJ iajs-1955 242 8 submodules submodule NOUN iajs-1955 242 9 . . PUNCT iajs-1955 243 1 then then ADV iajs-1955 243 2 by by ADP iajs-1955 243 3 prop(2.13 prop(2.13 NOUN iajs-1955 243 4 ) ) PUNCT iajs-1955 243 5 , , PUNCT iajs-1955 243 6 𝐸 𝐸 PROPN iajs-1955 243 7 is be AUX iajs-1955 243 8 a a DET iajs-1955 243 9 w w NOUN iajs-1955 243 10 - - PUNCT iajs-1955 243 11 closed closed ADJ iajs-1955 243 12 submodules submodule NOUN iajs-1955 243 13 for for ADP iajs-1955 243 14 each each DET iajs-1955 243 15 i=1,2 i=1,2 NOUN iajs-1955 243 16 , , PUNCT iajs-1955 243 17 . . PUNCT iajs-1955 243 18 . . PUNCT iajs-1955 243 19 . . PUNCT iajs-1955 244 1 . . PUNCT iajs-1955 245 1 but but CCONJ iajs-1955 245 2 m m VERB iajs-1955 245 3 satisfies satisfie NOUN iajs-1955 245 4 ( ( PUNCT iajs-1955 245 5 acc acc PROPN iajs-1955 245 6 ) ) PUNCT iajs-1955 245 7 on on ADP iajs-1955 245 8 wclosed wclose VERB iajs-1955 245 9 submodules submodule NOUN iajs-1955 245 10 , , PUNCT iajs-1955 245 11 then then ADV iajs-1955 245 12 there there PRON iajs-1955 245 13 exists exist VERB iajs-1955 245 14 a a DET iajs-1955 245 15 non non ADJ iajs-1955 245 16 zero zero NUM iajs-1955 245 17 integer integer NOUN iajs-1955 245 18 m m VERB iajs-1955 245 19 such such ADJ iajs-1955 245 20 that that SCONJ iajs-1955 245 21 𝐸 𝐸 PROPN iajs-1955 245 22 𝐸 𝐸 PROPN iajs-1955 245 23 for for ADP iajs-1955 245 24 all all DET iajs-1955 245 25 𝑛 𝑛 DET iajs-1955 245 26 𝑚. 𝑚. NOUN iajs-1955 245 27 hence hence ADV iajs-1955 245 28 m m VERB iajs-1955 245 29 satisfies satisfie NOUN iajs-1955 245 30 ( ( PUNCT iajs-1955 245 31 acc acc PROPN iajs-1955 245 32 ) ) PUNCT iajs-1955 245 33 on on ADP iajs-1955 245 34 closed closed ADJ iajs-1955 245 35 submodules submodule NOUN iajs-1955 245 36 . . PUNCT iajs-1955 246 1 ⟸ ⟸ NOUN iajs-1955 246 2 follows follow VERB iajs-1955 246 3 by by ADP iajs-1955 246 4 proposition proposition NOUN iajs-1955 246 5 ( ( PUNCT iajs-1955 246 6 4.5 4.5 NUM iajs-1955 246 7 ) ) PUNCT iajs-1955 246 8 . . PUNCT iajs-1955 247 1 the the DET iajs-1955 247 2 proof proof NOUN iajs-1955 247 3 the the DET iajs-1955 247 4 following follow VERB iajs-1955 247 5 proposition proposition NOUN iajs-1955 247 6 is be AUX iajs-1955 247 7 similar similar ADJ iajs-1955 247 8 to to ADP iajs-1955 247 9 proof proof NOUN iajs-1955 247 10 of of ADP iajs-1955 247 11 proposition proposition NOUN iajs-1955 247 12 ( ( PUNCT iajs-1955 247 13 4.8 4.8 NUM iajs-1955 247 14 ) ) PUNCT iajs-1955 247 15 . . PUNCT iajs-1955 248 1 proposition(4.9 proposition(4.9 PROPN iajs-1955 248 2 ) ) PUNCT iajs-1955 248 3 if if SCONJ iajs-1955 248 4 m m NOUN iajs-1955 248 5 is be AUX iajs-1955 248 6 a a DET iajs-1955 248 7 module module NOUN iajs-1955 248 8 , , PUNCT iajs-1955 248 9 and and CCONJ iajs-1955 248 10 𝐸 𝐸 PROPN iajs-1955 248 11 ⊇ ⊇ NOUN iajs-1955 248 12 𝐸 𝐸 PROPN iajs-1955 248 13 ⊇ ⊇ NOUN iajs-1955 248 14 . . PUNCT iajs-1955 248 15 . . PUNCT iajs-1955 248 16 . . PUNCT iajs-1955 249 1 be be AUX iajs-1955 249 2 a a DET iajs-1955 249 3 descending descend VERB iajs-1955 249 4 chain chain NOUN iajs-1955 249 5 of of ADP iajs-1955 249 6 submodules submodule NOUN iajs-1955 249 7 such such ADJ iajs-1955 249 8 that that SCONJ iajs-1955 249 9 each each DET iajs-1955 249 10 weak weak ADJ iajs-1955 249 11 essential essential ADJ iajs-1955 249 12 extension extension NOUN iajs-1955 249 13 of of ADP iajs-1955 249 14 𝐸 𝐸 PROPN iajs-1955 249 15 is be AUX iajs-1955 249 16 a a DET iajs-1955 249 17 completely completely ADV iajs-1955 249 18 essential essential ADJ iajs-1955 249 19 for for ADP iajs-1955 249 20 each each DET iajs-1955 249 21 i=1,2 i=1,2 NOUN iajs-1955 249 22 , , PUNCT iajs-1955 249 23 . . PUNCT iajs-1955 249 24 . . PUNCT iajs-1955 249 25 . . PUNCT iajs-1955 250 1 . . PUNCT iajs-1955 251 1 then then ADV iajs-1955 251 2 m m VERB iajs-1955 251 3 satisfies satisfie NOUN iajs-1955 251 4 ( ( PUNCT iajs-1955 251 5 dcc dcc PROPN iajs-1955 251 6 ) ) PUNCT iajs-1955 251 7 on on ADP iajs-1955 251 8 w w ADJ iajs-1955 251 9 - - PUNCT iajs-1955 251 10 closed closed ADJ iajs-1955 251 11 submodules submodule NOUN iajs-1955 251 12 if if SCONJ iajs-1955 251 13 and and CCONJ iajs-1955 251 14 only only ADV iajs-1955 251 15 if if SCONJ iajs-1955 251 16 m m NOUN iajs-1955 251 17 satisfies satisfie NOUN iajs-1955 251 18 ( ( PUNCT iajs-1955 251 19 dcc dcc PROPN iajs-1955 251 20 ) ) PUNCT iajs-1955 251 21 on on ADP iajs-1955 251 22 closed closed ADJ iajs-1955 251 23 submodules submodule NOUN iajs-1955 251 24 . . PUNCT iajs-1955 252 1 proposition(4.10 proposition(4.10 X iajs-1955 252 2 ) ) PUNCT iajs-1955 252 3 if if SCONJ iajs-1955 252 4 m m NOUN iajs-1955 252 5 is be AUX iajs-1955 252 6 a a DET iajs-1955 252 7 module module NOUN iajs-1955 252 8 , , PUNCT iajs-1955 252 9 and and CCONJ iajs-1955 252 10 d d PRON iajs-1955 252 11 be be AUX iajs-1955 252 12 a a DET iajs-1955 252 13 submodule submodule NOUN iajs-1955 252 14 of of ADP iajs-1955 252 15 m m PRON iajs-1955 252 16 such such ADJ iajs-1955 252 17 that that SCONJ iajs-1955 252 18 𝐷 𝐷 PROPN iajs-1955 252 19 𝑆𝑟𝑎𝑑 𝑆𝑟𝑎𝑑 PROPN iajs-1955 252 20 𝑀 𝑀 PROPN iajs-1955 252 21 ∩ ∩ ADJ iajs-1955 252 22 𝐾 𝐾 PROPN iajs-1955 252 23 , , PUNCT iajs-1955 252 24 where where SCONJ iajs-1955 252 25 k k PROPN iajs-1955 252 26 is be AUX iajs-1955 252 27 any any DET iajs-1955 252 28 w w NOUN iajs-1955 252 29 - - PUNCT iajs-1955 252 30 closed closed ADJ iajs-1955 252 31 submodule submodule NOUN iajs-1955 252 32 in in ADP iajs-1955 252 33 m. m. NOUN iajs-1955 252 34 if if SCONJ iajs-1955 252 35 𝑴 𝑴 NOUN iajs-1955 252 36 𝑫 𝑫 NOUN iajs-1955 252 37 satisfies satisfie NOUN iajs-1955 252 38 ( ( PUNCT iajs-1955 252 39 dcc dcc PROPN iajs-1955 252 40 ) ) PUNCT iajs-1955 252 41 on on ADP iajs-1955 252 42 w w ADJ iajs-1955 252 43 - - PUNCT iajs-1955 252 44 closed closed ADJ iajs-1955 252 45 submodules submodule NOUN iajs-1955 252 46 , , PUNCT iajs-1955 252 47 then then ADV iajs-1955 252 48 m m VERB iajs-1955 252 49 satisfies satisfie NOUN iajs-1955 252 50 ( ( PUNCT iajs-1955 252 51 dcc dcc PROPN iajs-1955 252 52 ) ) PUNCT iajs-1955 252 53 on on ADP iajs-1955 252 54 w w ADJ iajs-1955 252 55 - - PUNCT iajs-1955 252 56 closed closed ADJ iajs-1955 252 57 submodules submodule NOUN iajs-1955 252 58 . . PUNCT iajs-1955 253 1 proof proof NOUN iajs-1955 253 2 let let VERB iajs-1955 253 3 𝐸 𝐸 PROPN iajs-1955 253 4 ⊇ ⊇ NOUN iajs-1955 253 5 𝐸 𝐸 PROPN iajs-1955 253 6 ⊇ ⊇ NOUN iajs-1955 253 7 . . PUNCT iajs-1955 253 8 . . PUNCT iajs-1955 254 1 . . PUNCT iajs-1955 255 1 be be AUX