id sid tid token lemma pos iajs-2000 1 1 microsoft microsoft PROPN iajs-2000 1 2 word word NOUN iajs-2000 1 3 109 109 NUM iajs-2000 1 4 - - SYM iajs-2000 1 5 117 117 NUM iajs-2000 1 6 mathematics mathematic NOUN iajs-2000 1 7 | | ADV iajs-2000 1 8 109 109 NUM iajs-2000 1 9 ibn ibn PROPN iajs-2000 1 10 al al PROPN iajs-2000 1 11 - - PUNCT iajs-2000 1 12 haitham haitham PROPN iajs-2000 1 13 jour jour X iajs-2000 1 14 . . PROPN iajs-2000 2 1 for for ADP iajs-2000 2 2 pure pure ADJ iajs-2000 2 3 & & CCONJ iajs-2000 2 4 appl appl PROPN iajs-2000 2 5 . . PUNCT iajs-2000 3 1 sci sci PROPN iajs-2000 3 2 . . PROPN iajs-2000 3 3 ihjpas ihjpa VERB iajs-2000 3 4 https://doi.org/10.30526/31.3.2000 https://doi.org/10.30526/31.3.2000 NUM iajs-2000 3 5 vol vol NOUN iajs-2000 3 6 . . PROPN iajs-2000 3 7 31 31 NUM iajs-2000 3 8 ( ( PUNCT iajs-2000 3 9 3 3 NUM iajs-2000 3 10 ) ) PUNCT iajs-2000 3 11 2018 2018 NUM iajs-2000 3 12 we we PRON iajs-2000 3 13 - - PUNCT iajs-2000 3 14 prime prime NOUN iajs-2000 3 15 submodules submodule NOUN iajs-2000 3 16 and and CCONJ iajs-2000 3 17 we we PRON iajs-2000 3 18 - - PUNCT iajs-2000 3 19 semi semi ADJ iajs-2000 3 20 - - ADJ iajs-2000 3 21 prime prime ADJ iajs-2000 3 22 submodules submodule NOUN iajs-2000 3 23 saif saif PROPN iajs-2000 3 24 a. a. PROPN iajs-2000 3 25 hussin hussin PROPN iajs-2000 3 26 haibt haibt PROPN iajs-2000 3 27 k. k. PROPN iajs-2000 3 28 mohammadali mohammadali PROPN iajs-2000 3 29 department department PROPN iajs-2000 3 30 of of ADP iajs-2000 3 31 mathematics mathematics PROPN iajs-2000 3 32 , , PUNCT iajs-2000 3 33 college college NOUN iajs-2000 3 34 of of ADP iajs-2000 3 35 computer computer NOUN iajs-2000 3 36 science science NOUN iajs-2000 3 37 and and CCONJ iajs-2000 3 38 mathematics mathematic NOUN iajs-2000 3 39 , , PUNCT iajs-2000 3 40 tikrit tikrit NOUN iajs-2000 3 41 university university PROPN iajs-2000 3 42 , , PUNCT iajs-2000 3 43 iraq iraq PROPN iajs-2000 3 44 , , PUNCT iajs-2000 3 45 tikrit tikrit NOUN iajs-2000 3 46 saif.a19881988@gmail.com saif.a19881988@gmail.com PROPN iajs-2000 3 47 article article NOUN iajs-2000 3 48 history history NOUN iajs-2000 3 49 : : PUNCT iajs-2000 3 50 received receive VERB iajs-2000 3 51 5 5 NUM iajs-2000 3 52 august august PROPN iajs-2000 3 53 2018 2018 NUM iajs-2000 3 54 , , PUNCT iajs-2000 3 55 accepted accept VERB iajs-2000 3 56 18 18 NUM iajs-2000 3 57 september september PROPN iajs-2000 3 58 2018 2018 NUM iajs-2000 3 59 , , PUNCT iajs-2000 3 60 published publish VERB iajs-2000 3 61 december december PROPN iajs-2000 3 62 2018 2018 NUM iajs-2000 3 63 " " PUNCT iajs-2000 3 64 " " PUNCT iajs-2000 3 65 abstract abstract ADJ iajs-2000 3 66 " " PUNCT iajs-2000 3 67 in in ADP iajs-2000 3 68 this this DET iajs-2000 3 69 article article NOUN iajs-2000 3 70 , , PUNCT iajs-2000 3 71 " " PUNCT iajs-2000 3 72 we we PRON iajs-2000 3 73 introduce introduce VERB iajs-2000 3 74 the the DET iajs-2000 3 75 concept concept NOUN iajs-2000 3 76 of of ADP iajs-2000 3 77 a a DET iajs-2000 3 78 we we NOUN iajs-2000 3 79 - - PUNCT iajs-2000 3 80 prime prime NOUN iajs-2000 3 81 submodule submodule NOUN iajs-2000 3 82 " " PUNCT iajs-2000 3 83 , , PUNCT iajs-2000 3 84 as as ADP iajs-2000 3 85 a a DET iajs-2000 3 86 stronger strong ADJ iajs-2000 3 87 form form NOUN iajs-2000 3 88 of of ADP iajs-2000 3 89 a a DET iajs-2000 3 90 weakly weakly ADJ iajs-2000 3 91 prime prime ADJ iajs-2000 3 92 submodule submodule NOUN iajs-2000 3 93 . . PUNCT iajs-2000 4 1 and and CCONJ iajs-2000 4 2 as as ADP iajs-2000 4 3 a a DET iajs-2000 4 4 " " PUNCT iajs-2000 4 5 generalization generalization NOUN iajs-2000 4 6 of of ADP iajs-2000 4 7 we we NOUN iajs-2000 4 8 - - PUNCT iajs-2000 4 9 prime prime NOUN iajs-2000 4 10 submodule submodule NOUN iajs-2000 4 11 , , PUNCT iajs-2000 4 12 we we PRON iajs-2000 4 13 introduce introduce VERB iajs-2000 4 14 the the DET iajs-2000 4 15 concept concept NOUN iajs-2000 4 16 of of ADP iajs-2000 4 17 we we PRON iajs-2000 4 18 - - PUNCT iajs-2000 4 19 semi semi ADJ iajs-2000 4 20 - - ADJ iajs-2000 4 21 prime prime ADJ iajs-2000 4 22 submodule submodule NOUN iajs-2000 4 23 , , PUNCT iajs-2000 4 24 which which PRON iajs-2000 4 25 is be AUX iajs-2000 4 26 also also ADV iajs-2000 4 27 a a DET iajs-2000 4 28 stronger strong ADJ iajs-2000 4 29 form form NOUN iajs-2000 4 30 of of ADP iajs-2000 4 31 a a DET iajs-2000 4 32 weakly weakly ADJ iajs-2000 4 33 semi semi ADJ iajs-2000 4 34 - - ADJ iajs-2000 4 35 prime prime ADJ iajs-2000 4 36 submodule submodule NOUN iajs-2000 4 37 . . PUNCT iajs-2000 5 1 various various ADJ iajs-2000 5 2 basic basic ADJ iajs-2000 5 3 properties property NOUN iajs-2000 5 4 of of ADP iajs-2000 5 5 these these DET iajs-2000 5 6 two two NUM iajs-2000 5 7 concepts concept NOUN iajs-2000 5 8 are be AUX iajs-2000 5 9 discussed discuss VERB iajs-2000 5 10 . . PUNCT iajs-2000 6 1 furthermore furthermore ADV iajs-2000 6 2 , , PUNCT iajs-2000 6 3 the the DET iajs-2000 6 4 relationships relationship NOUN iajs-2000 6 5 between between ADP iajs-2000 6 6 " " PUNCT iajs-2000 6 7 we we PRON iajs-2000 6 8 - - PUNCT iajs-2000 6 9 prime prime NOUN iajs-2000 6 10 submodules submodule NOUN iajs-2000 6 11 and and CCONJ iajs-2000 6 12 weakly weakly ADJ iajs-2000 6 13 prime prime NOUN iajs-2000 6 14 submodules"and submodules"and PROPN iajs-2000 6 15 studied study VERB iajs-2000 6 16 . . PUNCT iajs-2000 7 1 on on ADP iajs-2000 7 2 the the DET iajs-2000 7 3 other other ADJ iajs-2000 7 4 hand hand NOUN iajs-2000 7 5 , , PUNCT iajs-2000 7 6 the the DET iajs-2000 7 7 relation relation NOUN iajs-2000 7 8 between between ADP iajs-2000 7 9 we we PRON iajs-2000 7 10 - - PUNCT iajs-2000 7 11 prime prime NOUN iajs-2000 7 12 submodules submodule NOUN iajs-2000 7 13 and and CCONJ iajs-2000 7 14 wesemi wesemi NOUN iajs-2000 7 15 prime prime PROPN iajs-2000 7 16 submodules submodule NOUN iajs-2000 7 17 are be AUX iajs-2000 7 18 consider consider VERB iajs-2000 7 19 . . PUNCT iajs-2000 8 1 "also"the "also"the DET iajs-2000 8 2 relation relation NOUN iajs-2000 8 3 of of ADP iajs-2000 8 4 " " PUNCT iajs-2000 8 5 we we PRON iajs-2000 8 6 – – PUNCT iajs-2000 8 7 sime sime PROPN iajs-2000 8 8 prime prime ADJ iajs-2000 8 9 submodules submodule NOUN iajs-2000 8 10 and and CCONJ iajs-2000 8 11 weakly weakly ADJ iajs-2000 8 12 semi semi ADJ iajs-2000 8 13 - - ADJ iajs-2000 8 14 prime prime ADJ iajs-2000 8 15 submodules submodule NOUN iajs-2000 8 16 " " PUNCT iajs-2000 8 17 are be AUX iajs-2000 8 18 explained explain VERB iajs-2000 8 19 . . PUNCT iajs-2000 9 1 behind behind ADP iajs-2000 9 2 that that PRON iajs-2000 9 3 , , PUNCT iajs-2000 9 4 some some DET iajs-2000 9 5 characterizations characterization NOUN iajs-2000 9 6 of of ADP iajs-2000 9 7 these these DET iajs-2000 9 8 concepts concept NOUN iajs-2000 9 9 are be AUX iajs-2000 9 10 investigated investigate VERB iajs-2000 9 11 " " PUNCT iajs-2000 9 12 . . PUNCT iajs-2000 10 1 keywords keyword NOUN iajs-2000 10 2 : : PUNCT iajs-2000 10 3 " " PUNCT iajs-2000 10 4 weakly weakly ADJ iajs-2000 10 5 prime prime ADJ iajs-2000 10 6 submodules submodule NOUN iajs-2000 10 7 , , PUNCT iajs-2000 10 8 weakly weakly ADJ iajs-2000 10 9 semi semi ADJ iajs-2000 10 10 - - ADJ iajs-2000 10 11 prime prime ADJ iajs-2000 10 12 submodules submodule NOUN iajs-2000 10 13 , , PUNCT iajs-2000 10 14 " " PUNCT iajs-2000 10 15 we we PRON iajs-2000 10 16 - - PUNCT iajs-2000 10 17 prime prime NOUN iajs-2000 10 18 submodules submodule NOUN iajs-2000 10 19 , , PUNCT iajs-2000 10 20 we we PRON iajs-2000 10 21 - - PUNCT iajs-2000 10 22 semi semi ADJ iajs-2000 10 23 - - ADJ iajs-2000 10 24 prime prime ADJ iajs-2000 10 25 submodules submodule NOUN iajs-2000 10 26 . . PUNCT iajs-2000 11 1 1 1 X iajs-2000 11 2 . . X iajs-2000 11 3 introduction introduction NOUN iajs-2000 11 4 " " PUNCT iajs-2000 11 5 " " PUNCT iajs-2000 11 6 weakly weakly ADJ iajs-2000 11 7 prime prime ADJ iajs-2000 11 8 submodule submodule NOUN iajs-2000 11 9 " " PUNCT iajs-2000 11 10 " " PUNCT iajs-2000 11 11 have have AUX iajs-2000 11 12 been be AUX iajs-2000 11 13 introduced introduce VERB iajs-2000 11 14 and and CCONJ iajs-2000 11 15 studied study VERB iajs-2000 11 16 " " PUNCT iajs-2000 11 17 by by ADP iajs-2000 11 18 hadi hadi PROPN iajs-2000 11 19 m. m. PROPN iajs-2000 11 20 a a PRON iajs-2000 11 21 in in ADP iajs-2000 11 22 [ [ X iajs-2000 11 23 1 1 NUM iajs-2000 11 24 ] ] PUNCT iajs-2000 11 25 , , PUNCT iajs-2000 11 26 where where SCONJ iajs-2000 11 27 " " PUNCT iajs-2000 11 28 a a DET iajs-2000 11 29 proper proper ADJ iajs-2000 11 30 submodule submodule NOUN iajs-2000 11 31 k k PROPN iajs-2000 11 32 of of ADP iajs-2000 11 33 an an DET iajs-2000 11 34 r r NOUN iajs-2000 11 35 - - PUNCT iajs-2000 11 36 module module NOUN iajs-2000 11 37 x x PRON iajs-2000 11 38 is be AUX iajs-2000 11 39 called call VERB iajs-2000 11 40 a a DET iajs-2000 11 41 weakly weakly ADJ iajs-2000 11 42 prime prime NOUN iajs-2000 11 43 , , PUNCT iajs-2000 11 44 if if SCONJ iajs-2000 11 45 " " PUNCT iajs-2000 11 46 wherever wherever SCONJ iajs-2000 11 47 0 0 NUM iajs-2000 11 48 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2000 11 49 ∈ ∈ PROPN iajs-2000 11 50 𝐾 𝐾 PROPN iajs-2000 11 51 , , PUNCT iajs-2000 11 52 where where SCONJ iajs-2000 11 53 " " PUNCT iajs-2000 11 54 π‘Ÿ π‘Ÿ X iajs-2000 11 55 ∈ ∈ PROPN iajs-2000 11 56 𝑅 𝑅 PROPN iajs-2000 11 57 , , PUNCT iajs-2000 11 58 π‘₯ π‘₯ DET iajs-2000 11 59 ∈ ∈ PROPN iajs-2000 11 60 𝑋 𝑋 PROPN iajs-2000 11 61 " " PUNCT iajs-2000 11 62 , , PUNCT iajs-2000 11 63 implies imply VERB iajs-2000 11 64 that that SCONJ iajs-2000 11 65 either either CCONJ iajs-2000 11 66 π‘₯ π‘₯ DET iajs-2000 11 67 ∈ ∈ PROPN iajs-2000 11 68 𝐾 𝐾 PROPN iajs-2000 11 69 or or CCONJ iajs-2000 11 70 π‘Ÿ π‘Ÿ PRON iajs-2000 11 71 ∈ ∈ PROPN iajs-2000 11 72 𝐾 𝐾 PROPN iajs-2000 11 73 : : PUNCT iajs-2000 11 74 𝑋 𝑋 PROPN iajs-2000 11 75 , , PUNCT iajs-2000 11 76 where where SCONJ iajs-2000 11 77 𝐾 𝐾 NOUN iajs-2000 11 78 : : PUNCT iajs-2000 11 79 𝑋 𝑋 PROPN iajs-2000 11 80 π‘Ž π‘Ž PROPN iajs-2000 11 81 ∈ ∈ PROPN iajs-2000 11 82 𝑅 𝑅 PROPN iajs-2000 11 83 ∢ ∢ NOUN iajs-2000 11 84 π‘Žπ‘‹ π‘Žπ‘‹ NOUN iajs-2000 11 85 𝐾 𝐾 NOUN iajs-2000 11 86 . . PUNCT iajs-2000 12 1 " " PUNCT iajs-2000 12 2 weakly weakly ADJ iajs-2000 12 3 semi semi ADJ iajs-2000 12 4 - - ADJ iajs-2000 12 5 prime prime ADJ iajs-2000 12 6 submodule submodule NOUN iajs-2000 12 7 have have AUX iajs-2000 12 8 been be AUX iajs-2000 12 9 introduced introduce VERB iajs-2000 12 10 and and CCONJ iajs-2000 12 11 studied study VERB iajs-2000 12 12 by by ADP iajs-2000 12 13 farzalipour farzalipour PROPN iajs-2000 12 14 f f PROPN iajs-2000 12 15 in in ADP iajs-2000 12 16 [ [ X iajs-2000 12 17 2 2 NUM iajs-2000 12 18 ] ] PUNCT iajs-2000 12 19 , , PUNCT iajs-2000 12 20 " " PUNCT iajs-2000 12 21 where where SCONJ iajs-2000 12 22 a a DET iajs-2000 12 23 proper proper ADJ iajs-2000 12 24 submodule submodule NOUN iajs-2000 12 25 k k PROPN iajs-2000 12 26 of of ADP iajs-2000 12 27 an an DET iajs-2000 12 28 r r NOUN iajs-2000 12 29 - - PUNCT iajs-2000 12 30 module module NOUN iajs-2000 12 31 " " PUNCT iajs-2000 12 32 x x NOUN iajs-2000 12 33 " " PUNCT iajs-2000 12 34 is be AUX iajs-2000 12 35 called call VERB iajs-2000 12 36 a a DET iajs-2000 12 37 weakly weakly ADJ iajs-2000 12 38 semi semi ADJ iajs-2000 12 39 - - ADJ iajs-2000 12 40 prime prime ADJ iajs-2000 12 41 if if SCONJ iajs-2000 12 42 " " PUNCT iajs-2000 12 43 wherever wherever SCONJ iajs-2000 12 44 0 0 NUM iajs-2000 12 45 π‘Ÿ π‘Ÿ X iajs-2000 12 46 π‘₯ π‘₯ PRON iajs-2000 12 47 ∈ ∈ PROPN iajs-2000 12 48 𝐾 𝐾 PROPN iajs-2000 12 49 , , PUNCT iajs-2000 12 50 where where SCONJ iajs-2000 12 51 π‘Ÿ π‘Ÿ X iajs-2000 12 52 ∈ ∈ PROPN iajs-2000 12 53 𝑅 𝑅 PROPN iajs-2000 12 54 , , PUNCT iajs-2000 12 55 π‘₯ π‘₯ PRON iajs-2000 12 56 ∈ ∈ PROPN iajs-2000 12 57 𝑋 𝑋 PROPN iajs-2000 12 58 , , PUNCT iajs-2000 12 59 implies imply VERB iajs-2000 12 60 that that SCONJ iajs-2000 12 61 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2000 12 62 ∈ ∈ PROPN iajs-2000 12 63 𝐾. 𝐾. PROPN iajs-2000 12 64 " " PUNCT iajs-2000 12 65 " " PUNCT iajs-2000 12 66 throughout throughout ADP iajs-2000 12 67 this this DET iajs-2000 12 68 note note NOUN iajs-2000 12 69 all all DET iajs-2000 12 70 rings ring NOUN iajs-2000 12 71 will will AUX iajs-2000 12 72 be be AUX iajs-2000 12 73 commutative commutative ADJ iajs-2000 12 74 with with ADP iajs-2000 12 75 identity identity NOUN iajs-2000 12 76 , , PUNCT iajs-2000 12 77 and and CCONJ iajs-2000 12 78 all all DET iajs-2000 12 79 r r NOUN iajs-2000 12 80 - - PUNCT iajs-2000 12 81 modules module NOUN iajs-2000 12 82 are be AUX iajs-2000 12 83 left leave VERB iajs-2000 12 84 unitary unitary ADJ iajs-2000 12 85 " " PUNCT iajs-2000 12 86 " " PUNCT iajs-2000 12 87 . . PUNCT iajs-2000 13 1 " " PUNCT iajs-2000 13 2 a a DET iajs-2000 13 3 proper proper ADJ iajs-2000 13 4 submodule submodule NOUN iajs-2000 13 5 k k PROPN iajs-2000 13 6 of of ADP iajs-2000 13 7 an an DET iajs-2000 13 8 r r NOUN iajs-2000 13 9 - - PUNCT iajs-2000 13 10 module module NOUN iajs-2000 13 11 x x PRON iajs-2000 13 12 is be AUX iajs-2000 13 13 said say VERB iajs-2000 13 14 to to PART iajs-2000 13 15 be be AUX iajs-2000 13 16 fully fully ADV iajs-2000 13 17 invariant invariant ADJ iajs-2000 13 18 if if SCONJ iajs-2000 13 19 𝑓 𝑓 DET iajs-2000 13 20 𝐾 𝐾 PROPN iajs-2000 13 21 𝐾 𝐾 PROPN iajs-2000 13 22 for for ADP iajs-2000 13 23 each each DET iajs-2000 13 24 𝑓 𝑓 DET iajs-2000 13 25 ∈ ∈ PROPN iajs-2000 13 26 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 13 27 𝑋 𝑋 NOUN iajs-2000 13 28 [ [ X iajs-2000 13 29 3 3 NUM iajs-2000 13 30 ] ] PUNCT iajs-2000 13 31 . . PUNCT iajs-2000 14 1 an an DET iajs-2000 14 2 r r NOUN iajs-2000 14 3 - - PUNCT iajs-2000 14 4 module module NOUN iajs-2000 14 5 m m NOUN iajs-2000 14 6 is be AUX iajs-2000 14 7 called call VERB iajs-2000 14 8 xinjective xinjective PROPN iajs-2000 14 9 " " PUNCT iajs-2000 14 10 , , PUNCT iajs-2000 14 11 if if SCONJ iajs-2000 14 12 for for ADP iajs-2000 14 13 " " PUNCT iajs-2000 14 14 every every DET iajs-2000 14 15 r r NOUN iajs-2000 14 16 - - PUNCT iajs-2000 14 17 homomorphism homomorphism NOUN iajs-2000 14 18 𝑔 𝑔 NOUN iajs-2000 14 19 : : PUNCT iajs-2000 14 20 𝑁 𝑁 PROPN iajs-2000 14 21 ⟢ ⟢ NOUN iajs-2000 14 22 𝑀 𝑀 NOUN iajs-2000 14 23 " " PUNCT iajs-2000 14 24 , , PUNCT iajs-2000 14 25 and and CCONJ iajs-2000 14 26 every every DET iajs-2000 14 27 r r NOUN iajs-2000 14 28 - - PUNCT iajs-2000 14 29 homomorphism homomorphism NOUN iajs-2000 14 30 𝑓 𝑓 X iajs-2000 14 31 : : PUNCT iajs-2000 14 32 𝑁 𝑁 PROPN iajs-2000 14 33 ⟢ ⟢ NOUN iajs-2000 14 34 𝑋 𝑋 NOUN iajs-2000 14 35 , , PUNCT iajs-2000 14 36 there there PRON iajs-2000 14 37 exists exist VERB iajs-2000 14 38 an an DET iajs-2000 14 39 r r NOUN iajs-2000 14 40 - - PUNCT iajs-2000 14 41 homomorphism homomorphism ADJ iajs-2000 14 42 β„Ž β„Ž NOUN iajs-2000 14 43 : : PUNCT iajs-2000 14 44 𝑋 𝑋 PROPN iajs-2000 14 45 ⟢ ⟢ PROPN iajs-2000 14 46 𝑀 𝑀 PROPN iajs-2000 14 47 , , PUNCT iajs-2000 14 48 " " PUNCT iajs-2000 14 49 where where SCONJ iajs-2000 14 50 n n PRON iajs-2000 14 51 is be AUX iajs-2000 14 52 an an DET iajs-2000 14 53 r r NOUN iajs-2000 14 54 - - PUNCT iajs-2000 14 55 module module NOUN iajs-2000 14 56 " " PUNCT iajs-2000 14 57 such such ADJ iajs-2000 14 58 that that SCONJ iajs-2000 14 59 β„Žπ‘œπ‘“ β„Žπ‘œπ‘“ NOUN iajs-2000 14 60 𝑔 𝑔 NOUN iajs-2000 15 1 [ [ X iajs-2000 15 2 5 5 NUM iajs-2000 15 3 ] ] PUNCT iajs-2000 15 4 " " PUNCT iajs-2000 15 5 . . PUNCT iajs-2000 16 1 " " PUNCT iajs-2000 16 2 an an DET iajs-2000 16 3 r r NOUN iajs-2000 16 4 - - PUNCT iajs-2000 16 5 module module NOUN iajs-2000 16 6 p p NOUN iajs-2000 16 7 is be AUX iajs-2000 16 8 called call VERB iajs-2000 16 9 xprojective xprojective ADJ iajs-2000 16 10 if if SCONJ iajs-2000 16 11 for for ADP iajs-2000 16 12 every every DET iajs-2000 16 13 r r NOUN iajs-2000 16 14 - - PUNCT iajs-2000 16 15 homomorphism homomorphism NOUN iajs-2000 16 16 𝑓 𝑓 X iajs-2000 16 17 : : PUNCT iajs-2000 16 18 𝑃 𝑃 NOUN iajs-2000 16 19 ⟢ ⟢ NOUN iajs-2000 16 20 𝑁 𝑁 PROPN iajs-2000 16 21 and and CCONJ iajs-2000 16 22 every every DET iajs-2000 16 23 r r NOUN iajs-2000 16 24 - - PUNCT iajs-2000 16 25 epimorphism epimorphism NOUN iajs-2000 16 26 𝑔 𝑔 NOUN iajs-2000 16 27 : : PUNCT iajs-2000 16 28 𝑀 𝑀 PROPN iajs-2000 16 29 ⟢ ⟢ NOUN iajs-2000 16 30 𝑁 𝑁 PROPN iajs-2000 16 31 , , PUNCT iajs-2000 16 32 there there PRON iajs-2000 16 33 exists exist VERB iajs-2000 16 34 an an DET iajs-2000 16 35 r r NOUN iajs-2000 16 36 - - PUNCT iajs-2000 16 37 homomorphism homomorphism ADJ iajs-2000 16 38 β„Ž β„Ž NOUN iajs-2000 16 39 : : PUNCT iajs-2000 16 40 𝑃 𝑃 NOUN iajs-2000 16 41 ⟢ ⟢ NOUN iajs-2000 16 42 𝑀 𝑀 NOUN iajs-2000 16 43 such such ADJ iajs-2000 16 44 that that PRON iajs-2000 16 45 π‘”π‘œβ„Ž π‘”π‘œβ„Ž VERB iajs-2000 17 1 𝑓 𝑓 PRON iajs-2000 18 1 [ [ X iajs-2000 18 2 5 5 NUM iajs-2000 18 3 ] ] PUNCT iajs-2000 18 4 . . PUNCT iajs-2000 19 1 an an DET iajs-2000 19 2 r r NOUN iajs-2000 19 3 - - PUNCT iajs-2000 19 4 module module NOUN iajs-2000 19 5 x x PRON iajs-2000 19 6 is be AUX iajs-2000 19 7 called call VERB iajs-2000 19 8 a a DET iajs-2000 19 9 scalar scalar ADJ iajs-2000 19 10 module module NOUN iajs-2000 19 11 " " PUNCT iajs-2000 19 12 " " PUNCT iajs-2000 19 13 if if SCONJ iajs-2000 19 14 for for ADP iajs-2000 19 15 each each DET iajs-2000 19 16 " " PUNCT iajs-2000 19 17 𝑓 𝑓 PROPN iajs-2000 19 18 ∈ ∈ PROPN iajs-2000 19 19 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 19 20 𝑋 𝑋 NOUN iajs-2000 19 21 , , PUNCT iajs-2000 19 22 " " PUNCT iajs-2000 19 23 there there PRON iajs-2000 19 24 exists exist VERB iajs-2000 19 25 π‘Ÿ π‘Ÿ X iajs-2000 19 26 ∈ ∈ PROPN iajs-2000 19 27 𝑅 𝑅 PROPN iajs-2000 19 28 such such ADJ iajs-2000 19 29 that that SCONJ iajs-2000 19 30 𝑓 𝑓 DET iajs-2000 19 31 π‘š π‘š NOUN iajs-2000 19 32 π‘Ÿπ‘š π‘Ÿπ‘š ADP iajs-2000 19 33 for for ADP iajs-2000 19 34 each each DET iajs-2000 19 35 π‘š π‘š ADP iajs-2000 19 36 ∈ ∈ PROPN iajs-2000 19 37 𝑋 𝑋 NOUN iajs-2000 19 38 [ [ X iajs-2000 19 39 6 6 NUM iajs-2000 19 40 ] ] X iajs-2000 19 41 " " PUNCT iajs-2000 19 42 . . PUNCT iajs-2000 20 1 2 2 X iajs-2000 20 2 . . X iajs-2000 20 3 we we PRON iajs-2000 20 4 - - PUNCT iajs-2000 20 5 prime prime NOUN iajs-2000 20 6 submodules submodule NOUN iajs-2000 20 7 " " PUNCT iajs-2000 20 8 in in ADP iajs-2000 20 9 this this DET iajs-2000 20 10 section section NOUN iajs-2000 20 11 , , PUNCT iajs-2000 20 12 we we PRON iajs-2000 20 13 introduce introduce VERB iajs-2000 20 14 the the DET iajs-2000 20 15 concept concept NOUN iajs-2000 20 16 we we PRON iajs-2000 20 17 - - PUNCT iajs-2000 20 18 prime prime NOUN iajs-2000 20 19 submodule submodule NOUN iajs-2000 20 20 as as ADP iajs-2000 20 21 a a DET iajs-2000 20 22 stronger strong ADJ iajs-2000 20 23 form form NOUN iajs-2000 20 24 of of ADP iajs-2000 20 25 a a DET iajs-2000 20 26 weakly weakly ADJ iajs-2000 20 27 prime prime ADJ iajs-2000 20 28 submodule submodule NOUN iajs-2000 20 29 , , PUNCT iajs-2000 20 30 and and CCONJ iajs-2000 20 31 established establish VERB iajs-2000 20 32 some some PRON iajs-2000 20 33 of of ADP iajs-2000 20 34 its its PRON iajs-2000 20 35 basic basic ADJ iajs-2000 20 36 properties property NOUN iajs-2000 20 37 , , PUNCT iajs-2000 20 38 examples example NOUN iajs-2000 20 39 and and CCONJ iajs-2000 20 40 characterizations characterization NOUN iajs-2000 20 41 . . PUNCT iajs-2000 21 1 mathematics mathematic NOUN iajs-2000 21 2 | | ADV iajs-2000 21 3 110 110 NUM iajs-2000 21 4 ibn ibn PROPN iajs-2000 21 5 al al PROPN iajs-2000 21 6 - - PUNCT iajs-2000 21 7 haitham haitham PROPN iajs-2000 21 8 jour jour X iajs-2000 21 9 . . PROPN iajs-2000 22 1 for for ADP iajs-2000 22 2 pure pure ADJ iajs-2000 22 3 & & CCONJ iajs-2000 22 4 appl appl PROPN iajs-2000 22 5 . . PUNCT iajs-2000 23 1 sci sci PROPN iajs-2000 23 2 . . PROPN iajs-2000 23 3 ihjpas ihjpa VERB iajs-2000 23 4 https://doi.org/10.30526/31.3.2000 https://doi.org/10.30526/31.3.2000 NUM iajs-2000 23 5 vol vol NOUN iajs-2000 23 6 . . PROPN iajs-2000 23 7 31 31 NUM iajs-2000 23 8 ( ( PUNCT iajs-2000 23 9 3 3 NUM iajs-2000 23 10 ) ) SYM iajs-2000 23 11 2018 2018 NUM iajs-2000 23 12 definition definition NOUN iajs-2000 23 13 ( ( PUNCT iajs-2000 23 14 1 1 X iajs-2000 23 15 ) ) PUNCT iajs-2000 23 16 a a DET iajs-2000 23 17 proper proper ADJ iajs-2000 23 18 submodule submodule NOUN iajs-2000 23 19 k k PROPN iajs-2000 23 20 of of ADP iajs-2000 23 21 an an DET iajs-2000 23 22 r r NOUN iajs-2000 23 23 - - PUNCT iajs-2000 23 24 module module NOUN iajs-2000 23 25 x x PRON iajs-2000 23 26 is be AUX iajs-2000 23 27 said say VERB iajs-2000 23 28 to to PART iajs-2000 23 29 be be AUX iajs-2000 23 30 a a DET iajs-2000 23 31 weakly weakly ADJ iajs-2000 23 32 endo endo NOUN iajs-2000 23 33 - - PUNCT iajs-2000 23 34 prime prime NOUN iajs-2000 23 35 ( ( PUNCT iajs-2000 23 36 for for ADP iajs-2000 23 37 a a DET iajs-2000 23 38 short short ADJ iajs-2000 23 39 we we NOUN iajs-2000 23 40 - - PUNCT iajs-2000 23 41 prime prime NOUN iajs-2000 23 42 ) ) PUNCT iajs-2000 23 43 , , PUNCT iajs-2000 23 44 where where SCONJ iajs-2000 23 45 𝐸 𝐸 PROPN iajs-2000 23 46 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 23 47 𝑋 𝑋 PROPN iajs-2000 23 48 , , PUNCT iajs-2000 23 49 if if SCONJ iajs-2000 23 50 wherever wherever ADV iajs-2000 23 51 , , PUNCT iajs-2000 23 52 0 0 NUM iajs-2000 23 53 πœ“ πœ“ PRON iajs-2000 23 54 π‘₯ π‘₯ X iajs-2000 23 55 ∈ ∈ PROPN iajs-2000 23 56 𝐾 𝐾 PROPN iajs-2000 23 57 , , PUNCT iajs-2000 23 58 where where SCONJ iajs-2000 23 59 πœ“ πœ“ PROPN iajs-2000 23 60 ∈ ∈ PROPN iajs-2000 23 61 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 23 62 𝑋 𝑋 PROPN iajs-2000 23 63 , , PUNCT iajs-2000 23 64 π‘₯ π‘₯ DET iajs-2000 23 65 ∈ ∈ PROPN iajs-2000 23 66 𝑋 𝑋 NOUN iajs-2000 23 67 , , PUNCT iajs-2000 23 68 implies imply VERB iajs-2000 23 69 that that SCONJ iajs-2000 23 70 either either CCONJ iajs-2000 23 71 π‘₯ π‘₯ DET iajs-2000 23 72 ∈ ∈ PROPN iajs-2000 23 73 𝐾 𝐾 PROPN iajs-2000 23 74 or or CCONJ iajs-2000 23 75 πœ“ πœ“ NOUN iajs-2000 23 76 π‘₯ π‘₯ X iajs-2000 24 1 𝐾. 𝐾. NOUN iajs-2000 24 2 " " PUNCT iajs-2000 24 3 and and CCONJ iajs-2000 24 4 an an DET iajs-2000 24 5 ideal ideal ADJ iajs-2000 24 6 i i PRON iajs-2000 24 7 of of ADP iajs-2000 24 8 a a DET iajs-2000 24 9 ring ring NOUN iajs-2000 24 10 r r NOUN iajs-2000 24 11 is be AUX iajs-2000 24 12 said say VERB iajs-2000 24 13 to to PART iajs-2000 24 14 be be AUX iajs-2000 24 15 a a DET iajs-2000 24 16 weakly weakly ADJ iajs-2000 24 17 endo endo NOUN iajs-2000 24 18 - - PUNCT iajs-2000 24 19 prime prime NOUN iajs-2000 24 20 ideal ideal NOUN iajs-2000 24 21 " " PUNCT iajs-2000 24 22 ( ( PUNCT iajs-2000 24 23 we we PRON iajs-2000 24 24 - - PUNCT iajs-2000 24 25 prime prime ADJ iajs-2000 24 26 ideal ideal NOUN iajs-2000 24 27 ) ) PUNCT iajs-2000 24 28 , , PUNCT iajs-2000 24 29 " " PUNCT iajs-2000 24 30 if if SCONJ iajs-2000 24 31 i i PRON iajs-2000 24 32 is be AUX iajs-2000 24 33 a a DET iajs-2000 24 34 we we NOUN iajs-2000 24 35 - - PUNCT iajs-2000 24 36 prime prime NOUN iajs-2000 24 37 as as ADP iajs-2000 24 38 an an DET iajs-2000 24 39 r r NOUN iajs-2000 24 40 - - PUNCT iajs-2000 24 41 submodule submodule NOUN iajs-2000 24 42 of of ADP iajs-2000 24 43 an an DET iajs-2000 24 44 r r NOUN iajs-2000 24 45 - - PUNCT iajs-2000 24 46 module module NOUN iajs-2000 24 47 r r NOUN iajs-2000 24 48 " " PUNCT iajs-2000 24 49 . . PUNCT iajs-2000 25 1 " " PUNCT iajs-2000 25 2 " " PUNCT iajs-2000 25 3 the the DET iajs-2000 25 4 following follow VERB iajs-2000 25 5 " " PUNCT iajs-2000 25 6 proposition proposition NOUN iajs-2000 25 7 gives give VERB iajs-2000 25 8 relation relation NOUN iajs-2000 25 9 of of ADP iajs-2000 25 10 " " PUNCT iajs-2000 25 11 we we PRON iajs-2000 25 12 - - PUNCT iajs-2000 25 13 prime prime NOUN iajs-2000 25 14 submodules submodule NOUN iajs-2000 25 15 and and CCONJ iajs-2000 25 16 weakly weakly ADJ iajs-2000 25 17 prime prime ADJ iajs-2000 25 18 submodules submodule NOUN iajs-2000 25 19 " " PUNCT iajs-2000 25 20 . . PUNCT iajs-2000 26 1 proposition proposition NOUN iajs-2000 26 2 ( ( PUNCT iajs-2000 26 3 2 2 NUM iajs-2000 26 4 ) ) PUNCT