iajs-1955 255 2 a a DET iajs-1955 255 3 descending descend VERB iajs-1955 255 4 chain chain NOUN iajs-1955 255 5 of of ADP iajs-1955 255 6 w w NOUN iajs-1955 255 7 - - PUNCT iajs-1955 255 8 closed closed ADJ iajs-1955 255 9 submodules submodule NOUN iajs-1955 255 10 in in ADP iajs-1955 255 11 m. m. NOUN iajs-1955 255 12 since since SCONJ iajs-1955 255 13 𝐸 𝐸 PROPN iajs-1955 255 14 is be AUX iajs-1955 255 15 a a DET iajs-1955 255 16 w w NOUN iajs-1955 255 17 - - PUNCT iajs-1955 255 18 closed closed ADJ iajs-1955 255 19 submodule submodule NOUN iajs-1955 255 20 in in ADP iajs-1955 255 21 m m PROPN iajs-1955 255 22 for for ADP iajs-1955 255 23 each each DET iajs-1955 255 24 i=1,2 i=1,2 NOUN iajs-1955 255 25 , , PUNCT iajs-1955 255 26 . . PUNCT iajs-1955 255 27 . . PUNCT iajs-1955 256 1 . . PUNCT iajs-1955 257 1 , , PUNCT iajs-1955 257 2 and and CCONJ iajs-1955 257 3 𝐷 𝐷 PROPN iajs-1955 257 4 𝑆𝑟𝑎𝑑 𝑆𝑟𝑎𝑑 PROPN iajs-1955 257 5 𝑀 𝑀 PROPN iajs-1955 257 6 ∩ ∩ NOUN iajs-1955 257 7 𝐸 𝐸 PROPN iajs-1955 257 8 then then ADV iajs-1955 257 9 by by ADP iajs-1955 257 10 corollary(2.32 corollary(2.32 NOUN iajs-1955 257 11 ) ) PUNCT iajs-1955 257 12 , , PUNCT iajs-1955 257 13 we we PRON iajs-1955 257 14 have have AUX iajs-1955 257 15 𝑬𝒊 𝑬𝒊 AUX iajs-1955 257 16 𝑫 𝑫 NOUN iajs-1955 257 17 is be AUX iajs-1955 257 18 a a DET iajs-1955 257 19 w w NOUN iajs-1955 257 20 - - PUNCT iajs-1955 257 21 closed closed ADJ iajs-1955 257 22 submodule submodule NOUN iajs-1955 257 23 in in ADP iajs-1955 257 24 𝑴 𝑴 PROPN iajs-1955 257 25 𝑫 𝑫 NOUN iajs-1955 257 26 for for ADP iajs-1955 257 27 each each DET iajs-1955 257 28 i=1,2 i=1,2 NOUN iajs-1955 257 29 , , PUNCT iajs-1955 257 30 . . PUNCT iajs-1955 257 31 . . PUNCT iajs-1955 257 32 . . PUNCT iajs-1955 258 1 . . PUNCT iajs-1955 259 1 hence hence ADV iajs-1955 259 2 ⊇ ⊇ PROPN iajs-1955 259 3 ⊇ ⊇ PROPN iajs-1955 259 4 . . PUNCT iajs-1955 259 5 . . PUNCT iajs-1955 259 6 . . PUNCT iajs-1955 260 1 , , PUNCT iajs-1955 260 2 is be AUX iajs-1955 260 3 a a DET iajs-1955 260 4 descending descend VERB iajs-1955 260 5 chain chain NOUN iajs-1955 260 6 of of ADP iajs-1955 260 7 w w NOUN iajs-1955 260 8 - - PUNCT iajs-1955 260 9 closed closed ADJ iajs-1955 260 10 submodules submodule NOUN iajs-1955 260 11 in in ADP iajs-1955 260 12 𝑴 𝑴 NOUN iajs-1955 260 13 𝑫 𝑫 NOUN iajs-1955 260 14 . . PUNCT iajs-1955 261 1 but but CCONJ iajs-1955 261 2 𝑴 𝑴 NOUN iajs-1955 261 3 𝑫 𝑫 AUX iajs-1955 261 4 satisfies satisfie NOUN iajs-1955 261 5 ( ( PUNCT iajs-1955 261 6 dcc dcc PROPN iajs-1955 261 7 ) ) PUNCT iajs-1955 261 8 on on ADP iajs-1955 261 9 w w ADJ iajs-1955 261 10 - - PUNCT iajs-1955 261 11 closed closed ADJ iajs-1955 261 12 submodules submodule NOUN iajs-1955 261 13 , , PUNCT iajs-1955 261 14 so so SCONJ iajs-1955 261 15 there there PRON iajs-1955 261 16 exists exist VERB iajs-1955 261 17 a a DET iajs-1955 261 18 positive positive ADJ iajs-1955 261 19 integer integer NOUN iajs-1955 261 20 m m VERB iajs-1955 261 21 such such ADJ iajs-1955 261 22 that that SCONJ iajs-1955 261 23 𝑬𝒏 𝑬𝒏 ADP iajs-1955 261 24 𝑫 𝑫 AUX iajs-1955 261 25 𝑬𝒎 𝑬𝒎 PROPN iajs-1955 261 26 𝑫 𝑫 NOUN iajs-1955 261 27 for for ADP iajs-1955 261 28 each each DET iajs-1955 261 29 𝑛 𝑛 DET iajs-1955 261 30 𝑚. 𝑚. NOUN iajs-1955 261 31 so so ADV iajs-1955 261 32 , , PUNCT iajs-1955 261 33 that that SCONJ iajs-1955 261 34 𝐸 𝐸 PROPN iajs-1955 261 35 𝐸 𝐸 PROPN iajs-1955 261 36 for for ADP iajs-1955 261 37 each each DET iajs-1955 261 38 𝑛 𝑛 DET iajs-1955 261 39 𝑚. 𝑚. NOUN iajs-1955 261 40 thus thus ADV iajs-1955 261 41 m m VERB iajs-1955 261 42 satisfies satisfie NOUN iajs-1955 261 43 ( ( PUNCT iajs-1955 261 44 dcc dcc PROPN iajs-1955 261 45 ) ) PUNCT iajs-1955 261 46 on on ADP iajs-1955 261 47 w w ADJ iajs-1955 261 48 - - PUNCT iajs-1955 261 49 closed closed ADJ iajs-1955 261 50 submodules submodule NOUN iajs-1955 261 51 . . PUNCT iajs-1955 262 1 proposition(4.11 proposition(4.11 NOUN iajs-1955 262 2 ) ) PUNCT iajs-1955 262 3 if if SCONJ iajs-1955 262 4 m m NOUN iajs-1955 262 5 is be AUX iajs-1955 262 6 a a DET iajs-1955 262 7 module module NOUN iajs-1955 262 8 , , PUNCT iajs-1955 262 9 and and CCONJ iajs-1955 262 10 d d PRON iajs-1955 262 11 be be AUX iajs-1955 262 12 a a DET iajs-1955 262 13 submodule submodule NOUN iajs-1955 262 14 of of ADP iajs-1955 262 15 m m PRON iajs-1955 262 16 such such ADJ iajs-1955 262 17 that that SCONJ iajs-1955 262 18 𝐷 𝐷 PROPN iajs-1955 262 19 𝑆𝑟𝑎𝑑 𝑆𝑟𝑎𝑑 PROPN iajs-1955 262 20 𝑀 𝑀 PROPN iajs-1955 262 21 ∩ ∩ ADJ iajs-1955 262 22 𝐾 𝐾 PROPN iajs-1955 262 23 , , PUNCT iajs-1955 262 24 where where SCONJ iajs-1955 262 25 k k PROPN iajs-1955 262 26 is be AUX iajs-1955 262 27 any any DET iajs-1955 262 28 w w NOUN iajs-1955 262 29 - - PUNCT iajs-1955 262 30 closed closed ADJ iajs-1955 262 31 submodule submodule NOUN iajs-1955 262 32 in in ADP iajs-1955 262 33 m. m. NOUN iajs-1955 262 34 if if SCONJ iajs-1955 262 35 𝑴 𝑴 NOUN iajs-1955 262 36 𝑫 𝑫 NOUN iajs-1955 262 37 satisfies satisfie NOUN iajs-1955 262 38 ( ( PUNCT iajs-1955 262 39 acc acc PROPN iajs-1955 262 40 ) ) PUNCT iajs-1955 262 41 on on ADP iajs-1955 262 42 w w ADJ iajs-1955 262 43 - - PUNCT iajs-1955 262 44 closed closed ADJ iajs-1955 262 45 submodules submodule NOUN iajs-1955 262 46 , , PUNCT iajs-1955 262 47 then then ADV iajs-1955 262 48 m m VERB iajs-1955 262 49 satisfies satisfie NOUN iajs-1955 262 50 ( ( PUNCT iajs-1955 262 51 acc acc PROPN iajs-1955 262 52 ) ) PUNCT iajs-1955 262 53 on on ADP iajs-1955 262 54 w w ADJ iajs-1955 262 55 - - PUNCT iajs-1955 262 56 closed closed ADJ iajs-1955 262 57 submodules submodule NOUN iajs-1955 262 58 . . PUNCT iajs-1955 263 1 proof proof NOUN iajs-1955 263 2 similar similar ADJ iajs-1955 263 3 to to ADP iajs-1955 263 4 proof proof NOUN iajs-1955 263 5 of of ADP iajs-1955 263 6 proposition proposition NOUN iajs-1955 263 7 ( ( PUNCT iajs-1955 263 8 4.10 4.10 NUM iajs-1955 263 9 ) ) PUNCT iajs-1955 263 10 . . PUNCT iajs-1955 264 1 proposition(4.12 proposition(4.12 X iajs-1955 264 2 ) ) PUNCT iajs-1955 264 3 if if SCONJ iajs-1955 264 4 𝑋 𝑋 PROPN iajs-1955 264 5 𝑋 𝑋 PROPN iajs-1955 264 6 ⨁ ⨁ PROPN iajs-1955 264 7 𝑋 𝑋 PROPN iajs-1955 264 8 is be AUX iajs-1955 264 9 a a DET iajs-1955 264 10 module module NOUN iajs-1955 264 11 , , PUNCT iajs-1955 264 12 where where SCONJ iajs-1955 264 13 𝑋 𝑋 NOUN iajs-1955 264 14 and and CCONJ iajs-1955 264 15 𝑋 𝑋 PROPN iajs-1955 264 16 are be AUX iajs-1955 264 17 submodules submodule NOUN iajs-1955 264 18 of of ADP iajs-1955 264 19 x x PRON iajs-1955 264 20 , , PUNCT iajs-1955 