iajs-2000 26 5 " " PUNCT iajs-2000 26 6 every every DET iajs-2000 26 7 we we NOUN iajs-2000 26 8 - - PUNCT iajs-2000 26 9 prime prime NOUN iajs-2000 26 10 submodule submodule NOUN iajs-2000 26 11 of of ADP iajs-2000 26 12 an an DET iajs-2000 26 13 r r NOUN iajs-2000 26 14 - - PUNCT iajs-2000 26 15 module module NOUN iajs-2000 26 16 " " PUNCT iajs-2000 26 17 x x NOUN iajs-2000 26 18 " " PUNCT iajs-2000 26 19 is be AUX iajs-2000 26 20 a a DET iajs-2000 26 21 weakly weakly ADJ iajs-2000 26 22 prime prime ADJ iajs-2000 26 23 submodule submodule NOUN iajs-2000 26 24 of of ADP iajs-2000 26 25 x x NOUN iajs-2000 26 26 " " PUNCT iajs-2000 26 27 . . PUNCT iajs-2000 27 1 proof proof NOUN iajs-2000 27 2 " " PUNCT iajs-2000 27 3 " " PUNCT iajs-2000 27 4 assume assume VERB iajs-2000 27 5 that that SCONJ iajs-2000 27 6 k k PROPN iajs-2000 27 7 is be AUX iajs-2000 27 8 a a DET iajs-2000 27 9 we we NOUN iajs-2000 27 10 - - PUNCT iajs-2000 27 11 prime prime NOUN iajs-2000 27 12 submodule submodule NOUN iajs-2000 27 13 of of ADP iajs-2000 27 14 x x PROPN iajs-2000 27 15 , , PUNCT iajs-2000 27 16 and and CCONJ iajs-2000 27 17 " " PUNCT iajs-2000 27 18 0 0 NUM iajs-2000 27 19 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2000 27 20 ∈ ∈ PROPN iajs-2000 27 21 𝐾 𝐾 PROPN iajs-2000 27 22 , , PUNCT iajs-2000 27 23 where where SCONJ iajs-2000 27 24 π‘Ÿ π‘Ÿ X iajs-2000 27 25 ∈ ∈ PROPN iajs-2000 27 26 𝑅 𝑅 PROPN iajs-2000 27 27 , , PUNCT iajs-2000 27 28 π‘₯ π‘₯ DET iajs-2000 27 29 ∈ ∈ PROPN iajs-2000 27 30 𝑋 𝑋 PROPN iajs-2000 27 31 , , PUNCT iajs-2000 27 32 with with ADP iajs-2000 27 33 π‘₯ π‘₯ PROPN iajs-2000 27 34 βˆ‰ βˆ‰ PROPN iajs-2000 27 35 𝐾 𝐾 PROPN iajs-2000 27 36 " " PUNCT iajs-2000 27 37 . . PUNCT iajs-2000 28 1 " " PUNCT iajs-2000 28 2 now now ADV iajs-2000 28 3 , , PUNCT iajs-2000 28 4 let let VERB iajs-2000 28 5 πœ“ πœ“ PRON iajs-2000 28 6 : : PUNCT iajs-2000 28 7 𝑋 𝑋 PROPN iajs-2000 28 8 ⟢ ⟢ NOUN iajs-2000 28 9 𝑋 𝑋 PROPN iajs-2000 28 10 be be VERB iajs-2000 28 11 a a DET iajs-2000 28 12 mapping mapping NOUN iajs-2000 28 13 defined define VERB iajs-2000 28 14 by by ADP iajs-2000 28 15 πœ“ πœ“ PROPN iajs-2000 28 16 π‘₯ π‘₯ X iajs-2000 28 17 π‘Ÿπ‘₯ π‘Ÿπ‘₯ NOUN iajs-2000 28 18 for for ADP iajs-2000 28 19 all all PRON iajs-2000 28 20 π‘₯ π‘₯ DET iajs-2000 28 21 ∈ ∈ PROPN iajs-2000 28 22 𝑋. 𝑋. PROPN iajs-2000 28 23 clearly clearly ADV iajs-2000 28 24 πœ“ πœ“ ADP iajs-2000 28 25 ∈ ∈ PROPN iajs-2000 28 26 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 28 27 𝑋 𝑋 PROPN iajs-2000 28 28 . . PUNCT iajs-2000 29 1 in in ADP iajs-2000 29 2 fact fact NOUN iajs-2000 29 3 we we PRON iajs-2000 29 4 have have VERB iajs-2000 29 5 0 0 NUM iajs-2000 29 6 π‘Ÿπ‘₯ π‘Ÿπ‘₯ NOUN iajs-2000 29 7 πœ“ πœ“ NOUN iajs-2000 29 8 π‘₯ π‘₯ PRON iajs-2000 29 9 ∈ ∈ PROPN iajs-2000 29 10 𝐾 𝐾 PROPN iajs-2000 29 11 " " PUNCT iajs-2000 29 12 . . PUNCT iajs-2000 30 1 " " PUNCT iajs-2000 30 2 " " PUNCT iajs-2000 30 3 but but CCONJ iajs-2000 30 4 k k PROPN iajs-2000 30 5 is be AUX iajs-2000 30 6 a a DET iajs-2000 30 7 we we NOUN iajs-2000 30 8 - - PUNCT iajs-2000 30 9 prime prime NOUN iajs-2000 30 10 submodule submodule NOUN iajs-2000 30 11 of of ADP iajs-2000 30 12 x x PROPN iajs-2000 30 13 , , PUNCT iajs-2000 30 14 and and CCONJ iajs-2000 30 15 π‘₯ π‘₯ PROPN iajs-2000 30 16 βˆ‰ βˆ‰ PROPN iajs-2000 30 17 𝐾 𝐾 PROPN iajs-2000 30 18 " " PUNCT iajs-2000 30 19 , , PUNCT iajs-2000 30 20 implies imply VERB iajs-2000 30 21 that that SCONJ iajs-2000 30 22 πœ“ πœ“ PROPN iajs-2000 30 23 π‘₯ π‘₯ PROPN iajs-2000 30 24 𝐾 𝐾 PROPN iajs-2000 30 25 , , PUNCT iajs-2000 30 26 hence hence ADV iajs-2000 30 27 π‘Ÿπ‘₯ π‘Ÿπ‘₯ ADP iajs-2000 30 28 𝐾 𝐾 PROPN iajs-2000 30 29 , , PUNCT iajs-2000 30 30 so so SCONJ iajs-2000 30 31 π‘Ÿ π‘Ÿ X iajs-2000 30 32 ∈ ∈ PROPN iajs-2000 30 33 𝐾 𝐾 PROPN iajs-2000 30 34 : : PUNCT iajs-2000 30 35 𝑋 𝑋 NOUN iajs-2000 30 36 . . PUNCT iajs-2000 31 1 " " PUNCT iajs-2000 31 2 therefore therefore ADV iajs-2000 31 3 k k PROPN iajs-2000 31 4 is be AUX iajs-2000 31 5 a a DET iajs-2000 31 6 weakly weakly ADJ iajs-2000 31 7 prime prime ADJ iajs-2000 31 8 submodule submodule NOUN iajs-2000 31 9 of of ADP iajs-2000 31 10 x x NOUN iajs-2000 31 11 " " PUNCT iajs-2000 31 12 . . PUNCT iajs-2000 32 1 the the DET iajs-2000 32 2 converse converse NOUN iajs-2000 32 3 of of ADP iajs-2000 32 4 proposition proposition NOUN iajs-2000 32 5 ( ( PUNCT iajs-2000 32 6 2 2 NUM iajs-2000 32 7 ) ) PUNCT iajs-2000 32 8 " " PUNCT iajs-2000 32 9 " " PUNCT iajs-2000 32 10 is be AUX iajs-2000 32 11 not not PART iajs-2000 32 12 true true ADJ iajs-2000 32 13 in in ADP iajs-2000 32 14 general general ADJ iajs-2000 32 15 , , PUNCT iajs-2000 32 16 as as SCONJ iajs-2000 32 17 the the DET iajs-2000 32 18 following follow VERB iajs-2000 32 19 example example NOUN iajs-2000 32 20 shows show VERB iajs-2000 32 21 " " PUNCT iajs-2000 32 22 . . PUNCT iajs-2000 33 1 example example NOUN iajs-2000 33 2 ( ( PUNCT iajs-2000 33 3 3 3 NUM iajs-2000 33 4 ) ) PUNCT iajs-2000 33 5 " " PUNCT iajs-2000 33 6 " " PUNCT iajs-2000 33 7 let let VERB iajs-2000 33 8 𝑋 𝑋 NOUN iajs-2000 33 9 𝑍 𝑍 PROPN iajs-2000 33 10 ⨁ ⨁ PROPN iajs-2000 33 11 𝑍 𝑍 NOUN iajs-2000 33 12 " " PUNCT iajs-2000 33 13 and and CCONJ iajs-2000 33 14 r r NOUN iajs-2000 33 15 = = SYM iajs-2000 33 16 z z PROPN iajs-2000 33 17 , , PUNCT iajs-2000 33 18 𝐾 𝐾 PROPN iajs-2000 33 19 〈 〈 PROPN iajs-2000 33 20 0βŒͺ⨁3𝑍. 0βŒͺ⨁3𝑍. NOUN iajs-2000 33 21 clearly clearly ADV iajs-2000 33 22 k k X iajs-2000 33 23 " " PUNCT iajs-2000 33 24 is be AUX iajs-2000 33 25 a a DET iajs-2000 33 26 weakly weakly ADJ iajs-2000 33 27 prime prime ADJ iajs-2000 33 28 submodule submodule NOUN iajs-2000 33 29 of of ADP iajs-2000 33 30 x x PRON iajs-2000 33 31 , , PUNCT iajs-2000 33 32 but but CCONJ iajs-2000 33 33 k k PROPN iajs-2000 33 34 is be AUX iajs-2000 33 35 not not PART iajs-2000 33 36 we we PRON iajs-2000 33 37 - - PUNCT iajs-2000 33 38 prime prime NOUN iajs-2000 33 39 submodule submodule NOUN iajs-2000 33 40 of of ADP iajs-2000 33 41 x x NOUN iajs-2000 33 42 " " PUNCT iajs-2000 33 43 . . PUNCT iajs-2000 34 1 since since SCONJ iajs-2000 34 2 we we PRON iajs-2000 34 3 define define VERB iajs-2000 34 4 πœ“ πœ“ NOUN iajs-2000 34 5 : : PUNCT iajs-2000 34 6 𝑋 𝑋 PROPN iajs-2000 34 7 ⟢ ⟢ NOUN iajs-2000 34 8 𝑋 𝑋 NOUN iajs-2000 34 9 " " PUNCT iajs-2000 34 10 by by ADP iajs-2000 34 11 πœ“ πœ“ NOUN iajs-2000 34 12 π‘Ž π‘Ž PROPN iajs-2000 34 13 , , PUNCT iajs-2000 34 14 𝑏 𝑏 PROPN iajs-2000 34 15 0 0 NUM iajs-2000 34 16 , , PUNCT iajs-2000 34 17 𝑏 𝑏 PROPN iajs-2000 34 18 for for ADP iajs-2000 34 19 all all DET iajs-2000 34 20 π‘Ž π‘Ž NOUN iajs-2000 34 21 , , PUNCT iajs-2000 34 22 𝑏 𝑏 PROPN iajs-2000 34 23 ∈ ∈ PROPN iajs-2000 34 24 𝑋 𝑋 PROPN iajs-2000 34 25 " " PUNCT iajs-2000 34 26 . . PUNCT iajs-2000 35 1 clearly clearly ADV iajs-2000 35 2 πœ“ πœ“ X iajs-2000 35 3 ∈ ∈ PROPN iajs-2000 35 4 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 35 5 𝑋 𝑋 PROPN iajs-2000 35 6 . . PUNCT iajs-2000 36 1 now now ADV iajs-2000 36 2 0 0 NUM iajs-2000 36 3 , , PUNCT iajs-2000 36 4 0 0 NUM iajs-2000 36 5 πœ“ πœ“ PROPN iajs-2000 36 6 1 1 NUM iajs-2000 36 7 , , PUNCT iajs-2000 36 8 3 3 NUM iajs-2000 36 9 0 0 NUM iajs-2000 36 10 , , PUNCT iajs-2000 36 11 3 3 NUM iajs-2000 36 12 ∈ ∈ PROPN iajs-2000 36 13 𝐾 𝐾 PROPN iajs-2000 36 14 , , PUNCT iajs-2000 36 15 but but CCONJ iajs-2000 36 16 1 1 NUM iajs-2000 36 17 , , PUNCT iajs-2000 36 18 3 3 NUM iajs-2000 36 19 βˆ‰ βˆ‰ PROPN iajs-2000 36 20 𝐾 𝐾 PROPN iajs-2000 36 21 and and CCONJ iajs-2000 36 22 πœ“ πœ“ ADP iajs-2000 36 23 𝑋 𝑋 PROPN iajs-2000 36 24 0 0 NUM iajs-2000 36 25 ⨁𝑍 ⨁𝑍 PROPN iajs-2000 36 26 β‰° β‰° PROPN iajs-2000 36 27 𝐾 𝐾 PROPN iajs-2000 36 28 " " PUNCT iajs-2000 36 29 . . PUNCT iajs-2000 37 1 " " PUNCT iajs-2000 37 2 " " PUNCT iajs-2000 37 3 the the DET iajs-2000 37 4 converse converse NOUN iajs-2000 37 5 of of ADP iajs-2000 37 6 proposition proposition NOUN iajs-2000 37 7 ( ( PUNCT iajs-2000 37 8 2 2 NUM iajs-2000 37 9 ) ) PUNCT iajs-2000 37 10 " " PUNCT iajs-2000 37 11 " " PUNCT iajs-2000 37 12 is be AUX iajs-2000 37 13 true true ADJ iajs-2000 37 14 in in ADP iajs-2000 37 15 the the DET iajs-2000 37 16 class class NOUN iajs-2000 37 17 of of ADP iajs-2000 37 18 cyclic cyclic ADJ iajs-2000 37 19 r r NOUN iajs-2000 37 20 - - PUNCT iajs-2000 37 21 modules module NOUN iajs-2000 37 22 , , PUNCT iajs-2000 37 23 as as SCONJ iajs-2000 37 24 the the DET iajs-2000 37 25 following follow VERB iajs-2000 37 26 proposition proposition NOUN iajs-2000 37 27 shows show VERB iajs-2000 37 28 " " PUNCT iajs-2000 37 29 . . PUNCT iajs-2000 38 1 proposition proposition NOUN iajs-2000 38 2 ( ( PUNCT iajs-2000 38 3 4 4 NUM iajs-2000 38 4 ) ) PUNCT iajs-2000 38 5 " " PUNCT iajs-2000 38 6 " " PUNCT iajs-2000 38 7 let let VERB iajs-2000 38 8 x x PRON iajs-2000 38 9 be be AUX iajs-2000 38 10 a a DET iajs-2000 38 11 cyclic cyclic ADJ iajs-2000 38 12 r r NOUN iajs-2000 38 13 - - PUNCT iajs-2000 38 14 module module NOUN iajs-2000 38 15 , , PUNCT iajs-2000 38 16 and and CCONJ iajs-2000 38 17 " " PUNCT iajs-2000 38 18 k k X iajs-2000 38 19 is be AUX iajs-2000 38 20 a a DET iajs-2000 38 21 " " PUNCT iajs-2000 38 22 proper proper ADJ iajs-2000 38 23 submodule submodule NOUN iajs-2000 38 24 of of ADP iajs-2000 38 25 x x SYM iajs-2000 38 26 such such ADJ iajs-2000 38 27 that that SCONJ iajs-2000 38 28 k k PROPN iajs-2000 38 29 is be AUX iajs-2000 38 30 a a DET iajs-2000 38 31 weakly weakly ADJ iajs-2000 38 32 prime prime ADJ iajs-2000 38 33 submodule submodule NOUN iajs-2000 38 34 of of ADP iajs-2000 38 35 x. x. PROPN iajs-2000 38 36 "then "then PROPN iajs-2000 39 1 k k PROPN iajs-2000 39 2 is be AUX iajs-2000 39 3 a a DET iajs-2000 39 4 we we NOUN iajs-2000 39 5 - - PUNCT iajs-2000 39 6 prime prime NOUN iajs-2000 39 7 submodule submodule NOUN iajs-2000 39 8 of of ADP iajs-2000 39 9 x x NOUN iajs-2000 39 10 " " PUNCT iajs-2000 39 11 . . PUNCT iajs-2000 40 1 proof proof NOUN iajs-2000 40 2 " " PUNCT iajs-2000 40 3 " " PUNCT iajs-2000 40 4 assume assume VERB iajs-2000 40 5 that that SCONJ iajs-2000 40 6 k k PROPN iajs-2000 40 7 is be AUX iajs-2000 40 8 a a DET iajs-2000 40 9 weakly weakly ADJ iajs-2000 40 10 prime prime ADJ iajs-2000 40 11 submodule submodule NOUN iajs-2000 40 12 of of ADP iajs-2000 40 13 cyclic cyclic ADJ iajs-2000 40 14 r r NOUN iajs-2000 40 15 - - PUNCT iajs-2000 40 16 module module NOUN iajs-2000 40 17 x x NOUN iajs-2000 40 18 " " PUNCT iajs-2000 40 19 , , PUNCT iajs-2000 40 20 " " PUNCT iajs-2000 40 21 where where SCONJ iajs-2000 40 22 𝑋 𝑋 PROPN iajs-2000 40 23 π‘…π‘š π‘…π‘š PROPN iajs-2000 40 24 , , PUNCT iajs-2000 40 25 π‘š π‘š PROPN iajs-2000 40 26 ∈ ∈ PROPN iajs-2000 40 27 𝑋 𝑋 PROPN iajs-2000 40 28 " " PUNCT iajs-2000 40 29 . . PUNCT iajs-2000 41 1 " " PUNCT iajs-2000 41 2 suppose suppose VERB iajs-2000 41 3 that that SCONJ iajs-2000 41 4 0 0 NUM iajs-2000 41 5 πœ“ πœ“ PRON iajs-2000 41 6 π‘₯ π‘₯ X iajs-2000 41 7 ∈ ∈ PROPN iajs-2000 41 8 𝐾 𝐾 PROPN iajs-2000 41 9 , , PUNCT iajs-2000 41 10 where where SCONJ iajs-2000 41 11 πœ“ πœ“ PROPN iajs-2000 41 12 ∈ ∈ PROPN iajs-2000 41 13 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 41 14 𝑋 𝑋 PROPN iajs-2000 41 15 , , PUNCT iajs-2000 41 16 π‘₯ π‘₯ DET iajs-2000 41 17 ∈ ∈ PROPN iajs-2000 41 18 𝑋 𝑋 NOUN iajs-2000 41 19 and and CCONJ iajs-2000 41 20 π‘₯ π‘₯ PROPN iajs-2000 41 21 βˆ‰ βˆ‰ PROPN iajs-2000 41 22 𝐾 𝐾 PROPN iajs-2000 41 23 " " PUNCT iajs-2000 41 24 . . PUNCT iajs-2000 42 1 " " PUNCT iajs-2000 42 2 now now ADV iajs-2000 42 3 , , PUNCT iajs-2000 42 4 let let VERB iajs-2000 42 5 𝑦 𝑦 PRON iajs-2000 42 6 ∈ ∈ PROPN iajs-2000 42 7 𝑋 𝑋 PROPN iajs-2000 42 8 , , PUNCT iajs-2000 42 9 then then ADV iajs-2000 42 10 𝑦 𝑦 NOUN iajs-2000 42 11 π‘Ÿπ‘š π‘Ÿπ‘š NOUN iajs-2000 42 12 and and CCONJ iajs-2000 42 13 π‘₯ π‘₯ NOUN iajs-2000 42 14 π‘Ÿ π‘Ÿ NOUN iajs-2000 42 15 π‘š π‘š NOUN iajs-2000 42 16 for for ADP iajs-2000 42 17 some some DET iajs-2000 42 18 π‘Ÿ π‘Ÿ NOUN iajs-2000 42 19 , , PUNCT iajs-2000 42 20 π‘Ÿ π‘Ÿ X iajs-2000 42 21 ∈ ∈ PROPN iajs-2000 42 22 𝑅 𝑅 PROPN iajs-2000 42 23 " " PUNCT iajs-2000 42 24 . . PUNCT iajs-2000 43 1 " " PUNCT iajs-2000 43 2 thus thus ADV iajs-2000 43 3 , , PUNCT iajs-2000 43 4 0 0 NUM iajs-2000 43 5 πœ“ πœ“ PART iajs-2000 43 6 π‘₯ π‘₯ VERB iajs-2000 43 7 π‘Ÿ π‘Ÿ X iajs-2000 43 8 πœ“ πœ“ X iajs-2000 43 9 π‘š π‘š PROPN iajs-2000 43 10 ∈ ∈ PROPN iajs-2000 43 11 𝐾 𝐾 PROPN iajs-2000 43 12 , , PUNCT iajs-2000 43 13 but but CCONJ iajs-2000 43 14 k k PROPN iajs-2000 43 15 is be AUX iajs-2000 43 16 a a DET iajs-2000 43 17 weakly weakly ADJ iajs-2000 43 18 prime prime ADJ iajs-2000 43 19 submodule submodule NOUN iajs-2000 43 20 of of ADP iajs-2000 43 21 x x PRON iajs-2000 43 22 , , PUNCT iajs-2000 43 23 then then ADV iajs-2000 43 24 either either CCONJ iajs-2000 43 25 π‘Ÿ π‘Ÿ PRON iajs-2000 43 26 ∈ ∈ PROPN iajs-2000 43 27 𝐾 𝐾 PROPN iajs-2000 43 28 : : PUNCT iajs-2000 43 29 𝑋 𝑋 NOUN iajs-2000 43 30 or or CCONJ iajs-2000 43 31 πœ‘ πœ‘ PROPN iajs-2000 43 32 π‘š π‘š PROPN iajs-2000 43 33 ∈ ∈ PROPN iajs-2000 43 34 𝐾 𝐾 PROPN iajs-2000 43 35 " " PUNCT iajs-2000 43 36 . . PUNCT iajs-2000 44 1 " " PUNCT iajs-2000 44 2 but but CCONJ iajs-2000 44 3 π‘Ÿ π‘Ÿ X iajs-2000 44 4 βˆ‰ βˆ‰ PROPN iajs-2000 44 5 𝐾 𝐾 PROPN iajs-2000 44 6 : : PUNCT iajs-2000 44 7 𝑋 𝑋 NOUN iajs-2000 44 8 for for ADP iajs-2000 44 9 π‘₯ π‘₯ PART iajs-2000 44 10 π‘Ÿ π‘Ÿ NOUN iajs-2000 44 11 π‘š π‘š X iajs-2000 44 12 βˆ‰ βˆ‰ X iajs-2000 44 13 𝐾. 𝐾. PROPN iajs-2000 44 14 hence hence ADV iajs-2000 44 15 πœ“ πœ“ VERB iajs-2000 44 16 π‘š π‘š PROPN iajs-2000 44 17 ∈ ∈ PROPN iajs-2000 44 18 𝐾 𝐾 PROPN iajs-2000 44 19 , , PUNCT iajs-2000 44 20 hence hence ADV iajs-2000 44 21 πœ“ πœ“ X iajs-2000 44 22 𝑦 𝑦 PROPN iajs-2000 44 23 π‘Ÿπœ“ π‘Ÿπœ“ NOUN iajs-2000 44 24 π‘š π‘š NOUN iajs-2000 44 25 ∈ ∈ NOUN iajs-2000 44 26 𝐾. 𝐾. NOUN iajs-2000 44 27 therefore therefore ADV iajs-2000 44 28 πœ“ πœ“ VERB iajs-2000 44 29 𝑋 𝑋 PROPN iajs-2000 44 30 𝐾 𝐾 PROPN iajs-2000 44 31 " " PUNCT iajs-2000 44 32 . . PUNCT iajs-2000 45 1 corollary corollary ADJ iajs-2000 45 2 ( ( PUNCT iajs-2000 45 3 5 5 NUM iajs-2000 45 4 ) ) PUNCT iajs-2000 45 5 let let VERB iajs-2000 45 6 k k PRON iajs-2000 45 7 be be AUX iajs-2000 45 8 a a DET iajs-2000 45 9 proper proper ADJ iajs-2000 45 10 submodule submodule NOUN iajs-2000 45 11 of of ADP iajs-2000 45 12 a a DET iajs-2000 45 13 cyclic cyclic ADJ iajs-2000 45 14 r r NOUN iajs-2000 45 15 - - PUNCT iajs-2000 45 16 module module NOUN iajs-2000 45 17 x x NOUN iajs-2000 45 18 " " PUNCT iajs-2000 45 19 . . PUNCT iajs-2000 46 1 " " PUNCT iajs-2000 46 2 then then ADV iajs-2000 46 3 k k PROPN iajs-2000 46 4 is be AUX iajs-2000 46 5 a a DET iajs-2000 46 6 we we NOUN iajs-2000 46 7 - - PUNCT iajs-2000 46 8 prime prime NOUN iajs-2000 46 9 if if SCONJ iajs-2000 47 1 and and CCONJ iajs-2000 47 2 only only ADV iajs-2000 47 3 if if SCONJ iajs-2000 47 4 k k PROPN iajs-2000 47 5 is be AUX iajs-2000 47 6 a a DET iajs-2000 47 7 weakly weakly ADJ iajs-2000 47 8 prime prime ADJ iajs-2000 47 9 submodule submodule NOUN iajs-2000 47 10 of of ADP iajs-2000 47 11 x. x. NOUN iajs-2000 47 12 proposition proposition NOUN iajs-2000 47 13 ( ( PUNCT iajs-2000 47 14 6 6 X iajs-2000 47 15 ) ) PUNCT iajs-2000 47 16 let let VERB iajs-2000 47 17 x x PRON iajs-2000 47 18 be be AUX iajs-2000 47 19 a a DET iajs-2000 47 20 faithful faithful ADJ iajs-2000 47 21 r r NOUN iajs-2000 47 22 - - PUNCT iajs-2000 47 23 module module NOUN iajs-2000 47 24 " " PUNCT iajs-2000 47 25 , , PUNCT iajs-2000 47 26 " " PUNCT iajs-2000 47 27 and and CCONJ iajs-2000 47 28 k k PROPN iajs-2000 47 29 is be AUX iajs-2000 47 30 a a DET iajs-2000 47 31 we we NOUN iajs-2000 47 32 - - PUNCT iajs-2000 47 33 prime prime NOUN iajs-2000 47 34 submodule submodule NOUN iajs-2000 47 35 of of ADP iajs-2000 47 36 x x NOUN iajs-2000 47 37 " " PUNCT iajs-2000 47 38 . . PUNCT iajs-2000 48 1 " " PUNCT iajs-2000 48 2 then then ADV iajs-2000 48 3 𝐾 𝐾 PROPN iajs-2000 48 4 : : PUNCT iajs-2000 48 5 𝑋 𝑋 NOUN iajs-2000 48 6 is be AUX iajs-2000 48 7 a a DET iajs-2000 48 8 we we NOUN iajs-2000 48 9 - - PUNCT iajs-2000 48 10 prime prime ADJ iajs-2000 48 11 ideal ideal NOUN iajs-2000 48 12 of of ADP iajs-2000 48 13 r. r. PROPN iajs-2000 48 14 mathematics mathematics PROPN iajs-2000 48 15 | | ADV iajs-2000 48 16 111 111 NUM iajs-2000 48 17 ibn ibn PROPN iajs-2000 48 18 al al PROPN iajs-2000 48 19 - - PUNCT iajs-2000 48 20 haitham haitham PROPN iajs-2000 48 21 jour jour X iajs-2000 48 22 . . PROPN iajs-2000 49 1 for for ADP iajs-2000 49 2 pure pure ADJ iajs-2000 49 3 & & CCONJ iajs-2000 49 4 appl appl PROPN iajs-2000 49 5 . . PUNCT iajs-2000 50 1 sci sci PROPN iajs-2000 50 2 . . PROPN iajs-2000 50 3 ihjpas ihjpa VERB iajs-2000 50 4 https://doi.org/10.30526/31.3.2000 https://doi.org/10.30526/31.3.2000 NUM iajs-2000 50 5 vol vol NOUN iajs-2000 50 6 . . PROPN iajs-2000 50 7 31 31 NUM iajs-2000 50 8 ( ( PUNCT iajs-2000 50 9 3 3 NUM iajs-2000 50 10 ) ) SYM iajs-2000 50 11 2018 2018 NUM iajs-2000 50 12 proof proof NOUN iajs-2000 50 13 " " PUNCT iajs-2000 50 14 since since SCONJ iajs-2000 50 15 k k PROPN iajs-2000 50 16 is be AUX iajs-2000 50 17 a a DET iajs-2000 50 18 we we NOUN iajs-2000 50 19 - - PUNCT iajs-2000 50 20 prime prime NOUN iajs-2000 50 21 submodule submodule NOUN iajs-2000 50 22 of of ADP iajs-2000 50 23 x x NOUN iajs-2000 50 24 " " PUNCT iajs-2000 50 25 , , PUNCT iajs-2000 50 26 " " PUNCT iajs-2000 50 27 then then ADV iajs-2000 50 28 by by ADP iajs-2000 50 29 proposition proposition NOUN iajs-2000 50 30 ( ( PUNCT iajs-2000 50 31 2.2 2.2 NUM iajs-2000 50 32 ) ) PUNCT iajs-2000 50 33 , , PUNCT iajs-2000 50 34 k k PROPN iajs-2000 50 35 is be AUX iajs-2000 50 36 a a DET iajs-2000 50 37 weakly weakly ADJ iajs-2000 50 38 prime prime ADJ iajs-2000 50 39 submodule submodule NOUN iajs-2000 50 40 of of ADP iajs-2000 50 41 x x NOUN iajs-2000 50 42 " " PUNCT iajs-2000 50 43 . . PUNCT iajs-2000 51 1 " " PUNCT iajs-2000 51 2 hence hence ADV iajs-2000 51 3 by by ADP iajs-2000 51 4 [ [ X iajs-2000 51 5 1 1 NUM iajs-2000 51 6 , , PUNCT iajs-2000 51 7 prop.2.4 prop.2.4 PROPN iajs-2000 51 8 ] ] PUNCT iajs-2000 51 9 " " PUNCT iajs-2000 51 10 , , PUNCT iajs-2000 51 11 we we PRON iajs-2000 51 12 get get VERB iajs-2000 51 13 𝐾 𝐾 NOUN iajs-2000 51 14 : : PUNCT iajs-2000 51 15 𝑋 𝑋 PROPN iajs-2000 51 16 " " PUNCT iajs-2000 51 17 is be AUX iajs-2000 51 18 a a DET iajs-2000 51 19 weakly weakly ADJ iajs-2000 51 20 prime prime ADJ iajs-2000 51 21 ideal ideal NOUN iajs-2000 51 22 of of ADP iajs-2000 51 23 r. r. PROPN iajs-2000 51 24 but but CCONJ iajs-2000 51 25 r r NOUN iajs-2000 51 26 is be AUX iajs-2000 51 27 a a DET iajs-2000 51 28 cyclic cyclic ADJ iajs-2000 51 29 r r NOUN iajs-2000 51 30 - - PUNCT iajs-2000 51 31 module module NOUN iajs-2000 51 32 " " PUNCT iajs-2000 51 33 , , PUNCT iajs-2000 51 34 " " PUNCT iajs-2000 51 35 then then ADV iajs-2000 51 36 by by ADP iajs-2000 51 37 proposition proposition NOUN iajs-2000 51 38 ( ( PUNCT iajs-2000 51 39 2.4 2.4 NUM iajs-2000 51 40 ) ) PUNCT iajs-2000 51 41 , , PUNCT iajs-2000 51 42 we we PRON iajs-2000 51 43 get get VERB iajs-2000 51 44 𝐾 𝐾 NOUN iajs-2000 51 45 : : PUNCT iajs-2000 51 46 𝑋 𝑋 NOUN iajs-2000 51 47 is be AUX iajs-2000 51 48 a a DET iajs-2000 51 49 we we NOUN iajs-2000 51 50 - - PUNCT iajs-2000 51 51 prime prime ADJ iajs-2000 51 52 ideal ideal NOUN iajs-2000 51 53 of of ADP iajs-2000 51 54 r r NOUN iajs-2000 51 55 " " PUNCT iajs-2000 51 56 . . PUNCT iajs-2000 52 1 " " PUNCT iajs-2000 52 2 we we PRON iajs-2000 52 3 need need VERB iajs-2000 52 4 to to PART iajs-2000 52 5 recall recall VERB iajs-2000 52 6 the the DET iajs-2000 52 7 following following ADJ iajs-2000 52 8 result result NOUN iajs-2000 52 9 before before SCONJ iajs-2000 52 10 we we PRON iajs-2000 52 11 introduce introduce VERB iajs-2000 52 12 the the DET iajs-2000 52 13 next next ADJ iajs-2000 52 14 proposition proposition NOUN iajs-2000 52 15 " " PUNCT iajs-2000 52 16 . . PUNCT iajs-2000 53 1 lemma lemma PROPN iajs-2000 53 2 ( ( PUNCT iajs-2000 53 3 7 7 NUM iajs-2000 53 4 ) ) PUNCT iajs-2000 53 5 [ [ X iajs-2000 53 6 3 3 NUM iajs-2000 53 7 ] ] PUNCT iajs-2000 53 8 " " PUNCT iajs-2000 53 9 let let VERB iajs-2000 53 10 n n PRON iajs-2000 53 11 and and CCONJ iajs-2000 53 12 k k PROPN iajs-2000 53 13 be be AUX iajs-2000 53 14 " " PUNCT iajs-2000 53 15 " " PUNCT iajs-2000 53 16 two two NUM iajs-2000 53 17 submodules submodule NOUN iajs-2000 53 18 of of ADP iajs-2000 53 19 an an DET iajs-2000 53 20 r r NOUN iajs-2000 53 21 - - PUNCT iajs-2000 53 22 module module NOUN iajs-2000 53 23 x x NOUN iajs-2000 53 24 , , PUNCT iajs-2000 53 25 then then ADV iajs-2000 53 26 1 1 NUM iajs-2000 53 27 . . PUNCT iajs-2000 54 1 " " PUNCT iajs-2000 54 2 if if SCONJ iajs-2000 54 3 𝑁 𝑁 PROPN iajs-2000 54 4 𝐾 𝐾 PROPN iajs-2000 54 5 , , PUNCT iajs-2000 54 6 then then ADV iajs-2000 54 7 𝑁 𝑁 PROPN iajs-2000 54 8 : : PUNCT iajs-2000 54 9 𝑋 𝑋 PROPN iajs-2000 54 10 𝐾 𝐾 PROPN iajs-2000 54 11 : : PUNCT iajs-2000 54 12 𝑋 𝑋 PROPN iajs-2000 54 13 " " PUNCT iajs-2000 54 14 . . PUNCT iajs-2000 55 1 2 2 X iajs-2000 55 2 . . PUNCT iajs-2000 55 3 " " PUNCT iajs-2000 55 4 if if SCONJ iajs-2000 55 5 𝑁 𝑁 PROPN iajs-2000 55 6 𝐾 𝐾 PROPN iajs-2000 55 7 , , PUNCT iajs-2000 55 8 then then ADV iajs-2000 55 9 𝑁 𝑁 PROPN iajs-2000 55 10 : : PUNCT iajs-2000 55 11 𝑋 𝑋 PROPN iajs-2000 55 12 𝑁 𝑁 PROPN iajs-2000 55 13 : : PUNCT iajs-2000 55 14 𝐾 𝐾 NOUN iajs-2000 55 15 " " PUNCT iajs-2000 55 16 . . PUNCT iajs-2000 56 1 " " PUNCT iajs-2000 56 2 " " PUNCT iajs-2000 56 3 the the DET iajs-2000 56 4 following follow VERB iajs-2000 56 5 proposition proposition NOUN iajs-2000 56 6 is be AUX iajs-2000 56 7 a a DET iajs-2000 56 8 characterization characterization NOUN iajs-2000 56 9 of of ADP iajs-2000 56 10 a a DET iajs-2000 56 11 we we NOUN iajs-2000 56 12 - - PUNCT iajs-2000 56 13 prime prime ADJ iajs-2000 56 14 submodules submodule NOUN iajs-2000 56 15 " " PUNCT iajs-2000 56 16 . . PUNCT iajs-2000 57 1 proposition proposition NOUN iajs-2000 57 2 ( ( PUNCT iajs-2000 57 3 8) 8) NUM iajs-2000 57 4 " " PUNCT iajs-2000 57 5 let let VERB iajs-2000 57 6 k k PRON iajs-2000 57 7 be be AUX iajs-2000 57 8 a a DET iajs-2000 57 9 proper proper ADJ iajs-2000 57 10 fully fully ADV iajs-2000 57 11 invariant invariant ADJ iajs-2000 57 12 submodule submodule NOUN iajs-2000 57 13 of of ADP iajs-2000 57 14 an an DET iajs-2000 57 15 r r NOUN iajs-2000 57 16 - - PUNCT iajs-2000 57 17 module module NOUN iajs-2000 57 18 x x NOUN iajs-2000 57 19 " " PUNCT iajs-2000 57 20 . . PUNCT iajs-2000 58 1 " " PUNCT iajs-2000 58 2 then then ADV iajs-2000 58 3 k k PROPN iajs-2000 58 4 is be AUX iajs-2000 58 5 a a DET iajs-2000 58 6 we we NOUN iajs-2000 58 7 - - PUNCT iajs-2000 58 8 prime prime NOUN iajs-2000 58 9 submodule submodule NOUN iajs-2000 58 10 of of ADP iajs-2000 58 11 x x SYM iajs-2000 58 12 if if SCONJ iajs-2000 58 13 and and CCONJ iajs-2000 58 14 only only ADV iajs-2000 58 15 if if SCONJ iajs-2000 58 16 " " PUNCT iajs-2000 58 17 𝐾 𝐾 NOUN iajs-2000 58 18 : : PUNCT iajs-2000 58 19 πœ“ πœ“ PROPN iajs-2000 58 20 𝑋 𝑋 PROPN iajs-2000 58 21 𝐾 𝐾 PROPN iajs-2000 58 22 : : PUNCT iajs-2000 58 23 πœ“ πœ“ PROPN iajs-2000 58 24 𝐻 𝐻 PROPN iajs-2000 58 25 for for ADP iajs-2000 58 26 all all PRON iajs-2000 58 27 πœ“ πœ“ PRON iajs-2000 58 28 ∈ ∈ PROPN iajs-2000 58 29 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 58 30 𝑋 𝑋 PROPN iajs-2000 58 31 and and CCONJ iajs-2000 58 32 " " PUNCT iajs-2000 58 33 a a DET iajs-2000 58 34 non non ADJ iajs-2000 58 35 - - ADJ iajs-2000 58 36 zero zero NUM iajs-2000 58 37 submodule submodule NOUN iajs-2000 58 38 h h NOUN iajs-2000 58 39 of of ADP iajs-2000 58 40 x x PUNCT iajs-2000 58 41 with with ADP iajs-2000 58 42 𝐾 𝐾 PROPN iajs-2000 58 43 𝐻 𝐻 PROPN iajs-2000 58 44 " " PUNCT iajs-2000 58 45 . . PUNCT iajs-2000 59 1 proof proof NOUN iajs-2000 59 2 " " PUNCT iajs-2000 59 3 ⟹ ⟹ NUM iajs-2000 59 4 assume assume VERB iajs-2000 59 5 that that SCONJ iajs-2000 59 6 k k PROPN iajs-2000 59 7 is be AUX iajs-2000 59 8 a a DET iajs-2000 59 9 we we NOUN iajs-2000 59 10 - - PUNCT iajs-2000 59 11 prime prime NOUN iajs-2000 59 12 submodule submodule NOUN iajs-2000 59 13 of of ADP iajs-2000 59 14 x x PRON iajs-2000 