264 21 provided provide VERB iajs-1955 264 22 that that SCONJ iajs-1955 264 23 ann ann PROPN iajs-1955 264 24 𝑋 𝑋 PROPN iajs-1955 264 25 + + CCONJ iajs-1955 264 26 ann ann PROPN iajs-1955 264 27 𝑋 𝑋 PROPN iajs-1955 264 28 = = SYM iajs-1955 264 29 r r NOUN iajs-1955 264 30 , , PUNCT iajs-1955 264 31 and and CCONJ iajs-1955 264 32 all all DET iajs-1955 264 33 weak weak ADJ iajs-1955 264 34 essential essential ADJ iajs-1955 264 35 extensions extension NOUN iajs-1955 264 36 of of ADP iajs-1955 264 37 𝐸 𝐸 PROPN iajs-1955 264 38 ⨁ ⨁ PROPN iajs-1955 264 39 𝑋 𝑋 PROPN iajs-1955 264 40 ( ( PUNCT iajs-1955 264 41 or or CCONJ iajs-1955 264 42 𝑋 𝑋 PROPN iajs-1955 264 43 ⨁ ⨁ PROPN iajs-1955 264 44 𝐸 𝐸 PROPN iajs-1955 264 45             SPACE iajs-1955 264 46 177 177 NUM iajs-1955 264 47     SPACE iajs-1955 264 48 | | ADV iajs-1955 264 49     SPACE iajs-1955 264 50 mathmatics mathmatic NOUN iajs-1955 264 51 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 264 52     SPACE iajs-1955 264 53 2018 2018 NUM iajs-1955 264 54 ) ) PUNCT iajs-1955 264 55 عام عام ADP iajs-1955 264 56 2العدد 2العدد NUM iajs-1955 264 57 ( ( PUNCT iajs-1955 264 58 31لمجلد 31لمجلد NUM iajs-1955 264 59 ا ا NOUN iajs-1955 264 60 مجلة مجلة NOUN iajs-1955 264 61 إبن إبن VERB iajs-1955 264 62 الهيثم الهيثم ADJ iajs-1955 264 63 للعلوم للعلوم NOUN iajs-1955 264 64 الصرفة الصرفة NOUN iajs-1955 265 1 و و PRON iajs-1955 265 2 التطبيقية التطبيقية ADJ iajs-1955 265 3 ibn ibn PROPN iajs-1955 265 4 al al PROPN iajs-1955 265 5 - - PUNCT iajs-1955 265 6 haitham haitham PROPN iajs-1955 265 7 jour jour X iajs-1955 265 8 . . PROPN iajs-1955 265 9 for for ADP iajs-1955 265 10 pure pure ADJ iajs-1955 265 11 & & CCONJ iajs-1955 265 12 appl appl PROPN iajs-1955 265 13 . . PUNCT iajs-1955 266 1 sci sci PROPN iajs-1955 266 2 . . PUNCT iajs-1955 266 3 vol vol NOUN iajs-1955 266 4 . . PROPN iajs-1955 267 1 31 31 NUM iajs-1955 267 2 ( ( PUNCT iajs-1955 267 3 2 2 NUM iajs-1955 267 4 ) ) PUNCT iajs-1955 267 5 2018 2018 NUM iajs-1955 267 6 ) ) PUNCT iajs-1955 267 7 are be AUX iajs-1955 267 8 completely completely ADV iajs-1955 267 9 essential essential ADJ iajs-1955 267 10 modules module NOUN iajs-1955 267 11 where where SCONJ iajs-1955 267 12 𝐸 𝐸 PROPN iajs-1955 267 13 is be AUX iajs-1955 267 14 a a DET iajs-1955 267 15 non non ADJ iajs-1955 267 16 zero zero NUM iajs-1955 267 17 w w ADV iajs-1955 267 18 - - PUNCT iajs-1955 267 19 closed closed ADJ iajs-1955 267 20 submodule submodule NOUN iajs-1955 267 21 in in ADP iajs-1955 267 22 𝑋 𝑋 PROPN iajs-1955 267 23 ( ( PUNCT iajs-1955 267 24 or or CCONJ iajs-1955 267 25 𝑋 𝑋 PROPN iajs-1955 267 26 ) ) PUNCT iajs-1955 267 27 for for ADP iajs-1955 267 28 each each DET iajs-1955 267 29 i=1,2 i=1,2 NOUN iajs-1955 267 30 , , PUNCT iajs-1955 267 31 . . PUNCT iajs-1955 267 32 . . PUNCT iajs-1955 267 33 . . PUNCT iajs-1955 267 34 . . PUNCT iajs-1955 268 1 if if SCONJ iajs-1955 268 2 x x PRON iajs-1955 268 3 satisfies satisfie NOUN iajs-1955 268 4 ( ( PUNCT iajs-1955 268 5 dcc dcc PROPN iajs-1955 268 6 ) ) PUNCT iajs-1955 268 7 on on ADP iajs-1955 268 8 w w ADJ iajs-1955 268 9 - - PUNCT iajs-1955 268 10 closed closed ADJ iajs-1955 268 11 submodules submodule NOUN iajs-1955 268 12 , , PUNCT iajs-1955 268 13 then then ADV iajs-1955 268 14 𝑋 𝑋 PROPN iajs-1955 268 15 ( ( PUNCT iajs-1955 268 16 𝑜𝑟 𝑜𝑟 ADP iajs-1955 268 17 𝑋 𝑋 NOUN iajs-1955 268 18 ) ) PUNCT iajs-1955 268 19 satisfies satisfie NOUN iajs-1955 268 20 ( ( PUNCT iajs-1955 268 21 dcc dcc PROPN iajs-1955 268 22 ) ) PUNCT iajs-1955 268 23 on on ADP iajs-1955 268 24 nonzero nonzero PROPN iajs-1955 268 25 w w PROPN iajs-1955 268 26 - - PUNCT iajs-1955 268 27 closed closed ADJ iajs-1955 268 28 submodules submodule NOUN iajs-1955 268 29 . . PUNCT iajs-1955 269 1 proof proof NOUN iajs-1955 269 2 let let VERB iajs-1955 269 3 𝐸 𝐸 PROPN iajs-1955 269 4 ⊇ ⊇ NOUN iajs-1955 269 5 𝐸 𝐸 PROPN iajs-1955 269 6 ⊇ ⊇ NOUN iajs-1955 269 7 . . PUNCT iajs-1955 269 8 . . PUNCT iajs-1955 269 9 . . PUNCT iajs-1955 270 1 " " PUNCT iajs-1955 270 2 be be AUX iajs-1955 270 3 a a DET iajs-1955 270 4 descending descend VERB iajs-1955 270 5 chain chain NOUN iajs-1955 270 6 " " PUNCT iajs-1955 270 7 of of ADP iajs-1955 270 8 a a DET iajs-1955 270 9 non non ADJ iajs-1955 270 10 - - ADJ iajs-1955 270 11 zero zero NUM iajs-1955 270 12 w w ADJ iajs-1955 270 13 - - PUNCT iajs-1955 270 14 closed closed ADJ iajs-1955 270 15 submodules submodule NOUN iajs-1955 270 16 of of ADP iajs-1955 270 17 𝑋 𝑋 PROPN iajs-1955 270 18 . . PUNCT iajs-1955 271 1 if if SCONJ iajs-1955 271 2 𝑋 𝑋 PROPN iajs-1955 271 3 is be AUX iajs-1955 271 4 equal equal ADJ iajs-1955 271 5 to to ADP iajs-1955 271 6 zero zero NUM iajs-1955 271 7 , , PUNCT iajs-1955 271 8 then then ADV iajs-1955 271 9 x=𝑋 x=𝑋 PROPN iajs-1955 271 10 and and CCONJ iajs-1955 271 11 this this PRON iajs-1955 271 12 , , PUNCT iajs-1955 271 13 implies imply VERB iajs-1955 271 14 that that SCONJ iajs-1955 271 15 𝑋 𝑋 PROPN iajs-1955 271 16 satisfies satisfie NOUN iajs-1955 271 17 ( ( PUNCT iajs-1955 271 18 dcc dcc PROPN iajs-1955 271 19 ) ) PUNCT iajs-1955 271 20 on on ADP iajs-1955 271 21 non non ADJ iajs-1955 271 22 - - ADJ iajs-1955 271 23 zero zero NUM iajs-1955 271 24 w w ADJ iajs-1955 271 25 - - PUNCT iajs-1955 271 26 closed closed ADJ iajs-1955 271 27 submodules submodule NOUN iajs-1955 271 28 . . PUNCT iajs-1955 272 1 otherwise otherwise ADV iajs-1955 272 2 , , PUNCT iajs-1955 272 3 since since SCONJ iajs-1955 272 4 𝐸 𝐸 PROPN iajs-1955 272 5 is be AUX iajs-1955 272 6 a a DET iajs-1955 272 7 non non ADJ iajs-1955 272 8 - - ADJ iajs-1955 272 9 zero zero NUM iajs-1955 272 10 w w ADJ iajs-1955 272 11 - - PUNCT iajs-1955 272 12 closed closed ADJ iajs-1955 272 13 submodule submodule NOUN iajs-1955 272 14 in in ADP iajs-1955 272 15 𝑋 𝑋 PROPN iajs-1955 272 16 , , PUNCT iajs-1955 272 17 and and CCONJ iajs-1955 272 18 𝑋 𝑋 NOUN iajs-1955 272 19 is be AUX iajs-1955 272 20 a a DET iajs-1955 272 21 wclosed wclose VERB iajs-1955 272 22 in in ADP iajs-1955 272 23 𝑋 𝑋 PROPN iajs-1955 272 24 , , PUNCT iajs-1955 272 25 so so ADV iajs-1955 272 26 by by ADP iajs-1955 272 27 proposition(2.26 proposition(2.26 NOUN iajs-1955 272 28 ) ) PUNCT iajs-1955 272 29 , , PUNCT iajs-1955 272 30 𝐸 𝐸 PROPN iajs-1955 272 31 ⨁ ⨁ PROPN iajs-1955 272 32 𝑋 𝑋 PROPN iajs-1955 272 33 is be AUX iajs-1955 272 34 a a DET iajs-1955 272 35 w w NOUN iajs-1955 272 36 - - PUNCT iajs-1955 272 37 closed closed ADJ iajs-1955 272 38 submodule submodule NOUN iajs-1955 272 39 in in ADP iajs-1955 272 