59 15 , , PUNCT iajs-2000 59 16 and and CCONJ iajs-2000 59 17 " " PUNCT iajs-2000 59 18 h h NOUN iajs-2000 59 19 " " PUNCT iajs-2000 59 20 is be AUX iajs-2000 59 21 a a DET iajs-2000 59 22 non non ADJ iajs-2000 59 23 - - ADJ iajs-2000 59 24 zero zero NUM iajs-2000 59 25 submodule submodule NOUN iajs-2000 59 26 of of ADP iajs-2000 59 27 x x SYM iajs-2000 59 28 such such ADJ iajs-2000 59 29 that that SCONJ iajs-2000 59 30 𝐾 𝐾 PROPN iajs-2000 59 31 𝐻 𝐻 PROPN iajs-2000 59 32 " " PUNCT iajs-2000 59 33 . . PUNCT iajs-2000 60 1 " " PUNCT iajs-2000 60 2 let let VERB iajs-2000 60 3 πœ“ πœ“ X iajs-2000 60 4 ∈ ∈ PROPN iajs-2000 60 5 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 60 6 𝑋 𝑋 PROPN iajs-2000 60 7 , , PUNCT iajs-2000 60 8 then then ADV iajs-2000 60 9 by by ADP iajs-2000 60 10 lemma lemma PROPN iajs-2000 60 11 ( ( PUNCT iajs-2000 60 12 2.7)(2 2.7)(2 NOUN iajs-2000 60 13 ) ) PUNCT iajs-2000 60 14 we we PRON iajs-2000 60 15 have have VERB iajs-2000 60 16 𝐾 𝐾 NOUN iajs-2000 60 17 : : PUNCT iajs-2000 60 18 πœ“ πœ“ PROPN iajs-2000 60 19 𝑋 𝑋 PROPN iajs-2000 60 20 𝐾 𝐾 PROPN iajs-2000 60 21 : : PUNCT iajs-2000 60 22 πœ“ πœ“ PROPN iajs-2000 60 23 𝐻 𝐻 PROPN iajs-2000 60 24 , , PUNCT iajs-2000 60 25 since since SCONJ iajs-2000 60 26 𝐾 𝐾 PROPN iajs-2000 60 27 𝐻 𝐻 PROPN iajs-2000 60 28 , , PUNCT iajs-2000 60 29 " " PUNCT iajs-2000 60 30 then then ADV iajs-2000 60 31 there there PRON iajs-2000 60 32 exists exist VERB iajs-2000 60 33 π‘₯ π‘₯ DET iajs-2000 60 34 ∈ ∈ PROPN iajs-2000 60 35 𝐻 𝐻 PROPN iajs-2000 60 36 and and CCONJ iajs-2000 60 37 π‘₯ π‘₯ PROPN iajs-2000 60 38 βˆ‰ βˆ‰ PROPN iajs-2000 60 39 𝐾 𝐾 PROPN iajs-2000 60 40 " " PUNCT iajs-2000 60 41 . . PUNCT iajs-2000 61 1 " " PUNCT iajs-2000 61 2 now now ADV iajs-2000 61 3 , , PUNCT iajs-2000 61 4 " " PUNCT iajs-2000 61 5 suppose suppose VERB iajs-2000 61 6 that that SCONJ iajs-2000 61 7 b b PROPN iajs-2000 61 8 is be AUX iajs-2000 61 9 a a DET iajs-2000 61 10 non non ADJ iajs-2000 61 11 - - ADJ iajs-2000 61 12 zero zero NUM iajs-2000 61 13 element element NOUN iajs-2000 61 14 in in ADP iajs-2000 61 15 " " PUNCT iajs-2000 61 16 𝐾 𝐾 NOUN iajs-2000 61 17 : : PUNCT iajs-2000 61 18 πœ“ πœ“ PROPN iajs-2000 61 19 𝐻 𝐻 PROPN iajs-2000 61 20 , , PUNCT iajs-2000 61 21 then then ADV iajs-2000 61 22 0 0 NUM iajs-2000 61 23 π‘πœ“ π‘πœ“ ADJ iajs-2000 61 24 𝐻 𝐻 PROPN iajs-2000 61 25 𝐾 𝐾 PROPN iajs-2000 61 26 , , PUNCT iajs-2000 61 27 implies imply VERB iajs-2000 61 28 that that SCONJ iajs-2000 61 29 0 0 NUM iajs-2000 62 1 π‘πœ“ π‘πœ“ NOUN iajs-2000 62 2 π‘₯ π‘₯ DET iajs-2000 62 3 ∈ ∈ PROPN iajs-2000 62 4 𝐾 𝐾 PROPN iajs-2000 62 5 , , PUNCT iajs-2000 62 6 where where SCONJ iajs-2000 62 7 π‘₯ π‘₯ DET iajs-2000 62 8 ∈ ∈ PROPN iajs-2000 62 9 𝐻 𝐻 PROPN iajs-2000 62 10 𝑋 𝑋 PROPN iajs-2000 62 11 " " PUNCT iajs-2000 62 12 . . PUNCT iajs-2000 63 1 " " PUNCT iajs-2000 63 2 define define VERB iajs-2000 63 3 πœ“ πœ“ NOUN iajs-2000 63 4 : : PUNCT iajs-2000 63 5 𝑋 𝑋 PROPN iajs-2000 63 6 ⟢ ⟢ NOUN iajs-2000 63 7 𝑋 𝑋 NOUN iajs-2000 63 8 by by ADP iajs-2000 63 9 πœ“ πœ“ PROPN iajs-2000 63 10 𝑦 𝑦 X iajs-2000 63 11 π‘πœ“ π‘πœ“ PROPN iajs-2000 63 12 𝑦 𝑦 NOUN iajs-2000 63 13 for for ADP iajs-2000 63 14 all all DET iajs-2000 63 15 𝑦 𝑦 PRON iajs-2000 63 16 ∈ ∈ PROPN iajs-2000 63 17 𝑋 𝑋 NOUN iajs-2000 63 18 , , PUNCT iajs-2000 63 19 clearly clearly ADV iajs-2000 63 20 πœ“ πœ“ ADP iajs-2000 63 21 ∈ ∈ PROPN iajs-2000 63 22 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 63 23 𝑋 𝑋 PROPN iajs-2000 63 24 , , PUNCT iajs-2000 63 25 also also ADV iajs-2000 63 26 0 0 NUM iajs-2000 63 27 π‘πœ“ π‘πœ“ ADJ iajs-2000 63 28 π‘₯ π‘₯ X iajs-2000 63 29 πœ“ πœ“ VERB iajs-2000 63 30 π‘₯ π‘₯ X iajs-2000 63 31 ∈ ∈ PROPN iajs-2000 63 32 𝐾. 𝐾. PROPN iajs-2000 64 1 but but CCONJ iajs-2000 64 2 k k PROPN iajs-2000 64 3 is be AUX iajs-2000 64 4 a a DET iajs-2000 64 5 we we NOUN iajs-2000 64 6 - - PUNCT iajs-2000 64 7 prime prime NOUN iajs-2000 64 8 submodule submodule NOUN iajs-2000 64 9 of of ADP iajs-2000 64 10 x x PROPN iajs-2000 64 11 , , PUNCT iajs-2000 64 12 and and CCONJ iajs-2000 64 13 π‘₯ π‘₯ PROPN iajs-2000 64 14 βˆ‰ βˆ‰ PROPN iajs-2000 64 15 𝐾 𝐾 PROPN iajs-2000 64 16 , , PUNCT iajs-2000 64 17 then then ADV iajs-2000 64 18 πœ“ πœ“ PROPN iajs-2000 64 19 𝑋 𝑋 PROPN iajs-2000 64 20 𝐾 𝐾 PROPN iajs-2000 64 21 , , PUNCT iajs-2000 64 22 implies imply VERB iajs-2000 64 23 that that SCONJ iajs-2000 64 24 π‘πœ“ π‘πœ“ PROPN iajs-2000 64 25 𝑋 𝑋 PROPN iajs-2000 64 26 𝐾 𝐾 PROPN iajs-2000 64 27 and and CCONJ iajs-2000 64 28 hence hence ADV iajs-2000 64 29 𝑏 𝑏 PROPN iajs-2000 64 30 ∈ ∈ PROPN iajs-2000 64 31 𝐾 𝐾 PROPN iajs-2000 64 32 : : PUNCT iajs-2000 64 33 πœ“ πœ“ PROPN iajs-2000 64 34 𝑋 𝑋 PROPN iajs-2000 64 35 . . PUNCT iajs-2000 65 1 thus thus ADV iajs-2000 65 2 𝐾 𝐾 PROPN iajs-2000 65 3 : : PUNCT iajs-2000 65 4 πœ“ πœ“ PROPN iajs-2000 65 5 𝐻 𝐻 PROPN iajs-2000 65 6 𝐾 𝐾 PROPN iajs-2000 65 7 : : PUNCT iajs-2000 65 8 πœ“ πœ“ PROPN iajs-2000 65 9 𝑋 𝑋 PROPN iajs-2000 65 10 , , PUNCT iajs-2000 65 11 and and CCONJ iajs-2000 65 12 it it PRON iajs-2000 65 13 follows follow VERB iajs-2000 65 14 that that SCONJ iajs-2000 65 15 𝐾 𝐾 PROPN iajs-2000 65 16 : : PUNCT iajs-2000 65 17 πœ“ πœ“ PROPN iajs-2000 65 18 𝑋 𝑋 PROPN iajs-2000 65 19 𝐾 𝐾 PROPN iajs-2000 65 20 : : PUNCT iajs-2000 65 21 πœ“ πœ“ VERB iajs-2000 65 22 𝐻 𝐻 PROPN iajs-2000 65 23 " " PUNCT iajs-2000 65 24 . . PUNCT iajs-2000 65 25 " " PUNCT iajs-2000 66 1 ⟸ ⟸ ADJ iajs-2000 66 2 assume assume VERB iajs-2000 66 3 that that SCONJ iajs-2000 66 4 0 0 NUM iajs-2000 66 5 πœ“ πœ“ PRON iajs-2000 66 6 π‘₯ π‘₯ X iajs-2000 66 7 ∈ ∈ PROPN iajs-2000 66 8 𝐾 𝐾 PROPN iajs-2000 66 9 " " PUNCT iajs-2000 66 10 , , PUNCT iajs-2000 66 11 " " PUNCT iajs-2000 66 12 where where SCONJ iajs-2000 66 13 π‘₯ π‘₯ PROPN iajs-2000 66 14 ∈ ∈ PROPN iajs-2000 66 15 𝑋 𝑋 NOUN iajs-2000 66 16 and and CCONJ iajs-2000 66 17 πœ“ πœ“ ADP iajs-2000 66 18 ∈ ∈ PROPN iajs-2000 66 19 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 66 20 𝑋 𝑋 PROPN iajs-2000 66 21 , , PUNCT iajs-2000 66 22 and and CCONJ iajs-2000 66 23 suppose suppose VERB iajs-2000 66 24 that that SCONJ iajs-2000 66 25 π‘₯ π‘₯ PROPN iajs-2000 66 26 βˆ‰ βˆ‰ PROPN iajs-2000 66 27 𝐾 𝐾 PROPN iajs-2000 66 28 " " PUNCT iajs-2000 66 29 , , PUNCT iajs-2000 66 30 we we PRON iajs-2000 66 31 want want VERB iajs-2000 66 32 to to PART iajs-2000 66 33 show show VERB iajs-2000 66 34 that that SCONJ iajs-2000 66 35 πœ“ πœ“ PROPN iajs-2000 66 36 𝑋 𝑋 PROPN iajs-2000 66 37 𝐾. 𝐾. PROPN iajs-2000 66 38 since since SCONJ iajs-2000 66 39 π‘₯ π‘₯ PROPN iajs-2000 66 40 βˆ‰ βˆ‰ PROPN iajs-2000 66 41 𝐾 𝐾 PROPN iajs-2000 66 42 , , PUNCT iajs-2000 66 43 then then ADV iajs-2000 66 44 𝐾 𝐾 PROPN iajs-2000 66 45 𝐾 𝐾 PROPN iajs-2000 66 46 𝑅π‘₯ 𝑅π‘₯ PROPN iajs-2000 66 47 , , PUNCT iajs-2000 66 48 where where SCONJ iajs-2000 66 49 𝐾 𝐾 PROPN iajs-2000 66 50 𝑅π‘₯ 𝑅π‘₯ PROPN iajs-2000 66 51 " " PUNCT iajs-2000 66 52 is be AUX iajs-2000 66 53 a a DET iajs-2000 66 54 non non ADJ iajs-2000 66 55 - - ADJ iajs-2000 66 56 zero zero NUM iajs-2000 66 57 submodule submodule NOUN iajs-2000 66 58 of of ADP iajs-2000 66 59 x x NOUN iajs-2000 66 60 " " PUNCT iajs-2000 66 61 . . PUNCT iajs-2000 67 1 thus thus ADV iajs-2000 67 2 by by ADP iajs-2000 67 3 our our PRON iajs-2000 67 4 hypothesis hypothesis NOUN iajs-2000 67 5 , , PUNCT iajs-2000 67 6 we we PRON iajs-2000 67 7 get get VERB iajs-2000 67 8 𝐾 𝐾 NOUN iajs-2000 67 9 : : PUNCT iajs-2000 67 10 πœ“ πœ“ PROPN iajs-2000 67 11 𝑋 𝑋 PROPN iajs-2000 67 12 𝐾 𝐾 PROPN iajs-2000 67 13 : : PUNCT iajs-2000 67 14 πœ“ πœ“ PROPN iajs-2000 67 15 𝐾 𝐾 PROPN iajs-2000 67 16 𝑅π‘₯ 𝑅π‘₯ PROPN iajs-2000 67 17 . . PUNCT iajs-2000 68 1 since since SCONJ iajs-2000 68 2 k k PROPN iajs-2000 68 3 is be AUX iajs-2000 68 4 a a DET iajs-2000 68 5 fully fully ADV iajs-2000 68 6 invariant invariant ADJ iajs-2000 68 7 , , PUNCT iajs-2000 68 8 then then ADV iajs-2000 68 9 πœ“ πœ“ PROPN iajs-2000 68 10 𝐾 𝐾 PROPN iajs-2000 68 11 𝐾 𝐾 PROPN iajs-2000 68 12 and and CCONJ iajs-2000 68 13 " " PUNCT iajs-2000 68 14 πœ“ πœ“ PROPN iajs-2000 68 15 𝑅π‘₯ 𝑅π‘₯ PROPN iajs-2000 68 16 𝐾 𝐾 PROPN iajs-2000 68 17 , , PUNCT iajs-2000 68 18 it it PRON iajs-2000 68 19 follows follow VERB iajs-2000 68 20 that that SCONJ iajs-2000 68 21 πœ“ πœ“ PROPN iajs-2000 68 22 𝐾 𝐾 PROPN iajs-2000 68 23 𝑅π‘₯ 𝑅π‘₯ PROPN iajs-2000 68 24 𝐾 𝐾 PROPN iajs-2000 68 25 " " PUNCT iajs-2000 68 26 . . PUNCT iajs-2000 69 1 hence hence ADV iajs-2000 69 2 𝐾 𝐾 PROPN iajs-2000 69 3 : : PUNCT iajs-2000 69 4 πœ“ πœ“ PROPN iajs-2000 69 5 𝐾 𝐾 PROPN iajs-2000 69 6 𝑅π‘₯ 𝑅π‘₯ PROPN iajs-2000 69 7 𝑅 𝑅 PROPN iajs-2000 69 8 , , PUNCT iajs-2000 69 9 therefore therefore ADV iajs-2000 69 10 1 1 NUM iajs-2000 69 11 ∈ ∈ PROPN iajs-2000 69 12 𝐾 𝐾 NOUN iajs-2000 69 13 : : PUNCT iajs-2000 69 14 πœ“ πœ“ PROPN iajs-2000 69 15 𝐾 𝐾 PROPN iajs-2000 69 16 𝑅π‘₯ 𝑅π‘₯ PROPN iajs-2000 69 17 , , PUNCT iajs-2000 69 18 implies imply VERB iajs-2000 69 19 that that SCONJ iajs-2000 69 20 1 1 NUM iajs-2000 69 21 ∈ ∈ PROPN iajs-2000 69 22 𝐾 𝐾 NOUN iajs-2000 69 23 : : PUNCT iajs-2000 69 24 πœ“ πœ“ PROPN iajs-2000 69 25 𝑋 𝑋 PROPN iajs-2000 69 26 , , PUNCT iajs-2000 69 27 hence hence ADV iajs-2000 69 28 πœ“ πœ“ SCONJ iajs-2000 69 29 𝑋 𝑋 PROPN iajs-2000 69 30 𝐾. 𝐾. PROPN iajs-2000 69 31 thus thus ADV iajs-2000 69 32 k k X iajs-2000 69 33 is be AUX iajs-2000 69 34 a a DET iajs-2000 69 35 we we NOUN iajs-2000 69 36 - - PUNCT iajs-2000 69 37 prime prime NOUN iajs-2000 69 38 submodule submodule NOUN iajs-2000 69 39 of of ADP iajs-2000 69 40 x x NOUN iajs-2000 69 41 " " PUNCT iajs-2000 69 42 . . PUNCT iajs-2000 70 1 proposition proposition NOUN iajs-2000 70 2 ( ( PUNCT iajs-2000 70 3 9 9 NUM iajs-2000 70 4 ) ) PUNCT iajs-2000 70 5 " " PUNCT iajs-2000 70 6 " " PUNCT iajs-2000 70 7 let let VERB iajs-2000 70 8 x x PRON iajs-2000 70 9 be be AUX iajs-2000 70 10 an an DET iajs-2000 70 11 r r NOUN iajs-2000 70 12 - - PUNCT iajs-2000 70 13 module module NOUN iajs-2000 70 14 , , PUNCT iajs-2000 70 15 and and CCONJ iajs-2000 70 16 l l NOUN iajs-2000 70 17 " " PUNCT iajs-2000 70 18 , , PUNCT iajs-2000 70 19 h h NOUN iajs-2000 70 20 are be AUX iajs-2000 70 21 " " PUNCT iajs-2000 70 22 submodules submodule NOUN iajs-2000 70 23 of of ADP iajs-2000 70 24 x x PRON iajs-2000 70 25 , , PUNCT iajs-2000 70 26 with with SCONJ iajs-2000 70 27 h h NOUN iajs-2000 70 28 is be AUX iajs-2000 70 29 a a DET iajs-2000 70 30 fully fully ADV iajs-2000 70 31 invariant invariant ADJ iajs-2000 70 32 submodule submodule NOUN iajs-2000 70 33 of of ADP iajs-2000 70 34 x x PUNCT iajs-2000 70 35 and and CCONJ iajs-2000 70 36 𝐻 𝐻 PROPN iajs-2000 70 37 𝐿 𝐿 PROPN iajs-2000 70 38 " " PUNCT iajs-2000 70 39 . . PUNCT iajs-2000 71 1 " " PUNCT iajs-2000 71 2 if if SCONJ iajs-2000 71 3 is be AUX iajs-2000 71 4 a a DET iajs-2000 71 5 we we NOUN iajs-2000 71 6 - - PUNCT iajs-2000 71 7 prime prime NOUN iajs-2000 71 8 submodule submodule NOUN iajs-2000 71 9 of of ADP iajs-2000 71 10 , , PUNCT iajs-2000 71 11 then then ADV iajs-2000 71 12 l l NOUN iajs-2000 71 13 is be AUX iajs-2000 71 14 a a DET iajs-2000 71 15 we we NOUN iajs-2000 71 16 - - PUNCT iajs-2000 71 17 prime prime NOUN iajs-2000 71 18 submodule submodule NOUN iajs-2000 71 19 of of ADP iajs-2000 71 20 x x NOUN iajs-2000 71 21 " " PUNCT iajs-2000 71 22 . . PUNCT iajs-2000 72 1 proof proof NOUN iajs-2000 72 2 " " PUNCT iajs-2000 72 3 assume assume VERB iajs-2000 72 4 that that SCONJ iajs-2000 72 5 0 0 NUM iajs-2000 72 6 πœ“ πœ“ PRON iajs-2000 72 7 π‘₯ π‘₯ X iajs-2000 72 8 ∈ ∈ PROPN iajs-2000 72 9 𝐿 𝐿 PROPN iajs-2000 72 10 , , PUNCT iajs-2000 72 11 where where SCONJ iajs-2000 72 12 π‘₯ π‘₯ DET iajs-2000 72 13 ∈ ∈ PROPN iajs-2000 72 14 𝑋 𝑋 NOUN iajs-2000 72 15 and and CCONJ iajs-2000 72 16 πœ“ πœ“ ADP iajs-2000 72 17 ∈ ∈ PROPN iajs-2000 72 18 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 72 19 𝑋 𝑋 PROPN iajs-2000 72 20 . . PUNCT iajs-2000 73 1 if if SCONJ iajs-2000 73 2 π‘₯ π‘₯ PROPN iajs-2000 73 3 βˆ‰ βˆ‰ PROPN iajs-2000 73 4 𝐿 𝐿 PROPN iajs-2000 73 5 , , PUNCT iajs-2000 73 6 then then ADV iajs-2000 73 7 we we PRON iajs-2000 73 8 must must AUX iajs-2000 73 9 show show VERB iajs-2000 73 10 that that SCONJ iajs-2000 73 11 πœ“ πœ“ PROPN iajs-2000 73 12 𝑋 𝑋 PROPN iajs-2000 73 13 𝐿. 𝐿. VERB iajs-2000 73 14 define define VERB iajs-2000 73 15 πœ“ πœ“ NOUN iajs-2000 73 16 : : PUNCT iajs-2000 73 17 ⟢ ⟢ NOUN iajs-2000 73 18 by by ADP iajs-2000 73 19 πœ“ πœ“ PROPN iajs-2000 73 20 π‘₯ π‘₯ X iajs-2000 73 21 𝐻 𝐻 NOUN iajs-2000 73 22 πœ“ πœ“ NOUN iajs-2000 73 23 π‘₯ π‘₯ X iajs-2000 73 24 𝐻 𝐻 NOUN iajs-2000 73 25 for for ADP iajs-2000 73 26 all all DET iajs-2000 73 27 π‘₯ π‘₯ DET iajs-2000 73 28 ∈ ∈ PROPN iajs-2000 73 29 𝑋. 𝑋. PROPN iajs-2000 73 30 to to PART iajs-2000 73 31 prove prove VERB iajs-2000 73 32 that that SCONJ iajs-2000 73 33 πœ‘ πœ‘ PROPN iajs-2000 73 34 is be AUX iajs-2000 73 35 well well ADV iajs-2000 73 36 define define NOUN iajs-2000 73 37 , , PUNCT iajs-2000 73 38 suppose suppose VERB iajs-2000 73 39 that that SCONJ iajs-2000 73 40 π‘₯ π‘₯ PROPN iajs-2000 73 41 𝐻 𝐻 NOUN iajs-2000 73 42 π‘₯ π‘₯ NOUN iajs-2000 73 43 𝐻 𝐻 NOUN iajs-2000 73 44 where where SCONJ iajs-2000 73 45 π‘₯ π‘₯ NOUN iajs-2000 73 46 , , PUNCT iajs-2000 73 47 π‘₯ π‘₯ DET iajs-2000 73 48 ∈ ∈ PROPN iajs-2000 73 49 𝑋 𝑋 NOUN iajs-2000 73 50 , , PUNCT iajs-2000 73 51 then then ADV iajs-2000 73 52 π‘₯ π‘₯ VERB iajs-2000 73 53 π‘₯ π‘₯ NOUN iajs-2000 73 54 ∈ ∈ PROPN iajs-2000 73 55 𝐻 𝐻 NOUN iajs-2000 73 56 , , PUNCT iajs-2000 73 57 hence hence ADV iajs-2000 73 58 πœ“ πœ“ ADP iajs-2000 73 59 π‘₯ π‘₯ NOUN iajs-2000 73 60 π‘₯ π‘₯ X iajs-2000 73 61 ∈ ∈ NOUN iajs-2000 73 62 πœ“ πœ“ NOUN iajs-2000 73 63 𝐻 𝐻 NOUN iajs-2000 73 64 𝐻 𝐻 PROPN iajs-2000 73 65 because because SCONJ iajs-2000 73 66 h h NOUN iajs-2000 73 67 is be AUX iajs-2000 73 68 a a DET iajs-2000 73 69 fully fully ADV iajs-2000 73 70 invariant invariant ADJ iajs-2000 73 71 . . PUNCT iajs-2000 74 1 it it PRON iajs-2000 74 2 follows follow VERB iajs-2000 74 3 that that SCONJ iajs-2000 74 4 πœ“ πœ“ PROPN iajs-2000 74 5 π‘₯ π‘₯ X iajs-2000 74 6 πœ“ πœ“ X iajs-2000 74 7 π‘₯ π‘₯ X iajs-2000 74 8 ∈ ∈ PROPN iajs-2000 74 9 𝐻. 𝐻. PROPN iajs-2000 74 10 hence hence ADV iajs-2000 74 11 πœ“ πœ“ VERB iajs-2000 74 12 π‘₯ π‘₯ X iajs-2000 74 13 𝐻 𝐻 NOUN iajs-2000 74 14 πœ“ πœ“ NOUN iajs-2000 74 15 π‘₯ π‘₯ X iajs-2000 74 16 𝐻 𝐻 PROPN iajs-2000 74 17 , , PUNCT iajs-2000 74 18 implies imply VERB iajs-2000 74 19 that that SCONJ iajs-2000 74 20 πœ“ πœ“ PROPN iajs-2000 74 21 π‘₯ π‘₯ X iajs-2000 74 22 𝐻 𝐻 NOUN iajs-2000 74 23 πœ“ πœ“ NOUN iajs-2000 74 24 π‘₯ π‘₯ X iajs-2000 74 25 𝐻. 𝐻. PROPN iajs-2000 74 26 since since SCONJ iajs-2000 74 27 0 0 NUM iajs-2000 74 28 πœ“ πœ“ DET iajs-2000 74 29 π‘₯ π‘₯ X iajs-2000 74 30 ∈ ∈ PROPN iajs-2000 74 31 mathematics mathematic NOUN iajs-2000 74 32 | | ADV iajs-2000 74 33 112 112 NUM iajs-2000 74 34 ibn ibn PROPN iajs-2000 74 35 al al PROPN iajs-2000 74 36 - - PUNCT iajs-2000 74 37 haitham haitham PROPN iajs-2000 74 38 jour jour X iajs-2000 74 39 . . PROPN iajs-2000 75 1 for for ADP iajs-2000 75 2 pure pure ADJ iajs-2000 75 3 & & CCONJ iajs-2000 75 4 appl appl PROPN iajs-2000 75 5 . . PUNCT iajs-2000 76 1 sci sci PROPN iajs-2000 76 2 . . PROPN iajs-2000 76 3 ihjpas ihjpa VERB iajs-2000 76 4 https://doi.org/10.30526/31.3.2000 https://doi.org/10.30526/31.3.2000 NUM iajs-2000 76 5 vol vol NOUN iajs-2000 76 6 . . PROPN iajs-2000 76 7 31 31 NUM iajs-2000 76 8 ( ( PUNCT iajs-2000 76 9 3 3 NUM iajs-2000 76 10 ) ) PUNCT iajs-2000 76 11 2018 2018 NUM iajs-2000 76 12 𝐿 𝐿 PROPN iajs-2000 76 13 , , PUNCT iajs-2000 76 14 implies imply VERB iajs-2000 76 15 that that SCONJ iajs-2000 76 16 0 0 NUM iajs-2000 76 17 πœ“ πœ“ NOUN iajs-2000 76 18 π‘₯ π‘₯ X iajs-2000 76 19 𝐻 𝐻 NOUN iajs-2000 76 20 πœ“ πœ“ NOUN iajs-2000 76 21 π‘₯ π‘₯ X iajs-2000 76 22 𝐻 𝐻 NOUN iajs-2000 76 23 ∈ ∈ PROPN iajs-2000 76 24 𝐿 𝐿 NOUN iajs-2000 76 25 𝐻 𝐻 PROPN iajs-2000 76 26 . . PUNCT iajs-2000 77 1 but but CCONJ iajs-2000 77 2 is be AUX iajs-2000 77 3 a a DET iajs-2000 77 4 we we NOUN iajs-2000 77 5 - - PUNCT iajs-2000 77 6 prime prime NOUN iajs-2000 77 7 submodule submodule NOUN iajs-2000 77 8 of of ADP iajs-2000 77 9 , , PUNCT iajs-2000 77 10 and and CCONJ iajs-2000 77 11 π‘₯ π‘₯ DET iajs-2000 77 12 𝐻 𝐻 NOUN iajs-2000 77 13 βˆ‰ βˆ‰ PROPN iajs-2000 77 14 𝐿 𝐿 PROPN iajs-2000 77 15 𝐻 𝐻 PROPN iajs-2000 77 16 , , PUNCT iajs-2000 77 17 implies imply VERB iajs-2000 77 18 that that SCONJ iajs-2000 77 19 πœ“ πœ“ ADP iajs-2000 77 20 𝑋 𝑋 PROPN iajs-2000 77 21 𝐻 𝐻 PROPN iajs-2000 77 22 𝐿 𝐿 PROPN iajs-2000 77 23 𝐻 𝐻 PROPN iajs-2000 77 24 , , PUNCT iajs-2000 77 25 thus thus ADV iajs-2000 77 26 , , PUNCT iajs-2000 77 27 we we PRON iajs-2000 77 28 have have VERB iajs-2000 77 29 𝐿 𝐿 PROPN iajs-2000 77 30 𝐻 𝐻 PROPN iajs-2000 77 31 , , PUNCT iajs-2000 77 32 it it PRON iajs-2000 77 33 follows follow VERB iajs-2000 77 34 that that SCONJ iajs-2000 77 35 πœ“ πœ“ SCONJ iajs-2000 77 36 𝑋 𝑋 PROPN iajs-2000 77 37 𝐻 𝐻 PROPN iajs-2000 77 38 𝐿. 𝐿. VERB iajs-2000 77 39 thus thus ADV iajs-2000 77 40 πœ“ πœ“ ADP iajs-2000 77 41 𝑋 𝑋 PROPN iajs-2000 77 42 𝐿 𝐿 PROPN iajs-2000 77 43 " " PUNCT iajs-2000 77 44 . . PUNCT iajs-2000 78 1 hence hence ADV iajs-2000 78 2 " " PUNCT iajs-2000 78 3 l l NOUN iajs-2000 78 4 is be AUX iajs-2000 78 5 a a DET iajs-2000 78 6 we we NOUN iajs-2000 78 7 - - PUNCT iajs-2000 78 8 prime prime NOUN iajs-2000 78 9 submodule submodule NOUN iajs-2000 78 10 of of ADP iajs-2000 78 11 x x NOUN iajs-2000 78 12 " " PUNCT iajs-2000 78 13 . . PUNCT iajs-2000 79 1 proposition proposition NOUN iajs-2000 79 2 ( ( PUNCT iajs-2000 79 3 10 10 NUM iajs-2000 79 4 ) ) PUNCT iajs-2000 79 5 " " PUNCT iajs-2000 79 6 " " PUNCT iajs-2000 79 7 let let VERB iajs-2000 79 8 l l NOUN iajs-2000 79 9 and and CCONJ iajs-2000 79 10 k k PROPN iajs-2000 79 11 are be AUX iajs-2000 79 12 submodules submodule NOUN iajs-2000 79 13 of of ADP iajs-2000 79 14 an an DET iajs-2000 79 15 r r NOUN iajs-2000 79 16 - - PUNCT iajs-2000 79 17 module module NOUN iajs-2000 79 18 x x NOUN iajs-2000 79 19 " " PUNCT iajs-2000 79 20 , , PUNCT iajs-2000 79 21 " " PUNCT iajs-2000 79 22 with with ADP iajs-2000 79 23 l l NOUN iajs-2000 79 24 is be AUX iajs-2000 79 25 an an DET iajs-2000 79 26 x x NOUN iajs-2000 79 27 - - NOUN iajs-2000 79 28 injective injective ADJ iajs-2000 79 29 , , PUNCT iajs-2000 79 30 and and CCONJ iajs-2000 79 31 k k PROPN iajs-2000 79 32 is be AUX iajs-2000 79 33 a a DET iajs-2000 79 34 weprime weprime ADJ iajs-2000 79 35 submodule submodule NOUN iajs-2000 79 36 of of ADP iajs-2000 79 37 x x NOUN iajs-2000 79 38 " " PUNCT iajs-2000 79 39 . . PUNCT iajs-2000 80 1 " " PUNCT iajs-2000 80 2 then then ADV iajs-2000 80 3 either either CCONJ iajs-2000 80 4 𝐿 𝐿 PROPN iajs-2000 80 5 𝐾or 𝐾or PROPN iajs-2000 80 6 𝐾 𝐾 PROPN iajs-2000 80 7 ∩ ∩ ADJ iajs-2000 80 8 𝐿 𝐿 PROPN iajs-2000 80 9 is be AUX iajs-2000 80 10 a a DET iajs-2000 80 11 we we NOUN iajs-2000 80 12 - - PUNCT iajs-2000 80 13 prime prime NOUN iajs-2000 80 14 submodule submodule NOUN iajs-2000 80 15 of of ADP iajs-2000 80 16 l l PROPN iajs-2000 80 17 " " PUNCT iajs-2000 80 18 . . PUNCT iajs-2000 81 1 proof proof NOUN iajs-2000 81 2 " " PUNCT iajs-2000 81 3 assume assume VERB iajs-2000 81 4 that that SCONJ iajs-2000 81 5 𝐿 𝐿 PROPN iajs-2000 81 6 β‰° β‰° PROPN iajs-2000 81 7 𝐾 𝐾 PROPN iajs-2000 81 8 " " PUNCT iajs-2000 81 9 , , PUNCT iajs-2000 81 10 " " PUNCT iajs-2000 81 11 then then ADV iajs-2000 81 12 𝐾 𝐾 PROPN iajs-2000 81 13 ∩ ∩ ADJ iajs-2000 81 14 𝐿 𝐿 PROPN iajs-2000 81 15 is be AUX iajs-2000 81 16 a a DET iajs-2000 81 17 proper proper ADJ iajs-2000 81 18 submodule submodule NOUN iajs-2000 81 19 of of ADP iajs-2000 81 20 l l PROPN iajs-2000 81 21 " " PUNCT iajs-2000 81 22 . . PUNCT iajs-2000 82 1 now now ADV iajs-2000 82 2 , , PUNCT iajs-2000 82 3 let let VERB iajs-2000 82 4 0 0 NUM iajs-2000 82 5 πœ“ πœ“ VERB iajs-2000 82 6 π‘₯ π‘₯ X iajs-2000 82 7 ∈ ∈ PROPN iajs-2000 82 8 𝐾 𝐾 PROPN iajs-2000 82 9 ∩ ∩ ADJ iajs-2000 82 10 𝐿 𝐿 PROPN iajs-2000 82 11 , , PUNCT iajs-2000 82 12 where where SCONJ iajs-2000 82 13 π‘₯ π‘₯ DET iajs-2000 82 14 ∈ ∈ PROPN iajs-2000 82 15 𝐿 𝐿 PROPN iajs-2000 82 16 and and CCONJ iajs-2000 82 17 πœ“ πœ“ ADP iajs-2000 82 18 ∈ ∈ PROPN iajs-2000 82 19 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 82 20 𝐿 𝐿 PROPN iajs-2000 82 21 . . PUNCT iajs-2000 82 22 suppose suppose VERB iajs-2000 82 23 that that SCONJ iajs-2000 82 24 π‘₯ π‘₯ PROPN iajs-2000 82 25 βˆ‰ βˆ‰ PROPN iajs-2000 82 26 𝐾 𝐾 PROPN iajs-2000 82 27 ∩ ∩ ADJ iajs-2000 82 28 𝐿 𝐿 PROPN iajs-2000 82 29 , , PUNCT iajs-2000 82 30 then then ADV iajs-2000 82 31 π‘₯ π‘₯ PROPN iajs-2000 82 32 βˆ‰ βˆ‰ PROPN iajs-2000 82 33 𝐾 𝐾 PROPN iajs-2000 82 34 . . PUNCT iajs-2000 83 1 now now ADV iajs-2000 83 2 , , PUNCT iajs-2000 83 3 consider consider VERB iajs-2000 83 4 the the DET iajs-2000 83 5 following follow VERB iajs-2000 83 6 diagram diagram NOUN iajs-2000 83 7 , , PUNCT iajs-2000 83 8 " " PUNCT iajs-2000 83 9 where where SCONJ iajs-2000 83 10 𝔦 𝔦 X iajs-2000 83 11 is be AUX iajs-2000 83 12 the the DET iajs-2000 83 13 inclusion inclusion NOUN iajs-2000 83 14 map map NOUN iajs-2000 83 15 . . PUNCT iajs-2000 84 1 since since SCONJ iajs-2000 84 2 l l NOUN iajs-2000 84 3 is be AUX iajs-2000 84 4 an an DET iajs-2000 84 5 x x NOUN iajs-2000 84 6 - - NOUN iajs-2000 84 7 injective injective ADJ iajs-2000 84 8 then then ADV iajs-2000 84 9 there there PRON iajs-2000 84 10 exists exist VERB iajs-2000 84 11 " " PUNCT iajs-2000 84 12 πœ™ πœ™ NOUN iajs-2000 84 13 : : PUNCT iajs-2000 84 14 𝑋 𝑋 PROPN iajs-2000 84 15 ⟢ ⟢ NOUN iajs-2000 84 16 𝐿 𝐿 NOUN iajs-2000 84 17 such such ADJ iajs-2000 84 18 that that DET iajs-2000 84 19 πœ™π‘œπ”¦ πœ™π‘œπ”¦ NOUN iajs-2000 84 20 πœ“. πœ“. VERB iajs-2000 84 21 clearly clearly ADV iajs-2000 84 22 πœ™ πœ™ PRON iajs-2000 84 23 ∈ ∈ PROPN iajs-2000 84 24 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 84 25 𝑋 𝑋 PROPN iajs-2000 84 26 , , PUNCT iajs-2000 84 27 but but CCONJ iajs-2000 84 28 0 0 NUM iajs-2000 84 29 πœ“ πœ“ PRON iajs-2000 84 30 π‘₯ π‘₯ PROPN iajs-2000 84 31 πœ™π‘œπ”¦ πœ™π‘œπ”¦ VERB iajs-2000 84 32 π‘₯ π‘₯ NOUN iajs-2000 84 33 πœ™ πœ™ NOUN iajs-2000 84 34 π‘₯ π‘₯ X iajs-2000 84 35 ∈ ∈ PROPN iajs-2000 84 36 𝐾 𝐾 PROPN iajs-2000 84 37 , , PUNCT iajs-2000 84 38 implies imply VERB iajs-2000 84 39 that that SCONJ iajs-2000 85 1 0 0 NUM iajs-2000 85 2 πœ™ πœ™ NOUN iajs-2000 85 3 π‘₯ π‘₯ X iajs-2000 85 4 ∈ ∈ PROPN iajs-2000 85 5 𝐾. 𝐾. PROPN iajs-2000 85 6 but but CCONJ iajs-2000 85 7 k k PROPN iajs-2000 85 8 is be AUX iajs-2000 85 9 a a DET iajs-2000 85 10 we we NOUN iajs-2000 85 11 - - PUNCT iajs-2000 85 12 prime prime NOUN iajs-2000 85 13 submodule submodule NOUN iajs-2000 85 14 of of ADP iajs-2000 85 15 x x PROPN iajs-2000 85 16 and and CCONJ iajs-2000 85 17 π‘₯ π‘₯ PROPN iajs-2000 85 18 βˆ‰ βˆ‰ PROPN iajs-2000 85 19 𝐾 𝐾 PROPN iajs-2000 85 20 , , PUNCT iajs-2000 85 21 then then ADV iajs-2000 85 22 πœ™ πœ™ PROPN iajs-2000 85 23 𝑋 𝑋 PROPN iajs-2000 85 24 𝐾. 𝐾. PROPN iajs-2000 85 25 " " PUNCT iajs-2000 85 26 also also ADV iajs-2000 85 27 , , PUNCT iajs-2000 85 28 we we PRON iajs-2000 85 29 have have VERB iajs-2000 85 30 πœ“ πœ“ ADP iajs-2000 85 31 𝐿 𝐿 PROPN iajs-2000 85 32 πœ™π‘œπ”¦ πœ™π‘œπ”¦ NOUN iajs-2000 85 33 𝐿 𝐿 PROPN iajs-2000 85 34 πœ™ πœ™ PROPN iajs-2000 85 35 𝐿 𝐿 PROPN iajs-2000 85 36 𝐿 𝐿 PROPN iajs-2000 85 37 and and CCONJ iajs-2000 85 38 πœ“πœ“ πœ“πœ“ PROPN iajs-2000 85 39 𝐿 𝐿 PROPN iajs-2000 85 40 πœ™ πœ™ PROPN iajs-2000 85 41 𝐿 𝐿 PROPN iajs-2000 85 42 πœ™ πœ™ NOUN iajs-2000 85 43 𝑋 𝑋 PROPN iajs-2000 85 44 𝐾. 