40 x x PUNCT iajs-1955 272 41 for for ADP iajs-1955 272 42 each each DET iajs-1955 272 43 i=1,2 i=1,2 NOUN iajs-1955 272 44 , , PUNCT iajs-1955 272 45 . . PUNCT iajs-1955 272 46 . . PUNCT iajs-1955 273 1 . . PUNCT iajs-1955 274 1 , , PUNCT iajs-1955 274 2 𝐸 𝐸 PROPN iajs-1955 274 3 ⨁𝑋 ⨁𝑋 PROPN iajs-1955 274 4 ⊇ ⊇ ADJ iajs-1955 274 5 𝐸 𝐸 ADJ iajs-1955 274 6 ⨁𝑋 ⨁𝑋 VERB iajs-1955 274 7 ⊇. ⊇. NOUN iajs-1955 274 8 . . PUNCT iajs-1955 274 9 . . PUNCT iajs-1955 275 1 , , PUNCT iajs-1955 275 2 " " PUNCT iajs-1955 275 3 is be AUX iajs-1955 275 4 a a DET iajs-1955 275 5 descending descend VERB iajs-1955 275 6 chain chain NOUN iajs-1955 275 7 " " PUNCT iajs-1955 275 8 of of ADP iajs-1955 275 9 w w NOUN iajs-1955 275 10 - - PUNCT iajs-1955 275 11 closed closed ADJ iajs-1955 275 12 submodule submodule NOUN iajs-1955 275 13 in in ADP iajs-1955 275 14 x. x. PROPN iajs-1955 275 15 but but CCONJ iajs-1955 275 16 x x NUM iajs-1955 275 17 satisfies satisfie NOUN iajs-1955 275 18 ( ( PUNCT iajs-1955 275 19 dcc dcc PROPN iajs-1955 275 20 ) ) PUNCT iajs-1955 275 21 on on ADP iajs-1955 275 22 w w ADJ iajs-1955 275 23 - - PUNCT iajs-1955 275 24 closed closed ADJ iajs-1955 275 25 submodules submodule NOUN iajs-1955 275 26 , , PUNCT iajs-1955 275 27 then then ADV iajs-1955 275 28 there there PRON iajs-1955 275 29 exists exist VERB iajs-1955 275 30 a a DET iajs-1955 275 31 positive positive ADJ iajs-1955 275 32 integer integer NOUN iajs-1955 275 33 m m VERB iajs-1955 275 34 such such ADJ iajs-1955 275 35 that that SCONJ iajs-1955 275 36 𝐸 𝐸 PROPN iajs-1955 275 37 ⨁𝑋 ⨁𝑋 NOUN iajs-1955 275 38 𝐸 𝐸 ADJ iajs-1955 275 39 ⨁𝑋 ⨁𝑋 NOUN iajs-1955 275 40 for for ADP iajs-1955 275 41 all all DET iajs-1955 275 42 𝑛 𝑛 DET iajs-1955 275 43 𝑚. 𝑚. NOUN iajs-1955 275 44 thus thus ADV iajs-1955 275 45 𝐸 𝐸 PROPN iajs-1955 275 46 𝐸 𝐸 PROPN iajs-1955 275 47 for for ADP iajs-1955 275 48 all all DET iajs-1955 275 49 𝑛 𝑛 DET iajs-1955 275 50 𝑚. 𝑚. NOUN iajs-1955 275 51 thus thus ADV iajs-1955 275 52 𝑋 𝑋 ADJ iajs-1955 275 53 satisfies satisfie NOUN iajs-1955 275 54 ( ( PUNCT iajs-1955 275 55 dcc dcc PROPN iajs-1955 275 56 ) ) PUNCT iajs-1955 275 57 on on ADP iajs-1955 275 58 w w ADJ iajs-1955 275 59 - - PUNCT iajs-1955 275 60 closed closed ADJ iajs-1955 275 61 submodule submodule NOUN iajs-1955 275 62 . . PUNCT iajs-1955 276 1 similarly similarly ADV iajs-1955 276 2 we we PRON iajs-1955 276 3 can can AUX iajs-1955 276 4 prove prove VERB iajs-1955 276 5 that that SCONJ iajs-1955 276 6 𝑋 𝑋 PROPN iajs-1955 276 7 satisfies satisfie NOUN iajs-1955 276 8 ( ( PUNCT iajs-1955 276 9 dcc dcc PROPN iajs-1955 276 10 ) ) PUNCT iajs-1955 276 11 on on ADP iajs-1955 276 12 w w ADJ iajs-1955 276 13 - - PUNCT iajs-1955 276 14 closed closed ADJ iajs-1955 276 15 submodule submodule NOUN iajs-1955 276 16 . . PUNCT iajs-1955 277 1 proposition(4.13 proposition(4.13 X iajs-1955 277 2 ) ) PUNCT iajs-1955 278 1 if if SCONJ iajs-1955 278 2 𝑋 𝑋 PROPN iajs-1955 278 3 𝑋 𝑋 PROPN iajs-1955 278 4 ⨁ ⨁ PROPN iajs-1955 278 5 𝑋 𝑋 PROPN iajs-1955 278 6 is be AUX iajs-1955 278 7 a a DET iajs-1955 278 8 module module NOUN iajs-1955 278 9 , , PUNCT iajs-1955 278 10 where where SCONJ iajs-1955 278 11 𝑋 𝑋 NOUN iajs-1955 278 12 and and CCONJ iajs-1955 278 13 𝑋 𝑋 PROPN iajs-1955 278 14 are be AUX iajs-1955 278 15 submodules submodule NOUN iajs-1955 278 16 of of ADP iajs-1955 278 17 x x PRON iajs-1955 278 18 , , PUNCT iajs-1955 278 19 provided provide VERB iajs-1955 278 20 that that SCONJ iajs-1955 278 21 ann ann PROPN iajs-1955 278 22 𝑋 𝑋 PROPN iajs-1955 278 23 + + CCONJ iajs-1955 278 24 ann ann PROPN iajs-1955 278 25 𝑋 𝑋 PROPN iajs-1955 278 26 = = SYM iajs-1955 278 27 r r NOUN iajs-1955 278 28 , , PUNCT iajs-1955 278 29 and and CCONJ iajs-1955 278 30 all all DET iajs-1955 278 31 weak weak ADJ iajs-1955 278 32 essential essential ADJ iajs-1955 278 33 extensions extension NOUN iajs-1955 278 34 of of ADP iajs-1955 278 35 𝐸 𝐸 PROPN iajs-1955 278 36 ⨁ ⨁ PROPN iajs-1955 278 37 𝑋 𝑋 PROPN iajs-1955 278 38 ( ( PUNCT iajs-1955 278 39 or or CCONJ iajs-1955 278 40 𝑋 𝑋 PROPN iajs-1955 278 41 ⨁ ⨁ PROPN iajs-1955 278 42 𝐸 𝐸 PROPN iajs-1955 278 43 ) ) PUNCT iajs-1955 278 44 are be AUX iajs-1955 278 45 completely completely ADV iajs-1955 278 46 essential essential ADJ iajs-1955 278 47 modules module NOUN iajs-1955 278 48 where where SCONJ iajs-1955 278 49 𝐸 𝐸 PROPN iajs-1955 278 50 is be AUX iajs-1955 278 51 a a DET iajs-1955 278 52 non non ADJ iajs-1955 278 53 zero zero NUM iajs-1955 278 54 w w ADV iajs-1955 278 55 - - PUNCT iajs-1955 278 56 closed closed ADJ iajs-1955 278 57 submodule submodule NOUN iajs-1955 278 58 in in ADP iajs-1955 278 59 𝑋 𝑋 PROPN iajs-1955 278 60 ( ( PUNCT iajs-1955 278 61 or or CCONJ iajs-1955 278 62 𝑋 𝑋 PROPN iajs-1955 278 63 ) ) PUNCT iajs-1955 278 64 for for ADP iajs-1955 278 65 each each DET iajs-1955 278 66 i=1,2 i=1,2 NOUN iajs-1955 278 67 , , PUNCT iajs-1955 278 68 . . PUNCT iajs-1955 278 69 . . PUNCT iajs-1955 279 1 . . PUNCT iajs-1955 280 1 . . PUNCT iajs-1955 281 1 if if SCONJ iajs-1955 281 2 x x PRON iajs-1955 281 3 satisfies satisfie NOUN iajs-1955 281 4 ( ( PUNCT iajs-1955 281 5 acc acc PROPN iajs-1955 281 6 ) ) PUNCT iajs-1955 281 7 on on ADP iajs-1955 281 8 w w ADJ iajs-1955 281 9 - - PUNCT iajs-1955 281 10 closed closed ADJ iajs-1955 281 11 submodules submodule NOUN iajs-1955 281 12 , , PUNCT iajs-1955 281 13 then then ADV iajs-1955 281 14 𝑋 𝑋 PROPN iajs-1955 281 15 ( ( PUNCT iajs-1955 281 16 𝑜𝑟 𝑜𝑟 ADP iajs-1955 281 17 𝑋 𝑋 NOUN iajs-1955 281 18 ) ) PUNCT iajs-1955 281 19 satisfies satisfie NOUN iajs-1955 281 20 ( ( PUNCT iajs-1955 281 21 acc acc PROPN iajs-1955 281 22 ) ) PUNCT iajs-1955 281 23 on on ADP iajs-1955 281 24 non non ADJ iajs-1955 281 25 zero zero NUM iajs-1955 281 26 w w ADJ iajs-1955 281 27 - - PUNCT iajs-1955 281 28 closed closed ADJ iajs-1955 281 29 submodules submodule NOUN iajs-1955 281 30 . . PUNCT iajs-1955 282 1 proof proof NOUN iajs-1955 282 2 similarly similarly ADV iajs-1955 282 3 as as ADP iajs-1955 282 4 in in ADP iajs-1955 282 5 proposition proposition NOUN iajs-1955 282 6 ( ( PUNCT iajs-1955 282 7 4.12 4.12 NUM iajs-1955 282 8 ) ) PUNCT iajs-1955 282 9 . . PUNCT iajs-1955 283 1 we we PRON iajs-1955 283 2 end end VERB iajs-1955 283 3 this this DET iajs-1955 283 4 section section NOUN iajs-1955 283 5 by by ADP iajs-1955 283 6 the the DET iajs-1955 283 7 following follow VERB iajs-1955 283 8 propositions proposition NOUN iajs-1955 283 9 . . PUNCT iajs-1955 284 1 proposition(4.14 proposition(4.14 X iajs-1955 284 2 ) ) PUNCT iajs-1955 285 1 " " PUNCT iajs-1955 285 2 if if SCONJ iajs-1955 285 3 m m NOUN iajs-1955 285 4 is be AUX iajs-1955 285 5 a a DET iajs-1955 285 6 finitely finitely ADV iajs-1955 285 7 generated generate VERB iajs-1955 285 8 , , PUNCT iajs-1955 285 9 faithful faithful ADJ iajs-1955 285 10 and and CCONJ iajs-1955 285 11 multiplication multiplication NOUN iajs-1955 285 12 module module NOUN iajs-1955 285 13 , , PUNCT iajs-1955 285 14 and and CCONJ iajs-1955 285 15 m m PROPN iajs-1955 285 16 satisfies satisfie NOUN iajs-1955 285 17 condition condition NOUN iajs-1955 285 18 ∗ ∗ NOUN iajs-1955 285 19 " " PUNCT iajs-1955 285 20 , , PUNCT iajs-1955 285 21 then then ADV iajs-1955 285 22 m m VERB iajs-1955 285 23 satisfies satisfie NOUN iajs-1955 285 24 ( ( PUNCT iajs-1955 285 25 dcc dcc PROPN iajs-1955 285 26 ) ) PUNCT iajs-1955 285 27 on on ADP iajs-1955 285 28 w w ADJ iajs-1955 285 29 - - PUNCT iajs-1955 285 30 closed closed ADJ iajs-1955 285 31 submodules submodule NOUN iajs-1955 285 32 , , PUNCT iajs-1955 285 33 if if SCONJ iajs-1955 285 34 and and CCONJ iajs-1955 285 35 only only ADV iajs-1955 285 36 if if SCONJ iajs-1955 285 37 r r NOUN iajs-1955 285 38 satisfies satisfie NOUN iajs-1955 285 39 ( ( PUNCT iajs-1955 285 40 dcc dcc PROPN iajs-1955 285 41 ) ) PUNCT iajs-1955 285 42 on on ADP iajs-1955 285 43 w w ADJ iajs-1955 285 44 - - PUNCT iajs-1955 285 45 closed closed ADJ iajs-1955 285 46 ideals ideal NOUN iajs-1955 285 47 . . PUNCT iajs-1955 286 1 proof proof NOUN iajs-1955 286 2 ⟹ ⟹ ADV iajs-1955 286 3 let let VERB iajs-1955 286 4 𝐿 𝐿 PROPN iajs-1955 286 5 ⊇ ⊇ PROPN iajs-1955 286 6 𝐿 𝐿 PROPN iajs-1955 286 7 ⊇ ⊇ NOUN iajs-1955 286 8 . . PUNCT iajs-1955 286 9 . . PUNCT iajs-1955 287 1 . . PUNCT iajs-1955 288 1 , , PUNCT iajs-1955 288 2 " " PUNCT iajs-1955 288 3 be be AUX iajs-1955 288 4 a a DET iajs-1955 288 5 descending descend VERB iajs-1955 288 6 chain chain NOUN iajs-1955 288 7 " " PUNCT iajs-1955 288 8 of of ADP iajs-1955 288 9 w w NOUN iajs-1955 288 10 - - PUNCT iajs-1955 288 11 closed closed ADJ iajs-1955 288 12 ideals ideal NOUN iajs-1955 288 13 in in ADP iajs-1955 288 14 r. r. PROPN iajs-1955 288 15 since since SCONJ iajs-1955 288 16 𝐿 𝐿 PROPN iajs-1955 288 17 is be AUX iajs-1955 288 18 a a DET iajs-1955 288 19 w w NOUN iajs-1955 288 20 - - PUNCT iajs-1955 288 21 closed closed ADJ iajs-1955 288 22 ideal ideal NOUN iajs-1955 288 23 in in ADP iajs-1955 288 24 r r NOUN iajs-1955 288 25 for for ADP iajs-1955 288 26 each each DET iajs-1955 288 27 i=1,2 i=1,2 NOUN iajs-1955 288 28 , , PUNCT iajs-1955 288 29 . . PUNCT iajs-1955 288 30 . . PUNCT iajs-1955 288 31 . . PUNCT iajs-1955 288 32 . . PUNCT iajs-1955 289 1 then then ADV iajs-1955 289 2 by by ADP iajs-1955 289 3 cor(3.8 cor(3.8 X iajs-1955 289 4 ) ) PUNCT iajs-1955 289 5 𝐿 𝐿 PROPN iajs-1955 289 6 𝑀 𝑀 PROPN iajs-1955 289 7 is be AUX iajs-1955 289 8 a a DET iajs-1955 289 9 w w NOUN iajs-1955 289 10 - - PUNCT iajs-1955 289 11 closed closed ADJ iajs-1955 289 12 submodule submodule NOUN iajs-1955 289 13 in in ADP iajs-1955 289 14 m m PROPN iajs-1955 289 15 for for ADP iajs-1955 289 16 each each DET iajs-1955 289 17 i=1,2 i=1,2 NOUN iajs-1955 289 18 , , PUNCT iajs-1955 289 19 . . PUNCT iajs-1955 289 20 . . PUNCT iajs-1955 290 1 . . PUNCT iajs-1955 291 1 , , PUNCT iajs-1955 291 2 then then ADV iajs-1955 291 3 𝐿 𝐿 PROPN iajs-1955 291 4 𝑀 𝑀 PROPN iajs-1955 291 5 ⊇ ⊇ PROPN iajs-1955 291 6 𝐿 𝐿 PROPN iajs-1955 291 7 𝑀 𝑀 PROPN iajs-1955 291 8 ⊇ ⊇ NOUN iajs-1955 291 9 . . PUNCT iajs-1955 291 10 . . PUNCT iajs-1955 291 11 . . PUNCT iajs-1955 292 1 , , PUNCT iajs-1955 292 2 be be AUX iajs-1955 292 3 a a DET iajs-1955 292 4 " " PUNCT iajs-1955 292 5 descending descend VERB iajs-1955 292 6 chain chain NOUN iajs-1955 292 7 " " PUNCT iajs-1955 292 8 of of ADP iajs-1955 292 9 w w NOUN iajs-1955 292 10 - - PUNCT iajs-1955 292 11 closed closed ADJ iajs-1955 292 12 submodules submodule NOUN iajs-1955 292 13 in in ADP iajs-1955 292 14 m. m. NOUN iajs-1955 292 15 but but CCONJ iajs-1955 292 16 m m NOUN iajs-1955 292 17 satisfies satisfie NOUN iajs-1955 292 18 ( ( PUNCT iajs-1955 292 19 dcc dcc PROPN iajs-1955 292 20 ) ) PUNCT iajs-1955 292 21 on on ADP iajs-1955 292 22 w w ADJ iajs-1955 292 23 - - PUNCT iajs-1955 292 24 closed closed ADJ iajs-1955 292 25 submodules submodule NOUN iajs-1955 292 26 , , PUNCT iajs-1955 292 27 " " PUNCT iajs-1955 292 28 so so ADV iajs-1955 292 29 there there PRON iajs-1955 292 30 exists exist VERB iajs-1955 292 31 a a DET iajs-1955 292 32 positive positive ADJ iajs-1955 292 33 integer integer NOUN iajs-1955 292 34 m m VERB iajs-1955 292 35 such such ADJ iajs-1955 292 36 that that SCONJ iajs-1955 292 37 " " PUNCT iajs-1955 292 38 𝐿 𝐿 PROPN iajs-1955 292 39 𝑀 𝑀 PROPN iajs-1955 292 40 𝐿 𝐿 PROPN iajs-1955 292 41 𝑀 𝑀 PROPN iajs-1955 292 42 for for ADP iajs-1955 292 43 each each DET iajs-1955 292 44 𝑛 𝑛 DET iajs-1955 292 45 𝑚. 𝑚. NOUN iajs-1955 292 46 but but CCONJ iajs-1955 292 47 m m NOUN iajs-1955 292 48 is be AUX iajs-1955 292 49 a a DET iajs-1955 292 50 finitely finitely ADV iajs-1955 292 51 generated generate VERB iajs-1955 292 52 faithful faithful ADJ iajs-1955 292 53 and and CCONJ iajs-1955 292 54 multiplication multiplication NOUN iajs-1955 292 55 r r NOUN iajs-1955 292 56 - - PUNCT iajs-1955 292 57 module module NOUN iajs-1955 292 58 , , PUNCT iajs-1955 292 59 then then ADV iajs-1955 292 60 by by ADP iajs-1955 292 61 [ [ X iajs-1955 292 62 12 12 NUM iajs-1955 292 63 , , PUNCT iajs-1955 292 64 th(3.1 th(3.1 PROPN iajs-1955 292 65 ) ) PUNCT iajs-1955 292 66 ] ] PUNCT iajs-1955 292 67 , , PUNCT iajs-1955 292 68 𝐿 𝐿 PROPN iajs-1955 292 69 𝐿 𝐿 PROPN iajs-1955 292 70 foe foe NOUN iajs-1955 292 71 each each DET iajs-1955 292 72 𝑛 𝑛 DET iajs-1955 292 73 𝑚. 