𝐾. PROPN iajs-2000 85 45 hence hence ADV iajs-2000 85 46 πœ“ πœ“ ADP iajs-2000 85 47 𝐿 𝐿 PROPN iajs-2000 85 48 𝐾 𝐾 PROPN iajs-2000 85 49 ∩ ∩ ADJ iajs-2000 85 50 𝐿 𝐿 PROPN iajs-2000 85 51 , , PUNCT iajs-2000 85 52 it it PRON iajs-2000 85 53 follows follow VERB iajs-2000 85 54 that that SCONJ iajs-2000 85 55 𝐾 𝐾 PROPN iajs-2000 85 56 ∩ ∩ ADJ iajs-2000 85 57 𝐿 𝐿 PROPN iajs-2000 85 58 is be AUX iajs-2000 85 59 a a DET iajs-2000 85 60 we we NOUN iajs-2000 85 61 - - PUNCT iajs-2000 85 62 prime prime NOUN iajs-2000 85 63 submodule submodule NOUN iajs-2000 85 64 of of ADP iajs-2000 85 65 l l PROPN iajs-2000 85 66 " " PUNCT iajs-2000 85 67 . . PUNCT iajs-2000 86 1 proposition proposition NOUN iajs-2000 86 2 ( ( PUNCT iajs-2000 86 3 11 11 NUM iajs-2000 86 4 ) ) PUNCT iajs-2000 86 5 " " PUNCT iajs-2000 86 6 " " PUNCT iajs-2000 86 7 let let VERB iajs-2000 86 8 x x PRON iajs-2000 86 9 be be AUX iajs-2000 86 10 an an DET iajs-2000 86 11 r r PROPN iajs-2000 86 12 - - PROPN iajs-2000 86 13 module"and module"and PROPN iajs-2000 86 14 k k NOUN iajs-2000 86 15 , , PUNCT iajs-2000 86 16 l l NOUN iajs-2000 86 17 are be AUX iajs-2000 86 18 non non ADJ iajs-2000 86 19 - - ADJ iajs-2000 86 20 trivial trivial ADJ iajs-2000 86 21 submodules submodule NOUN iajs-2000 86 22 of of ADP iajs-2000 86 23 x x SYM iajs-2000 86 24 such such ADJ iajs-2000 86 25 that that SCONJ iajs-2000 86 26 l l NOUN iajs-2000 86 27 is be AUX iajs-2000 86 28 a a DET iajs-2000 86 29 we we NOUN iajs-2000 86 30 - - PUNCT iajs-2000 86 31 prime prime NOUN iajs-2000 86 32 submodule submodule NOUN iajs-2000 86 33 of of ADP iajs-2000 86 34 x"and x"and PROPN iajs-2000 86 35 ik ik PROPN iajs-2000 86 36 is be AUX iajs-2000 86 37 a a DET iajs-2000 86 38 non non ADJ iajs-2000 86 39 - - ADJ iajs-2000 86 40 zero zero NUM iajs-2000 86 41 submodule submodule NOUN iajs-2000 86 42 of of ADP iajs-2000 86 43 l l NOUN iajs-2000 86 44 for for ADP iajs-2000 86 45 some some DET iajs-2000 86 46 ideal ideal NOUN iajs-2000 86 47 i i PRON iajs-2000 86 48 of of ADP iajs-2000 86 49 r. r. PROPN iajs-2000 86 50 if if SCONJ iajs-2000 86 51 𝐼 𝐼 PROPN iajs-2000 86 52 𝐿 𝐿 PROPN iajs-2000 86 53 : : PUNCT iajs-2000 86 54 𝑋 𝑋 NOUN iajs-2000 86 55 then then ADV iajs-2000 86 56 𝐾 𝐾 PROPN iajs-2000 86 57 𝐿 𝐿 PROPN iajs-2000 86 58 " " PUNCT iajs-2000 86 59 . . PUNCT iajs-2000 87 1 proof proof NOUN iajs-2000 87 2 " " PUNCT iajs-2000 87 3 suppose suppose VERB iajs-2000 87 4 that that SCONJ iajs-2000 87 5 𝑦 𝑦 PROPN iajs-2000 87 6 ∈ ∈ PROPN iajs-2000 87 7 𝐾 𝐾 PROPN iajs-2000 87 8 , , PUNCT iajs-2000 87 9 since since SCONJ iajs-2000 87 10 𝐼 𝐼 PROPN iajs-2000 87 11 β‰° β‰° PROPN iajs-2000 87 12 𝐿 𝐿 PROPN iajs-2000 87 13 : : PUNCT iajs-2000 87 14 𝑋 𝑋 NOUN iajs-2000 87 15 , , PUNCT iajs-2000 87 16 then then ADV iajs-2000 87 17 there there PRON iajs-2000 87 18 exists exist VERB iajs-2000 87 19 𝑖 𝑖 X iajs-2000 87 20 ∈ ∈ PROPN iajs-2000 87 21 𝐼 𝐼 PROPN iajs-2000 87 22 and and CCONJ iajs-2000 87 23 𝑖 𝑖 PROPN iajs-2000 87 24 βˆ‰ βˆ‰ PROPN iajs-2000 87 25 𝐿 𝐿 PROPN iajs-2000 87 26 : : PUNCT iajs-2000 87 27 𝑋 𝑋 PROPN iajs-2000 87 28 " " PUNCT iajs-2000 87 29 . . PUNCT iajs-2000 88 1 " " PUNCT iajs-2000 88 2 now now ADV iajs-2000 88 3 , , PUNCT iajs-2000 88 4 let let VERB iajs-2000 88 5 πœ“ πœ“ PRON iajs-2000 88 6 : : PUNCT iajs-2000 88 7 𝑋 𝑋 PROPN iajs-2000 88 8 ⟢ ⟢ NOUN iajs-2000 88 9 𝑋 𝑋 NOUN iajs-2000 88 10 define define VERB iajs-2000 88 11 by by ADP iajs-2000 88 12 πœ“ πœ“ PROPN iajs-2000 88 13 π‘₯ π‘₯ PROPN iajs-2000 88 14 𝑖π‘₯ 𝑖π‘₯ INTJ iajs-2000 88 15 for for ADP iajs-2000 88 16 all all DET iajs-2000 88 17 submodule submodule NOUN iajs-2000 88 18 π‘₯ π‘₯ PRON iajs-2000 88 19 ∈ ∈ PROPN iajs-2000 88 20 𝑋 𝑋 PROPN iajs-2000 88 21 , , PUNCT iajs-2000 88 22 clearly clearly ADV iajs-2000 88 23 πœ“ πœ“ ADP iajs-2000 88 24 ∈ ∈ PROPN iajs-2000 88 25 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 88 26 𝑋 𝑋 PROPN iajs-2000 88 27 " " PUNCT iajs-2000 88 28 . . PUNCT iajs-2000 89 1 " " PUNCT iajs-2000 89 2 since since SCONJ iajs-2000 89 3 ik ik PROPN iajs-2000 89 4 " " PUNCT iajs-2000 89 5 is be AUX iajs-2000 89 6 a a DET iajs-2000 89 7 non non ADJ iajs-2000 89 8 - - ADJ iajs-2000 89 9 zero zero NUM iajs-2000 89 10 submodule submodule NOUN iajs-2000 89 11 of of ADP iajs-2000 89 12 l l PROPN iajs-2000 89 13 " " PUNCT iajs-2000 89 14 , , PUNCT iajs-2000 89 15 " " PUNCT iajs-2000 89 16 then then ADV iajs-2000 89 17 iy iy PROPN iajs-2000 89 18 is be AUX iajs-2000 89 19 a a DET iajs-2000 89 20 non non ADJ iajs-2000 89 21 - - ADJ iajs-2000 89 22 zero zero NUM iajs-2000 89 23 element element NOUN iajs-2000 89 24 in in ADP iajs-2000 89 25 k k NOUN iajs-2000 89 26 " " PUNCT iajs-2000 89 27 . . PUNCT iajs-2000 90 1 that that PRON iajs-2000 90 2 is be AUX iajs-2000 90 3 0 0 NUM iajs-2000 90 4 πœ“ πœ“ NOUN iajs-2000 90 5 𝑦 𝑦 X iajs-2000 90 6 𝑖𝑦 𝑖𝑦 NOUN iajs-2000 90 7 ∈ ∈ PROPN iajs-2000 90 8 𝐼𝐾 𝐼𝐾 PROPN iajs-2000 90 9 𝐿 𝐿 PROPN iajs-2000 90 10 , , PUNCT iajs-2000 90 11 implies imply VERB iajs-2000 90 12 that that SCONJ iajs-2000 90 13 0 0 NUM iajs-2000 90 14 𝑖𝑦 𝑖𝑦 NUM iajs-2000 90 15 ∈ ∈ PROPN iajs-2000 90 16 𝐿 𝐿 PROPN iajs-2000 90 17 , , PUNCT iajs-2000 90 18 but but CCONJ iajs-2000 90 19 " " PUNCT iajs-2000 90 20 l l NOUN iajs-2000 90 21 is be AUX iajs-2000 90 22 a a DET iajs-2000 90 23 we we NOUN iajs-2000 90 24 - - PUNCT iajs-2000 90 25 prime prime NOUN iajs-2000 90 26 submodule submodule NOUN iajs-2000 90 27 of of ADP iajs-2000 90 28 x x PROPN iajs-2000 90 29 , , PUNCT iajs-2000 90 30 and and CCONJ iajs-2000 90 31 𝑖𝑋 𝑖𝑋 VERB iajs-2000 90 32 πœ“ πœ“ ADP iajs-2000 90 33 𝑋 𝑋 PROPN iajs-2000 90 34 β‰° β‰° PROPN iajs-2000 90 35 𝐿 𝐿 PROPN iajs-2000 90 36 , , PUNCT iajs-2000 90 37 implies imply VERB iajs-2000 90 38 that that SCONJ iajs-2000 90 39 𝑦 𝑦 PROPN iajs-2000 90 40 ∈ ∈ NOUN iajs-2000 90 41 𝐿. 𝐿. VERB iajs-2000 90 42 thus thus ADV iajs-2000 90 43 𝐾 𝐾 PROPN iajs-2000 90 44 𝐿 𝐿 PROPN iajs-2000 90 45 " " PUNCT iajs-2000 90 46 . . PUNCT iajs-2000 91 1 proposition proposition NOUN iajs-2000 91 2 ( ( PUNCT iajs-2000 91 3 12 12 NUM iajs-2000 91 4 ) ) PUNCT iajs-2000 91 5 " " PUNCT iajs-2000 91 6 let let VERB iajs-2000 91 7 x x PRON iajs-2000 91 8 be be AUX iajs-2000 91 9 an an DET iajs-2000 91 10 r r NOUN iajs-2000 91 11 - - PUNCT iajs-2000 91 12 module module NOUN iajs-2000 91 13 and and CCONJ iajs-2000 91 14 πœ“ πœ“ NOUN iajs-2000 91 15 : : PUNCT iajs-2000 91 16 𝑋 𝑋 PROPN iajs-2000 91 17 ⟢ ⟢ NOUN iajs-2000 91 18 𝑋 𝑋 PROPN iajs-2000 91 19 be be VERB iajs-2000 91 20 an an DET iajs-2000 91 21 r r NOUN iajs-2000 91 22 - - PUNCT iajs-2000 91 23 homomorphism homomorphism NOUN iajs-2000 91 24 " " PUNCT iajs-2000 91 25 , , PUNCT iajs-2000 91 26 " " PUNCT iajs-2000 91 27 and and CCONJ iajs-2000 91 28 k k PROPN iajs-2000 91 29 be be AUX iajs-2000 91 30 a a DET iajs-2000 91 31 proper proper ADJ iajs-2000 91 32 fully fully ADV iajs-2000 91 33 invariant invariant ADJ iajs-2000 91 34 we we PRON iajs-2000 91 35 - - PUNCT iajs-2000 91 36 prime prime NOUN iajs-2000 91 37 submodule submodule NOUN iajs-2000 91 38 of of ADP iajs-2000 91 39 x x PUNCT iajs-2000 91 40 with with ADP iajs-2000 91 41 πœ“ πœ“ NOUN iajs-2000 91 42 𝑋 𝑋 PROPN iajs-2000 91 43 β‰° β‰° PROPN iajs-2000 91 44 𝐾. 𝐾. PROPN iajs-2000 91 45 then then ADV iajs-2000 91 46 πœ“ πœ“ PROPN iajs-2000 91 47 𝐾 𝐾 PROPN iajs-2000 91 48 is be AUX iajs-2000 91 49 a a DET iajs-2000 91 50 we we NOUN iajs-2000 91 51 - - PUNCT iajs-2000 91 52 prime prime NOUN iajs-2000 91 53 submodule submodule NOUN iajs-2000 91 54 of of ADP iajs-2000 91 55 x x NOUN iajs-2000 91 56 " " PUNCT iajs-2000 91 57 . . PUNCT iajs-2000 92 1 proof proof NOUN iajs-2000 92 2 " " PUNCT iajs-2000 92 3 clearly clearly ADV iajs-2000 92 4 πœ“ πœ“ ADP iajs-2000 92 5 𝐾 𝐾 PROPN iajs-2000 92 6 is be AUX iajs-2000 92 7 a a DET iajs-2000 92 8 proper proper ADJ iajs-2000 92 9 submodule submodule NOUN iajs-2000 92 10 of of ADP iajs-2000 92 11 " " PUNCT iajs-2000 92 12 x. x. NOUN iajs-2000 92 13 now now ADV iajs-2000 92 14 , , PUNCT iajs-2000 92 15 assume assume VERB iajs-2000 92 16 that that SCONJ iajs-2000 92 17 0 0 NUM iajs-2000 92 18 πœ™ πœ™ PRON iajs-2000 92 19 π‘₯ π‘₯ X iajs-2000 92 20 ∈ ∈ PROPN iajs-2000 92 21 πœ“ πœ“ X iajs-2000 92 22 𝐾 𝐾 PROPN iajs-2000 92 23 where where SCONJ iajs-2000 92 24 π‘₯ π‘₯ DET iajs-2000 92 25 ∈ ∈ PROPN iajs-2000 92 26 𝑋 𝑋 PROPN iajs-2000 92 27 , , PUNCT iajs-2000 92 28 πœ™ πœ™ PRON iajs-2000 92 29 ∈ ∈ PROPN iajs-2000 92 30 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 92 31 𝑋 𝑋 PROPN iajs-2000 92 32 . . PUNCT iajs-2000 93 1 if if SCONJ iajs-2000 93 2 π‘₯ π‘₯ PROPN iajs-2000 93 3 βˆ‰ βˆ‰ PROPN iajs-2000 93 4 πœ“ πœ“ PROPN iajs-2000 93 5 𝐾 𝐾 PROPN iajs-2000 93 6 , , PUNCT iajs-2000 93 7 then then ADV iajs-2000 93 8 πœ“ πœ“ PROPN iajs-2000 93 9 π‘₯ π‘₯ PROPN iajs-2000 93 10 βˆ‰ βˆ‰ PROPN iajs-2000 93 11 𝐾 𝐾 PROPN iajs-2000 93 12 , , PUNCT iajs-2000 93 13 " " PUNCT iajs-2000 93 14 it it PRON iajs-2000 93 15 follows follow VERB iajs-2000 93 16 that that SCONJ iajs-2000 93 17 " " PUNCT iajs-2000 93 18 π‘₯ π‘₯ PROPN iajs-2000 93 19 βˆ‰ βˆ‰ PROPN iajs-2000 93 20 𝐾 𝐾 PROPN iajs-2000 93 21 " " PUNCT iajs-2000 93 22 because because SCONJ iajs-2000 93 23 k k PROPN iajs-2000 93 24 is be AUX iajs-2000 93 25 a a DET iajs-2000 93 26 fully fully ADV iajs-2000 93 27 invariant invariant ADJ iajs-2000 93 28 submodule submodule NOUN iajs-2000 93 29 of of ADP iajs-2000 93 30 x x NOUN iajs-2000 93 31 " " PUNCT iajs-2000 93 32 . . PUNCT iajs-2000 94 1 " " PUNCT iajs-2000 94 2 we we PRON iajs-2000 94 3 must must AUX iajs-2000 94 4 prove prove VERB iajs-2000 94 5 that that SCONJ iajs-2000 94 6 πœ™ πœ™ PROPN iajs-2000 94 7 𝑋 𝑋 PROPN iajs-2000 94 8 πœ“ πœ“ PROPN iajs-2000 94 9 𝐾 𝐾 PROPN iajs-2000 94 10 . . PUNCT iajs-2000 95 1 since since SCONJ iajs-2000 95 2 0 0 NUM iajs-2000 95 3 πœ“π‘œπœ™ πœ“π‘œπœ™ NOUN iajs-2000 95 4 π‘₯ π‘₯ X iajs-2000 95 5 πœ“ πœ“ NOUN iajs-2000 95 6 πœ™ πœ™ NOUN iajs-2000 95 7 π‘₯ π‘₯ X iajs-2000 95 8 ∈ ∈ PROPN iajs-2000 95 9 𝐾. 𝐾. PROPN iajs-2000 96 1 " " PUNCT iajs-2000 96 2 that that PRON iajs-2000 96 3 is be AUX iajs-2000 96 4 0 0 NUM iajs-2000 96 5 πœ“ πœ“ NOUN iajs-2000 96 6 πœ™ πœ™ NOUN iajs-2000 96 7 π‘₯ π‘₯ X iajs-2000 96 8 ∈ ∈ PROPN iajs-2000 96 9 𝐾 𝐾 PROPN iajs-2000 96 10 " " PUNCT iajs-2000 96 11 . . PUNCT iajs-2000 97 1 " " PUNCT iajs-2000 97 2 but but CCONJ iajs-2000 97 3 k k PROPN iajs-2000 97 4 is be AUX iajs-2000 97 5 a a DET iajs-2000 97 6 we we NOUN iajs-2000 97 7 - - PUNCT iajs-2000 97 8 prime prime NOUN iajs-2000 97 9 submodule submodule NOUN iajs-2000 97 10 of of ADP iajs-2000 97 11 x x PROPN iajs-2000 97 12 , , PUNCT iajs-2000 97 13 and and CCONJ iajs-2000 97 14 π‘₯ π‘₯ PROPN iajs-2000 97 15 βˆ‰ βˆ‰ PROPN iajs-2000 97 16 𝐾 𝐾 PROPN iajs-2000 97 17 " " PUNCT iajs-2000 97 18 , , PUNCT iajs-2000 97 19 " " PUNCT iajs-2000 97 20 it it PRON iajs-2000 97 21 follows follow VERB iajs-2000 97 22 that that SCONJ iajs-2000 97 23 πœ“π‘œπœ™ πœ“π‘œπœ™ NOUN iajs-2000 97 24 𝑋 𝑋 PROPN iajs-2000 97 25 𝐾 𝐾 PROPN iajs-2000 97 26 , , PUNCT iajs-2000 97 27 implies imply VERB iajs-2000 97 28 that that SCONJ iajs-2000 97 29 πœ™ πœ™ PROPN iajs-2000 97 30 𝑋 𝑋 PROPN iajs-2000 97 31 πœ“ πœ“ PROPN iajs-2000 97 32 𝐾 𝐾 PROPN iajs-2000 97 33 . . PUNCT iajs-2000 98 1 hence hence ADV iajs-2000 98 2 πœ“ πœ“ ADP iajs-2000 98 3 𝐾 𝐾 PROPN iajs-2000 98 4 is be AUX iajs-2000 98 5 a a DET iajs-2000 98 6 we we NOUN iajs-2000 98 7 - - PUNCT iajs-2000 98 8 prime prime NOUN iajs-2000 98 9 submodule submodule NOUN iajs-2000 98 10 of of ADP iajs-2000 98 11 x x NOUN iajs-2000 98 12 " " PUNCT iajs-2000 98 13 . . PUNCT iajs-2000 99 1 3 3 X iajs-2000 99 2 . . X iajs-2000 99 3 we we PRON iajs-2000 99 4 - - PUNCT iajs-2000 99 5 semi semi ADJ iajs-2000 99 6 - - ADJ iajs-2000 99 7 prime prime ADJ iajs-2000 99 8 submodules submodule NOUN iajs-2000 99 9 " " PUNCT iajs-2000 99 10 " " PUNCT iajs-2000 99 11 in in ADP iajs-2000 99 12 this this DET iajs-2000 99 13 section section NOUN iajs-2000 99 14 , , PUNCT iajs-2000 99 15 we we PRON iajs-2000 99 16 introduce introduce VERB iajs-2000 99 17 the the DET iajs-2000 99 18 " " PUNCT iajs-2000 99 19 concept concept NOUN iajs-2000 99 20 " " PUNCT iajs-2000 99 21 of of ADP iajs-2000 99 22 we we PRON iajs-2000 99 23 - - PUNCT iajs-2000 99 24 semi semi ADJ iajs-2000 99 25 - - ADJ iajs-2000 99 26 prime prime ADJ iajs-2000 99 27 submodule submodule NOUN iajs-2000 99 28 as as ADP iajs-2000 99 29 a a DET iajs-2000 99 30 generalization generalization NOUN iajs-2000 99 31 of of ADP iajs-2000 99 32 " " PUNCT iajs-2000 99 33 a a DET iajs-2000 99 34 we we NOUN iajs-2000 99 35 - - PUNCT iajs-2000 99 36 prime prime ADJ iajs-2000 99 37 " " PUNCT iajs-2000 99 38 submodule submodule NOUN iajs-2000 99 39 and and CCONJ iajs-2000 99 40 " " PUNCT iajs-2000 99 41 stronger