𝑚. NOUN iajs-1955 292 74 therefore therefore ADV iajs-1955 292 75 r r NOUN iajs-1955 292 76 satisfies satisfie NOUN iajs-1955 292 77 ( ( PUNCT iajs-1955 292 78 dcc dcc PROPN iajs-1955 292 79 ) ) PUNCT iajs-1955 292 80 on on ADP iajs-1955 292 81 w w ADJ iajs-1955 292 82 - - PUNCT iajs-1955 292 83 closed closed ADJ iajs-1955 292 84 ideals ideal NOUN iajs-1955 292 85 . . PUNCT iajs-1955 293 1 ⟸ ⟸ ADV iajs-1955 293 2 let let VERB iajs-1955 293 3 𝐸 𝐸 PROPN iajs-1955 293 4 ⊇ ⊇ NOUN iajs-1955 293 5 𝐸 𝐸 PROPN iajs-1955 293 6 ⊇ ⊇ NOUN iajs-1955 293 7 . . PUNCT iajs-1955 293 8 . . PUNCT iajs-1955 294 1 . . PUNCT iajs-1955 295 1 , , PUNCT iajs-1955 295 2 be be AUX iajs-1955 295 3 a a DET iajs-1955 295 4 descending descend VERB iajs-1955 295 5 chain chain NOUN iajs-1955 295 6 of of ADP iajs-1955 295 7 w w NOUN iajs-1955 295 8 - - PUNCT iajs-1955 295 9 closed closed ADJ iajs-1955 295 10 submodules submodule NOUN iajs-1955 295 11 in in ADP iajs-1955 295 12 m. m. NOUN iajs-1955 295 13 since since SCONJ iajs-1955 295 14 m m PROPN iajs-1955 295 15 is be AUX iajs-1955 295 16 multiplication multiplication NOUN iajs-1955 295 17 module module NOUN iajs-1955 295 18 , , PUNCT iajs-1955 295 19 then then ADV iajs-1955 295 20 𝐸 𝐸 PROPN iajs-1955 295 21 𝐿 𝐿 PROPN iajs-1955 295 22 𝑀 𝑀 PROPN iajs-1955 295 23 for for ADP iajs-1955 295 24 some some DET iajs-1955 295 25 ideal ideal ADJ iajs-1955 295 26 𝐿 𝐿 PROPN iajs-1955 295 27 of of ADP iajs-1955 295 28 r r NOUN iajs-1955 295 29 ∀ ∀ X iajs-1955 295 30 i=1,2 i=1,2 ADJ iajs-1955 295 31 , , PUNCT iajs-1955 295 32 . . PUNCT iajs-1955 295 33 . . PUNCT iajs-1955 296 1 . . PUNCT iajs-1955 297 1 , , PUNCT iajs-1955 297 2 then then ADV iajs-1955 297 3 𝐿 𝐿 PROPN iajs-1955 297 4 𝑀 𝑀 PROPN iajs-1955 297 5 ⊇ ⊇ PROPN iajs-1955 297 6 𝐿 𝐿 PROPN iajs-1955 297 7 𝑀 𝑀 PROPN iajs-1955 297 8 ⊇ ⊇ NOUN iajs-1955 297 9 . . PUNCT iajs-1955 297 10 . . PUNCT iajs-1955 297 11 . . PUNCT iajs-1955 297 12 . . PUNCT iajs-1955 298 1 since since SCONJ iajs-1955 298 2 𝐸 𝐸 PROPN iajs-1955 298 3 is be AUX iajs-1955 298 4 a a DET iajs-1955 298 5 w w NOUN iajs-1955 298 6 - - PUNCT iajs-1955 298 7 closed closed ADJ iajs-1955 298 8 submodule submodule NOUN iajs-1955 298 9 in in ADP iajs-1955 298 10 m m PROPN iajs-1955 298 11 for for ADP iajs-1955 298 12 each each DET iajs-1955 298 13 i=1,2 i=1,2 NOUN iajs-1955 298 14 , , PUNCT iajs-1955 298 15 . . PUNCT iajs-1955 298 16 . . PUNCT iajs-1955 299 1 . . PUNCT iajs-1955 300 1 , , PUNCT iajs-1955 300 2 so so ADV iajs-1955 300 3 by by ADP iajs-1955 300 4 cor(3.8 cor(3.8 NUM iajs-1955 300 5 ) ) PUNCT iajs-1955 300 6 , , PUNCT iajs-1955 300 7 𝐿 𝐿 PROPN iajs-1955 300 8 is be AUX iajs-1955 300 9 a a DET iajs-1955 300 10 w w NOUN iajs-1955 300 11 - - PUNCT iajs-1955 300 12 closed closed ADJ iajs-1955 300 13 ideal ideal NOUN iajs-1955 300 14 in in ADP iajs-1955 300 15 r r NOUN iajs-1955 300 16 for for ADP iajs-1955 300 17 each each DET iajs-1955 300 18 i=1,2 i=1,2 NOUN iajs-1955 300 19 , , PUNCT iajs-1955 300 20 . . PUNCT iajs-1955 300 21 . . PUNCT iajs-1955 300 22 . . PUNCT iajs-1955 301 1 , , PUNCT iajs-1955 301 2 . . PUNCT iajs-1955 302 1 but but CCONJ iajs-1955 302 2 m m PROPN iajs-1955 302 3 is be AUX iajs-1955 302 4 a a DET iajs-1955 302 5 finitely finitely ADV iajs-1955 302 6 generated generate VERB iajs-1955 302 7 , , PUNCT iajs-1955 302 8 faithful faithful ADJ iajs-1955 302 9 and and CCONJ iajs-1955 302 10 multiplication multiplication NOUN iajs-1955 302 11 module module NOUN iajs-1955 302 12 , , PUNCT iajs-1955 302 13 then then ADV iajs-1955 302 14 by by ADP iajs-1955 302 15 [ [ X iajs-1955 302 16 12 12 NUM iajs-1955 302 17 , , PUNCT iajs-1955 302 18 th(3.1 th(3.1 PROPN iajs-1955 302 19 ) ) PUNCT iajs-1955 302 20 ] ] PUNCT iajs-1955 303 1 we we PRON iajs-1955 303 2 have have VERB iajs-1955 303 3 𝐿 𝐿 PROPN iajs-1955 303 4 ⊇ ⊇ PROPN iajs-1955 303 5 𝐿 𝐿 PROPN iajs-1955 303 6 ⊇ ⊇ NOUN iajs-1955 303 7 . . PUNCT iajs-1955 303 8 . . PUNCT iajs-1955 303 9 . . PUNCT iajs-1955 304 1 , , PUNCT iajs-1955 304 2 is be AUX iajs-1955 304 3 a a DET iajs-1955 304 4 " " PUNCT iajs-1955 304 5 descending descend VERB iajs-1955 304 6 chain chain NOUN iajs-1955 304 7 " " PUNCT iajs-1955 304 8 of of ADP iajs-1955 304 9 w w NOUN iajs-1955 304 10 - - PUNCT iajs-1955 304 11 closed closed ADJ iajs-1955 304 12 ideals ideal NOUN iajs-1955 304 13 in in ADP iajs-1955 304 14 r. r. PROPN iajs-1955 304 15 but but CCONJ iajs-1955 304 16 r r NOUN iajs-1955 304 17 satisfies satisfie NOUN iajs-1955 304 18 ( ( PUNCT iajs-1955 304 19 dcc dcc PROPN iajs-1955 304 20 ) ) PUNCT iajs-1955 304 21 on on ADP iajs-1955 304 22 w w ADJ iajs-1955 304 23 - - PUNCT iajs-1955 304 24 closed closed ADJ iajs-1955 304 25 ideals ideal NOUN iajs-1955 304 26 , , PUNCT iajs-1955 304 27 therefore therefore ADV iajs-1955 304 28 , , PUNCT iajs-1955 304 29 there there PRON iajs-1955 304 30 exists exist VERB iajs-1955 304 31 a a DET iajs-1955 304 32 positive positive ADJ iajs-1955 304 33 integer integer NOUN iajs-1955 304 34 m m VERB iajs-1955 304 35 such such ADJ iajs-1955 304 36 that that SCONJ iajs-1955 304 37 𝐿 𝐿 PROPN iajs-1955 304 38 𝑀 𝑀 PROPN iajs-1955 304 39 𝐿 𝐿 PROPN iajs-1955 304 40 𝑀 𝑀 PROPN iajs-1955 304 41 for for ADP iajs-1955 304 42 each each DET iajs-1955 304 43 𝑛 𝑛 PRON iajs-1955 304 44 𝑚 𝑚 NOUN iajs-1955 304 45 , , PUNCT iajs-1955 304 46 thus thus ADV iajs-1955 304 47 𝐸 𝐸 PROPN iajs-1955 304 48 𝐸 𝐸 PROPN iajs-1955 304 49 for for ADP iajs-1955 304 50 each each DET iajs-1955 304 51 𝑛 𝑛 PRON iajs-1955 304 52 𝑚. 𝑚. VERB iajs-1955 304 53 the the DET iajs-1955 304 54 proof proof NOUN iajs-1955 304 55 the the DET iajs-1955 304 56 following follow VERB iajs-1955 304 57 proposition proposition NOUN iajs-1955 304 58 is be AUX iajs-1955 304 59 similar similar ADJ iajs-1955 304 60 to to ADP iajs-1955 304 61 the the DET iajs-1955 304 62 proof proof NOUN iajs-1955 304 63 of of ADP iajs-1955 304 64 prop(4.14 prop(4.14 NOUN iajs-1955 304 65 ) ) PUNCT iajs-1955 304 66 , , PUNCT iajs-1955 304 67 hence hence ADV iajs-1955 304 68 we we PRON iajs-1955 304 69 omited omit VERB iajs-1955 304 70 . . PUNCT iajs-1955 304 71             SPACE iajs-1955 305 1 178 178 NUM iajs-1955 305 2     SPACE iajs-1955 305 3 | | ADV iajs-1955 305 4     SPACE iajs-1955 305 5 mathmatics mathmatic NOUN iajs-1955 305 6 5https://doi.org/10.30526/31.2.195 5https://doi.org/10.30526/31.2.195 ADJ iajs-1955 305 7     SPACE iajs-1955 305 8 2018 2018 NUM iajs-1955 305 9 ) ) PUNCT iajs-1955 306 1 عام عام ADP iajs-1955 306 2 2العدد 2العدد NUM iajs-1955 306 3 ( ( PUNCT iajs-1955 306 4 31لمجلد 31لمجلد NUM iajs-1955 306 5 ا ا NOUN iajs-1955 306 6 مجلة مجلة NOUN iajs-1955 306 7 إبن إبن VERB iajs-1955 306 8 الهيثم الهيثم ADJ iajs-1955 306 9 للعلوم للعلوم NOUN iajs-1955 306 10 الصرفة الصرفة NOUN iajs-1955 307 1 و و PRON iajs-1955 307 2 التطبيقية التطبيقية ADJ iajs-1955 307 3 ibn ibn PROPN iajs-1955 307 4 al al PROPN iajs-1955 307 5 - - PUNCT iajs-1955 307 6 haitham haitham PROPN iajs-1955 307 7 jour jour X iajs-1955 307 8 . . PROPN iajs-1955 307 9 for for ADP iajs-1955 307 10 pure pure ADJ iajs-1955 307 11 & & CCONJ iajs-1955 307 12 appl appl PROPN iajs-1955 307 13 . . PUNCT iajs-1955 308 1 sci sci PROPN iajs-1955 308 2 . . PUNCT iajs-1955 308 3 vol vol NOUN iajs-1955 308 4 . . PROPN iajs-1955 309 1 31 31 NUM iajs-1955 309 2 ( ( PUNCT iajs-1955 309 3 2 2 NUM iajs-1955 309 4 ) ) PUNCT iajs-1955 309 5 2018 2018 NUM iajs-1955 309 6 proposition(4.15 proposition(4.15 NOUN iajs-1955 309 7 ) ) PUNCT iajs-1955 309 8 " " PUNCT iajs-1955 309 9 if if SCONJ iajs-1955 309 10 m m NOUN iajs-1955 309 11 is be AUX iajs-1955 309 12 a a DET iajs-1955 309 13 finitely finitely ADV iajs-1955 309 14 generated generate VERB iajs-1955 309 15 , , PUNCT iajs-1955 309 16 faithful faithful ADJ iajs-1955 309 17 and and CCONJ iajs-1955 309 18 multiplication multiplication NOUN iajs-1955 309 19 module module NOUN iajs-1955 309 20 , , PUNCT iajs-1955 309 21 and and CCONJ iajs-1955 309 22 m m PROPN iajs-1955 309 23 satisfies satisfie NOUN iajs-1955 309 24 condition condition NOUN iajs-1955 309 25 ∗ ∗ NOUN iajs-1955 309 26 " " PUNCT iajs-1955 309 27 , , PUNCT iajs-1955 309 28 then then ADV iajs-1955 309 29 m m VERB iajs-1955 309 30 satisfies satisfie NOUN iajs-1955 309 31 ( ( PUNCT iajs-1955 309 32 acc acc PROPN iajs-1955 309 33 ) ) PUNCT iajs-1955 309 34 on on ADP iajs-1955 309 35 w w ADJ iajs-1955 309 36 - - PUNCT iajs-1955 309 37 closed closed ADJ iajs-1955 309 38 submodules submodule NOUN iajs-1955 309 39 , , PUNCT iajs-1955 309 40 if if SCONJ iajs-1955 309 41 and and CCONJ iajs-1955 309 42 only only ADV iajs-1955 309 43 if if SCONJ iajs-1955 309 44 r r NOUN iajs-1955 309 45 satisfies satisfie NOUN iajs-1955 309 46 ( ( PUNCT iajs-1955 309 47 acc acc PROPN iajs-1955 309 48 ) ) PUNCT iajs-1955 309 49 on on ADP iajs-1955 309 50 w w ADJ iajs-1955 309 51 - - PUNCT iajs-1955 309 52 closed closed ADJ iajs-1955 309 53 ideals ideal NOUN iajs-1955 309 54 . . PUNCT iajs-1955 310 1 references reference NOUN iajs-1955 310 2 1 1 NUM iajs-1955 310 3 . . PUNCT iajs-1955 311 1 goodearl goodearl PROPN iajs-1955 311 2 k.r k.r PROPN iajs-1955 311 3 . . PROPN iajs-1955 311 4 ( ( PUNCT iajs-1955 311 5 1976 1976 NUM iajs-1955 311 6 ) ) PUNCT iajs-1955 311 7 . . PUNCT iajs-1955 311 8 , , PUNCT iajs-1955 311 9 " " PUNCT iajs-1955 311 10 ring ring NOUN iajs-1955 311 11 theory theory NOUN iajs-1955 311 12 , , PUNCT iajs-1955 311 13 nonsingular nonsingular ADJ iajs-1955 311 14 rings ring NOUN iajs-1955 311 15 and and CCONJ iajs-1955 311 16 modules module NOUN iajs-1955 311 17 " " PUNCT iajs-1955 311 18 , , PUNCT iajs-1955 311 19 marcel marcel PROPN iajs-1955 311 20 , , PUNCT iajs-1955 311 21 dekker dekker PROPN iajs-1955 311 22 , , PUNCT iajs-1955 311 23 new new PROPN iajs-1955 311 24 york york PROPN iajs-1955 311 25 , , PUNCT iajs-1955 311 26 2 2 NUM iajs-1955 311 27 . . X iajs-1955 312 1 muna muna PROPN iajs-1955 312 2 , , PUNCT iajs-1955 312 3 a. a. NOUN iajs-1955 312 4 a. a. NOUN iajs-1955 312 5 ( ( PUNCT iajs-1955 312 6 2009 2009 NUM iajs-1955 312 7 ) ) PUNCT iajs-1955 312 8 , , PUNCT iajs-1955 312 9 " " PUNCT iajs-1955 312 10 weak weak ADJ iajs-1955 312 11 essential essential ADJ iajs-1955 312 12 submodules submodule NOUN iajs-1955 312 13 " " PUNCT iajs-1955 312 14 , , PUNCT iajs-1955 312 15 um um INTJ iajs-1955 312 16 - - PUNCT iajs-1955 312 17 salma salma NOUN iajs-1955 312 18 science science PROPN iajs-1955 312 19 journal journal PROPN iajs-1955 312 20 , , PUNCT iajs-1955 312 21 vol vol NOUN iajs-1955 312 22 . . PUNCT iajs-1955 313 1 6(1).214 6(1).214 NUM iajs-1955 313 2 - - SYM iajs-1955 313 3 222 222 NUM iajs-1955 313 4 . . PUNCT iajs-1955 314 1 3 3 NUM iajs-1955 314 2 . . X iajs-1955 314 3 muna muna PROPN iajs-1955 314 4 , , PUNCT iajs-1955 314 5 a. a. NOUN iajs-1955 314 6 a. a. PROPN iajs-1955 314 7 ( ( PUNCT iajs-1955 314 8 2016 2016 NUM iajs-1955 314 9 ) ) PUNCT iajs-1955 314 10 " " PUNCT iajs-1955 314 11 some some DET iajs-1955 314 12 results result VERB iajs-1955 314 13 on on ADP iajs-1955 314 14 weak weak ADJ iajs-1955 314 15 essential essential ADJ iajs-1955 314 16 submodules submodule NOUN iajs-1955 314 17 " " PUNCT iajs-1955 314 18 , , PUNCT iajs-1955 314 19 baghdad baghdad PROPN iajs-1955 314 20 science science PROPN iajs-1955 314 21 journal journal PROPN iajs-1955 314 22 to to ADP iajs-1955 314 23 apper apper NOUN iajs-1955 314 24 . . PUNCT iajs-1955 315 1 4 4 X iajs-1955 315 2 . . X iajs-1955 315 3 dauns daun NOUN iajs-1955 315 4 , , PUNCT iajs-1955 315 5 j. j. PROPN iajs-1955 315 6 and and CCONJ iajs-1955 315 7 b. b. PROPN iajs-1955 315 8 r. r. PROPN iajs-1955 315 9 mcdonald mcdonald PROPN iajs-1955 315 10 , , PUNCT iajs-1955 315 11 ( ( PUNCT iajs-1955 315 12 1980 1980 NUM iajs-1955 315 13 ) ) PUNCT iajs-1955 315 14 , , PUNCT iajs-1955 315 15 " " PUNCT iajs-1955 315 16 prime prime ADJ iajs-1955 315 17 modules module NOUN iajs-1955 315 18 , , PUNCT iajs-1955 315 19 and and CCONJ iajs-1955 315 20 one one NUM iajs-1955 315 21 - - PUNCT iajs-1955 315 22 sided sided ADJ iajs-1955 315 23 ideals ideal NOUN iajs-1955 315 24 in in ADP iajs-1955 315 25 rings ring NOUN iajs-1955 315 26 and and CCONJ iajs-1955 315 27 algebra algebra NOUN iajs-1955 315 28 iii iii PROPN iajs-1955 315 29 " " PUNCT iajs-1955 315 30 , , PUNCT iajs-1955 315 31 proceeding proceed VERB iajs-1955 315 32 of of ADP iajs-1955 315 33 third third PROPN iajs-1955 315 34 oklahoma oklahoma PROPN iajs-1955 315 35 , , PUNCT iajs-1955 315 36 conference conference NOUN iajs-1955 315 37 301 301 NUM iajs-1955 315 38 - - NUM iajs-1955 315 39 344 344 NUM iajs-1955 315 40 . . NOUN iajs-1955 315 41 5 5 NUM iajs-1955 315 42 . . NOUN iajs-1955 315 43 athab athab PROPN iajs-1955 315 44 e. e. PROPN iajs-1955 315 45 a. a. PROPN iajs-1955 315 46 , , PUNCT iajs-1955 315 47 ( ( PUNCT iajs-1955 315 48 1996 1996 NUM iajs-1955 315 49 ) ) PUNCT iajs-1955 315 50 " " PUNCT iajs-1955 315 51 semi semi ADJ iajs-1955 315 52 - - ADJ iajs-1955 315 53 prime prime ADJ iajs-1955 315 54 and and CCONJ iajs-1955 315 55 weak weak ADJ iajs-1955 315 56 semi semi ADJ iajs-1955 315 57 - - ADJ iajs-1955 315 58 prime prime ADJ iajs-1955 315 59 submodules submodule NOUN iajs-1955 315 60 " " PUNCT iajs-1955 315 61 m. m. PROPN iajs-1955 315 62 sc sc PROPN iajs-1955 315 63 . . PUNCT iajs-1955 316 1 thesis thesis PROPN iajs-1955 316 2 , , PUNCT iajs-1955 316 3 university university NOUN iajs-1955 316 4 of of ADP iajs-1955 316 5 baghdad baghdad PROPN iajs-1955 316 6 , , PUNCT iajs-1955 316 7 6 6 NUM iajs-1955 316 8 . . PUNCT iajs-1955 317 1 osfsky osfsky ADJ iajs-1955 317 2 , , PUNCT iajs-1955 317 3 b. b. PROPN iajs-1955 317 4 l. l. PROPN iajs-1955 317 5 ( ( PUNCT iajs-1955 317 6 1991 1991 NUM iajs-1955 317 7 ) ) PUNCT iajs-1955 317 8 , , PUNCT iajs-1955 317 9 , , PUNCT iajs-1955 317 10 " " PUNCT iajs-1955 317 11 aconstruction aconstruction NOUN iajs-1955 317 12 of of ADP iajs-1955 317 13 nonstandared nonstandare VERB iajs-1955 317 14 uniserial uniserial ADJ iajs-1955 317 15 modules module NOUN iajs-1955 317 16 over over ADP iajs-1955 317 17 valuation valuation NOUN iajs-1955 317 18 domain domain NOUN iajs-1955 317 19 " " PUNCT iajs-1955 317 20 , , PUNCT iajs-1955 317 21 bull bull PROPN iajs-1955 317 22 . . PUNCT iajs-1955 318 1 amer amer PROPN iajs-1955 318 2 , , PUNCT iajs-1955 318 3 math math NOUN iajs-1955 318 4 . . PUNCT iajs-1955 319 1 soc soc PROPN iajs-1955 319 2 . . PUNCT iajs-1955 320 1 25 25 NUM iajs-1955 320 2 , , PUNCT iajs-1955 320 3 89 89 NUM iajs-1955 320 4 - - SYM iajs-1955 320 5 97 97 NUM iajs-1955 320 6 . . PUNCT iajs-1955 321 1 7 7 X iajs-1955 321 2 . . X iajs-1955 321 3 barnard barnard NOUN iajs-1955 321 4 , , PUNCT iajs-1955 321 5 a. a. NOUN iajs-1955 321 6 ( ( PUNCT iajs-1955 321 7 1981 1981 NUM iajs-1955 321 8 ) ) PUNCT iajs-1955 321 9 , , PUNCT iajs-1955 321 10 " " PUNCT iajs-1955 321 11 multiplication multiplication NOUN iajs-1955 321 12 modules module NOUN iajs-1955 321 13 " " PUNCT iajs-1955 321 14 , , PUNCT iajs-1955 321 15 j. j. PROPN iajs-1955 321 16 algebra algebra PROPN iajs-1955 321 17 , , PUNCT iajs-1955 321 18 71 71 NUM iajs-1955 321 19 , , PUNCT iajs-1955 321 20 174 174 NUM iajs-1955 321 21 - - SYM iajs-1955 321 22 178 178 NUM iajs-1955 321 23 . . NOUN iajs-1955 322 1 8 8 NUM iajs-1955 322 2 . . X iajs-1955 323 1 kasch kasch PROPN iajs-1955 323 2 , , PUNCT iajs-1955 323 3 f. f. PROPN iajs-1955 323 4 ( ( PUNCT iajs-1955 323 5 1982 1982 NUM iajs-1955 323 6 ) ) PUNCT iajs-1955 323 7 " " PUNCT iajs-1955 323 8 modules module NOUN iajs-1955 323 9 and and CCONJ iajs-1955 323 10 rings ring NOUN iajs-1955 323 11 " " PUNCT iajs-1955 323 12 , , PUNCT iajs-1955 323 13 academic academic ADJ iajs-1955 323 14 press press NOUN iajs-1955 323 15 , , PUNCT iajs-1955 323 16 london london PROPN iajs-1955 323 17 , , PUNCT iajs-1955 323 18 . . PUNCT iajs-1955 324 1 9 9 X iajs-1955 324 2 . . X iajs-1955 324 3 abbas abbas PROPN iajs-1955 324 4 , , PUNCT iajs-1955 324 5 m.s.(1991 m.s.(1991 PROPN iajs-1955 324 6 ) ) PUNCT iajs-1955 324 7 , , PUNCT iajs-1955 324 8 " " PUNCT iajs-1955 324 9 on on ADP iajs-1955 324 10 fully fully ADV iajs-1955 324 11 stable stable ADJ iajs-1955 324 12 modules module NOUN iajs-1955 324 13 " " PUNCT iajs-1955 324 14 , , PUNCT iajs-1955 324 15 ph ph PROPN iajs-1955 324 16 . . PROPN iajs-1955 324 17 d. d. PROPN iajs-1955 324 18 thesis thesis PROPN iajs-1955 324 19 , , PUNCT iajs-1955 324 20 university university NOUN iajs-1955 324 21 of of ADP iajs-1955 324 22 baghdad baghdad PROPN iajs-1955 324 23 , , PUNCT iajs-1955 324 24 . . PUNCT iajs-1955 325 1 10 10 NUM iajs-1955 325 2 . . X iajs-1955 326 1 anderson anderson PROPN iajs-1955 326 2 , , PUNCT iajs-1955 326 3 f. f. PROPN iajs-1955 326 4 w. w. PROPN iajs-1955 326 5 and and CCONJ iajs-1955 326 6 fullar fullar PROPN iajs-1955 326 7 , , PUNCT iajs-1955 326 8 k. k. PROPN iajs-1955 326 9 r. r. PROPN iajs-1955 326 10 ( ( PUNCT iajs-1955 326 11 1974 1974 NUM iajs-1955 326 12 ) ) PUNCT iajs-1955 326 13 " " PUNCT iajs-1955 326 14 rings ring NOUN iajs-1955 326 15 and and CCONJ iajs-1955 326 16 categories category NOUN iajs-1955 326 17 of of ADP iajs-1955 326 18 modules module NOUN iajs-1955 326 19 " " PUNCT iajs-1955 326 20 , , PUNCT iajs-1955 326 21 springer springer NOUN iajs-1955 326 22 - - PUNCT iajs-1955 326 23 verlay verlay NOUN iajs-1955 326 24 , , PUNCT iajs-1955 326 25 new new ADJ iajs-1955 326 26 yourk yourk PROPN iajs-1955 326 27 , , PUNCT iajs-1955 326 28 heidelbrge heidelbrge NOUN iajs-1955 326 29 berlin berlin PROPN iajs-1955 326 30 , , PUNCT iajs-1955 326 31 . . PUNCT iajs-1955 327 1 11 11 NUM iajs-1955 327 2 . . PUNCT iajs-1955 328 1 shahib shahib NOUN iajs-1955 328 2 , , PUNCT iajs-1955 328 3 l. l. PROPN iajs-1955 328 4 h.(2012 h.(2012 PROPN iajs-1955 328 5 ) ) PUNCT iajs-1955 328 6 " " PUNCT iajs-1955 328 7 extending extend VERB iajs-1955 328 8 , , PUNCT iajs-1955 328 9 iniectivity iniectivity NOUN iajs-1955 328 10 and and CCONJ iajs-1955 328 11 chain chain NOUN iajs-1955 328 12 condition condition NOUN iajs-1955 328 13 on on ADP iajs-1955 328 14 y y PROPN iajs-1955 328 15 - - PUNCT iajs-1955 328 16 closed close VERB iajs-1955 328 17 submodules submodule NOUN iajs-1955 328 18 " " PUNCT iajs-1955 328 19 , , PUNCT iajs-1955 328 20 m. m. PROPN iajs-1955 328 21 sc sc PROPN iajs-1955 328 22 . . PUNCT iajs-1955 329 1 thesis thesis PROPN iajs-1955 329 2 , , PUNCT iajs-1955 329 3 university university NOUN iajs-1955 329 4 of of ADP iajs-1955 329 5 baghdad baghdad PROPN iajs-1955 329 6 , , PUNCT iajs-1955 329 7 . . PUNCT iajs-1955 330 1 12 12 NUM iajs-1955 330 2 . . PUNCT iajs-1955 331 1 al al PROPN iajs-1955 331 2 - - PUNCT iajs-1955 331 3 bast bast NOUN iajs-1955 331 4 , , PUNCT iajs-1955 331 5 z. z. PROPN iajs-1955 331 6 a. a. PROPN iajs-1955 331 7 and and CCONJ iajs-1955 331 8 smith smith PROPN iajs-1955 331 9 , , PUNCT iajs-1955 331 10 p. p. PROPN iajs-1955 331 11 f. f. PROPN iajs-1955 331 12 ( ( PUNCT iajs-1955 331 13 1988 1988 NUM iajs-1955 331 14 ) ) PUNCT iajs-1955 331 15 “ " PUNCT iajs-1955 331 16 multiplication multiplication NOUN iajs-1955 331 17 modules module NOUN iajs-1955 331 18 " " PUNCT iajs-1955 331 19 , , PUNCT iajs-1955 331 20 communication communication NOUN iajs-1955 331 21 in in ADP iajs-1955 331 22 algebra algebra NOUN iajs-1955 331 23 , , PUNCT iajs-1955 331 24 vol vol NOUN iajs-1955 331 25 . . PROPN iajs-1955 331 26 10 10 NUM iajs-1955 331 27 , , PUNCT iajs-1955 331 28 , , PUNCT iajs-1955 331 29 755 755 NUM iajs-1955 331 30 - - SYM iajs-1955 331 31 779 779 NUM iajs-1955 331 32 . . PROPN iajs-1955 331 33 13 13 NUM iajs-1955 331 34 . . PUNCT iajs-1955 332 1 ahmed ahmed PROPN iajs-1955 332 2 a. a. PROPN iajs-1955 332 3 a. a. PROPN iajs-1955 332 4 ( ( PUNCT iajs-1955 332 5 1992 1992 NUM iajs-1955 332 6 ) ) PUNCT iajs-1955 332 7 . . PUNCT iajs-1955 332 8 " " PUNCT iajs-1955 333 1 submodule submodule NOUN iajs-1955 333 2 of of ADP iajs-1955 333 3 multiplication multiplication NOUN iajs-1955 333 4 modules module NOUN iajs-1955 333 5 " " PUNCT iajs-1955 333 6 , , PUNCT iajs-1955 333 7 m. m. PROPN iajs-1955 333 8 sc sc PROPN iajs-1955 333 9 . . PUNCT iajs-1955 334 1 thesis thesis PROPN iajs-1955 334 2 , , PUNCT iajs-1955 334 3 university university NOUN iajs-1955 334 4 of of ADP iajs-1955 334 5 baghdad baghdad PROPN iajs-1955 334 6 . . PUNCT