strong ADJ iajs-2000 99 42 form form NOUN iajs-2000 99 43 of of ADP iajs-2000 99 44 a a DET iajs-2000 99 45 weakly weakly ADJ iajs-2000 99 46 semi semi ADJ iajs-2000 99 47 - - ADJ iajs-2000 99 48 prime prime ADJ iajs-2000 99 49 " " PUNCT iajs-2000 99 50 submodule submodule NOUN iajs-2000 99 51 and and CCONJ iajs-2000 99 52 give give VERB iajs-2000 99 53 some some DET iajs-2000 99 54 basic basic ADJ iajs-2000 99 55 properties property NOUN iajs-2000 99 56 " " PUNCT iajs-2000 99 57 , , PUNCT iajs-2000 99 58 " " PUNCT iajs-2000 99 59 examples example NOUN iajs-2000 99 60 and and CCONJ iajs-2000 99 61 characterizations characterization NOUN iajs-2000 99 62 of of ADP iajs-2000 99 63 this this DET iajs-2000 99 64 concept concept NOUN iajs-2000 99 65 " " PUNCT iajs-2000 99 66 . . PUNCT iajs-2000 100 1 mathematics mathematic NOUN iajs-2000 100 2 | | ADV iajs-2000 100 3 113 113 NUM iajs-2000 100 4 ibn ibn PROPN iajs-2000 100 5 al al PROPN iajs-2000 100 6 - - PUNCT iajs-2000 100 7 haitham haitham PROPN iajs-2000 100 8 jour jour X iajs-2000 100 9 . . PROPN iajs-2000 100 10 for for ADP iajs-2000 100 11 pure pure ADJ iajs-2000 100 12 & & CCONJ iajs-2000 100 13 appl appl PROPN iajs-2000 100 14 . . PUNCT iajs-2000 101 1 sci sci PROPN iajs-2000 101 2 . . PROPN iajs-2000 101 3 ihjpas ihjpa VERB iajs-2000 101 4 https://doi.org/10.30526/31.3.2000 https://doi.org/10.30526/31.3.2000 NUM iajs-2000 101 5 vol vol NOUN iajs-2000 101 6 . . PROPN iajs-2000 101 7 31 31 NUM iajs-2000 101 8 ( ( PUNCT iajs-2000 101 9 3 3 NUM iajs-2000 101 10 ) ) SYM iajs-2000 101 11 2018 2018 NUM iajs-2000 101 12 definition definition NOUN iajs-2000 101 13 ( ( PUNCT iajs-2000 101 14 13 13 NUM iajs-2000 101 15 ) ) PUNCT iajs-2000 101 16 " " PUNCT iajs-2000 101 17 " " PUNCT iajs-2000 101 18 a a DET iajs-2000 101 19 proper proper ADJ iajs-2000 101 20 submodule submodule NOUN iajs-2000 101 21 k k PROPN iajs-2000 101 22 of of ADP iajs-2000 101 23 an an DET iajs-2000 101 24 r r NOUN iajs-2000 101 25 - - PUNCT iajs-2000 101 26 module module NOUN iajs-2000 101 27 " " PUNCT iajs-2000 101 28 x x NOUN iajs-2000 101 29 " " PUNCT iajs-2000 101 30 is be AUX iajs-2000 101 31 said say VERB iajs-2000 101 32 to to PART iajs-2000 101 33 be be AUX iajs-2000 101 34 a a DET iajs-2000 101 35 weakly weakly ADJ iajs-2000 101 36 endo endo NOUN iajs-2000 101 37 semi semi ADJ iajs-2000 101 38 - - ADJ iajs-2000 101 39 prime prime ADJ iajs-2000 101 40 submodule submodule NOUN iajs-2000 101 41 of of ADP iajs-2000 101 42 x x X iajs-2000 101 43 ( ( PUNCT iajs-2000 101 44 for for ADP iajs-2000 101 45 a a DET iajs-2000 101 46 short short ADJ iajs-2000 101 47 we we PRON iajs-2000 101 48 - - PUNCT iajs-2000 101 49 semi semi ADV iajs-2000 101 50 - - ADJ iajs-2000 101 51 prime prime ADJ iajs-2000 101 52 ) ) PUNCT iajs-2000 101 53 " " PUNCT iajs-2000 101 54 , , PUNCT iajs-2000 101 55 where where SCONJ iajs-2000 101 56 𝐸 𝐸 PROPN iajs-2000 101 57 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 101 58 𝑋 𝑋 PROPN iajs-2000 101 59 , , PUNCT iajs-2000 101 60 if if SCONJ iajs-2000 101 61 , , PUNCT iajs-2000 101 62 wherever wherever SCONJ iajs-2000 101 63 0 0 NUM iajs-2000 101 64 ψ ψ NOUN iajs-2000 101 65 π‘₯ π‘₯ PRON iajs-2000 101 66 ∈ ∈ PROPN iajs-2000 101 67 𝐾 𝐾 PROPN iajs-2000 101 68 , , PUNCT iajs-2000 101 69 where where SCONJ iajs-2000 101 70 π‘₯ π‘₯ DET iajs-2000 101 71 ∈ ∈ PROPN iajs-2000 101 72 𝑋 𝑋 PROPN iajs-2000 101 73 and and CCONJ iajs-2000 101 74 ψ ψ NOUN iajs-2000 101 75 ∈ ∈ PROPN iajs-2000 101 76 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 101 77 𝑋 𝑋 PROPN iajs-2000 101 78 " " PUNCT iajs-2000 101 79 , , PUNCT iajs-2000 101 80 " " PUNCT iajs-2000 101 81 implies imply VERB iajs-2000 101 82 that that SCONJ iajs-2000 101 83 ψ ψ ADP iajs-2000 101 84 π‘š π‘š X iajs-2000 101 85 ∈ ∈ VERB iajs-2000 101 86 𝐾. 𝐾. NOUN iajs-2000 101 87 " " PUNCT iajs-2000 101 88 and and CCONJ iajs-2000 101 89 an an DET iajs-2000 101 90 ideal ideal ADJ iajs-2000 101 91 i i PRON iajs-2000 101 92 of of ADP iajs-2000 101 93 a a DET iajs-2000 101 94 ring ring NOUN iajs-2000 101 95 r r NOUN iajs-2000 101 96 is be AUX iajs-2000 101 97 said say VERB iajs-2000 101 98 to to PART iajs-2000 101 99 be be AUX iajs-2000 101 100 a a DET iajs-2000 101 101 " " PUNCT iajs-2000 101 102 weakly weakly ADJ iajs-2000 101 103 endo endo NOUN iajs-2000 101 104 semi-"prime semi-"prime ADJ iajs-2000 101 105 ideal ideal NOUN iajs-2000 101 106 of of ADP iajs-2000 101 107 r r NOUN iajs-2000 101 108 , , PUNCT iajs-2000 101 109 if if SCONJ iajs-2000 101 110 i i PRON iajs-2000 101 111 is be AUX iajs-2000 101 112 a a DET iajs-2000 101 113 " " PUNCT iajs-2000 101 114 weakly weakly ADJ iajs-2000 101 115 endo endo NOUN iajs-2000 101 116 semi-"prime semi-"prime NOUN iajs-2000 101 117 as as ADP iajs-2000 101 118 an an DET iajs-2000 101 119 r r NOUN iajs-2000 101 120 - - PUNCT iajs-2000 101 121 submodule submodule NOUN iajs-2000 101 122 of of ADP iajs-2000 101 123 r r NOUN iajs-2000 101 124 - - PUNCT iajs-2000 101 125 module module NOUN iajs-2000 101 126 r r NOUN iajs-2000 101 127 " " PUNCT iajs-2000 101 128 . . PUNCT iajs-2000 102 1 proposition proposition NOUN iajs-2000 102 2 ( ( PUNCT iajs-2000 102 3 14 14 NUM iajs-2000 102 4 ) ) PUNCT iajs-2000 102 5 " " PUNCT iajs-2000 102 6 every every DET iajs-2000 102 7 we we NOUN iajs-2000 102 8 - - PUNCT iajs-2000 102 9 prime prime NOUN iajs-2000 102 10 submodule submodule NOUN iajs-2000 102 11 of of ADP iajs-2000 102 12 an an DET iajs-2000 102 13 r r NOUN iajs-2000 102 14 - - PUNCT iajs-2000 102 15 module module NOUN iajs-2000 102 16 x x NOUN iajs-2000 102 17 " " PUNCT iajs-2000 102 18 " " PUNCT iajs-2000 102 19 is be AUX iajs-2000 102 20 a a DET iajs-2000 102 21 we we PRON iajs-2000 102 22 - - PUNCT iajs-2000 102 23 semi semi ADJ iajs-2000 102 24 - - ADJ iajs-2000 102 25 prime prime ADJ iajs-2000 102 26 submodule submodule NOUN iajs-2000 102 27 of of ADP iajs-2000 102 28 x x NOUN iajs-2000 102 29 " " PUNCT iajs-2000 102 30 . . PUNCT iajs-2000 103 1 proof proof NOUN iajs-2000 103 2 " " PUNCT iajs-2000 103 3 let let VERB iajs-2000 103 4 k k PRON iajs-2000 103 5 be be AUX iajs-2000 103 6 a a DET iajs-2000 103 7 we we NOUN iajs-2000 103 8 - - PUNCT iajs-2000 103 9 prime prime NOUN iajs-2000 103 10 submodule submodule NOUN iajs-2000 103 11 of of ADP iajs-2000 103 12 x x NOUN iajs-2000 103 13 " " PUNCT iajs-2000 103 14 , , PUNCT iajs-2000 103 15 " " PUNCT iajs-2000 103 16 and and CCONJ iajs-2000 103 17 0 0 NUM iajs-2000 103 18 ψ ψ NOUN iajs-2000 103 19 π‘₯ π‘₯ PRON iajs-2000 103 20 ∈ ∈ PROPN iajs-2000 103 21 𝐾 𝐾 PROPN iajs-2000 103 22 " " PUNCT iajs-2000 103 23 , , PUNCT iajs-2000 103 24 where where SCONJ iajs-2000 103 25 π‘₯ π‘₯ DET iajs-2000 103 26 ∈ ∈ PROPN iajs-2000 103 27 𝑋 𝑋 PROPN iajs-2000 103 28 , , PUNCT iajs-2000 103 29 ψ ψ ADP iajs-2000 103 30 ∈ ∈ PROPN iajs-2000 103 31 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 103 32 𝑋 𝑋 PROPN iajs-2000 103 33 . . PUNCT iajs-2000 104 1 since since SCONJ iajs-2000 104 2 " " PUNCT iajs-2000 104 3 k k X iajs-2000 104 4 is be AUX iajs-2000 104 5 a a DET iajs-2000 104 6 we we NOUN iajs-2000 104 7 - - PUNCT iajs-2000 104 8 prime prime NOUN iajs-2000 104 9 submodule submodule NOUN iajs-2000 104 10 , , PUNCT iajs-2000 104 11 and and CCONJ iajs-2000 104 12 0 0 NUM iajs-2000 104 13 ψ ψ NOUN iajs-2000 104 14 ψ ψ X iajs-2000 104 15 π‘₯ π‘₯ PRON iajs-2000 104 16 ∈ ∈ PROPN iajs-2000 104 17 𝐾 𝐾 NOUN iajs-2000 104 18 " " PUNCT iajs-2000 104 19 , , PUNCT iajs-2000 104 20 then then ADV iajs-2000 104 21 " " PUNCT iajs-2000 104 22 either either CCONJ iajs-2000 104 23 ψ ψ X iajs-2000 104 24 π‘₯ π‘₯ PRON iajs-2000 104 25 ∈ ∈ PROPN iajs-2000 104 26 𝐾 𝐾 NOUN iajs-2000 104 27 or or CCONJ iajs-2000 104 28 ψ ψ ADP iajs-2000 104 29 𝑋 𝑋 PROPN iajs-2000 104 30 𝐾 𝐾 PROPN iajs-2000 104 31 " " PUNCT iajs-2000 104 32 . . PUNCT iajs-2000 105 1 " " PUNCT iajs-2000 105 2 thus thus ADV iajs-2000 105 3 in in ADP iajs-2000 105 4 any any DET iajs-2000 105 5 case case NOUN iajs-2000 105 6 ψ ψ ADP iajs-2000 105 7 π‘₯ π‘₯ X iajs-2000 105 8 ∈ ∈ PROPN iajs-2000 105 9 𝐾 𝐾 PROPN iajs-2000 105 10 " " PUNCT iajs-2000 105 11 . . PUNCT iajs-2000 106 1 " " PUNCT iajs-2000 106 2 hence hence ADV iajs-2000 106 3 k k PROPN iajs-2000 106 4 is be AUX iajs-2000 106 5 a a DET iajs-2000 106 6 we we PRON iajs-2000 106 7 - - PUNCT iajs-2000 106 8 semi semi ADJ iajs-2000 106 9 - - ADJ iajs-2000 106 10 prime prime ADJ iajs-2000 106 11 submodule submodule NOUN iajs-2000 106 12 of of ADP iajs-2000 106 13 x x NOUN iajs-2000 106 14 " " PUNCT iajs-2000 106 15 . . PUNCT iajs-2000 107 1 " " PUNCT iajs-2000 107 2 " " PUNCT iajs-2000 107 3 the the DET iajs-2000 107 4 converse converse NOUN iajs-2000 107 5 of of ADP iajs-2000 107 6 proposition proposition NOUN iajs-2000 107 7 ( ( PUNCT iajs-2000 107 8 3.2 3.2 NUM iajs-2000 107 9 ) ) PUNCT iajs-2000 107 10 is be AUX iajs-2000 107 11 not not PART iajs-2000 107 12 true true ADJ iajs-2000 107 13 in in ADP iajs-2000 107 14 general general ADJ iajs-2000 107 15 " " PUNCT iajs-2000 107 16 , , PUNCT iajs-2000 107 17 " " PUNCT iajs-2000 107 18 as as SCONJ iajs-2000 107 19 the the DET iajs-2000 107 20 following follow VERB iajs-2000 107 21 example example NOUN iajs-2000 107 22 shows show VERB iajs-2000 107 23 that that SCONJ iajs-2000 107 24 " " PUNCT iajs-2000 107 25 . . PUNCT iajs-2000 108 1 example example NOUN iajs-2000 108 2 ( ( PUNCT iajs-2000 108 3 15 15 NUM iajs-2000 108 4 ) ) PUNCT iajs-2000 108 5 " " PUNCT iajs-2000 108 6 " " PUNCT iajs-2000 108 7 let let VERB iajs-2000 108 8 x x X iajs-2000 108 9 = = NOUN iajs-2000 108 10 z z NOUN iajs-2000 108 11 and and CCONJ iajs-2000 108 12 r r NOUN iajs-2000 108 13 = = PROPN iajs-2000 108 14 z z PROPN iajs-2000 108 15 , , PUNCT iajs-2000 108 16 k=10z k=10z PROPN iajs-2000 108 17 as as ADP iajs-2000 108 18 a a DET iajs-2000 108 19 " " PUNCT iajs-2000 108 20 z z NOUN iajs-2000 108 21 - - PUNCT iajs-2000 108 22 module module NOUN iajs-2000 108 23 of of ADP iajs-2000 108 24 x. x. NOUN iajs-2000 108 25 " " PUNCT iajs-2000 108 26 then then ADV iajs-2000 108 27 k k PROPN iajs-2000 108 28 is be AUX iajs-2000 108 29 a a DET iajs-2000 108 30 we we PRON iajs-2000 108 31 - - PUNCT iajs-2000 108 32 semi semi NOUN iajs-2000 108 33 - - ADJ iajs-2000 108 34 prime prime ADJ iajs-2000 108 35 but but CCONJ iajs-2000 108 36 not not PART iajs-2000 108 37 weprime weprime NOUN iajs-2000 108 38 submodule submodule NOUN iajs-2000 108 39 of of ADP iajs-2000 108 40 x x PRON iajs-2000 108 41 , , PUNCT iajs-2000 108 42 since since SCONJ iajs-2000 108 43 if if SCONJ iajs-2000 108 44 we we PRON iajs-2000 108 45 defined define VERB iajs-2000 108 46 ψ ψ NOUN iajs-2000 108 47 : : PUNCT iajs-2000 108 48 𝑍 𝑍 NOUN iajs-2000 108 49 ⟢ ⟢ NOUN iajs-2000 108 50 𝑍 𝑍 NOUN iajs-2000 108 51 by by ADP iajs-2000 108 52 ψ ψ X iajs-2000 108 53 π‘₯ π‘₯ PRON iajs-2000 108 54 π‘₯ π‘₯ NOUN iajs-2000 108 55 , , PUNCT iajs-2000 108 56 ψ ψ X iajs-2000 108 57 ∈ ∈ PROPN iajs-2000 108 58 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 108 59 𝑋 𝑋 PROPN iajs-2000 108 60 and and CCONJ iajs-2000 108 61 0 0 NUM iajs-2000 108 62 2ψ 2ψ NUM iajs-2000 108 63 5 5 NUM iajs-2000 108 64 10 10 NUM iajs-2000 108 65 ∈ ∈ PROPN iajs-2000 108 66 𝐾 𝐾 PROPN iajs-2000 108 67 , , PUNCT iajs-2000 108 68 but but CCONJ iajs-2000 108 69 5 5 NUM iajs-2000 108 70 βˆ‰ βˆ‰ PROPN iajs-2000 108 71 𝐾 𝐾 PROPN iajs-2000 108 72 and and CCONJ iajs-2000 108 73 ψ ψ ADP iajs-2000 108 74 𝑍 𝑍 PROPN iajs-2000 108 75 𝑍 𝑍 NOUN iajs-2000 108 76 β‰° β‰° PROPN iajs-2000 108 77 𝐾 𝐾 PROPN iajs-2000 108 78 10𝑍 10𝑍 NUM iajs-2000 108 79 , , PUNCT iajs-2000 108 80 hence hence ADV iajs-2000 108 81 k k PROPN iajs-2000 108 82 is be AUX iajs-2000 108 83 not not PART iajs-2000 108 84 we we PRON iajs-2000 108 85 - - PUNCT iajs-2000 108 86 prime prime NOUN iajs-2000 108 87 submodule submodule NOUN iajs-2000 108 88 of of ADP iajs-2000 108 89 x. x. PROPN iajs-2000 109 1 but but CCONJ iajs-2000 109 2 k k PROPN iajs-2000 109 3 is be AUX iajs-2000 109 4 a a DET iajs-2000 109 5 we we PRON iajs-2000 109 6 - - PUNCT iajs-2000 109 7 semi semi ADV iajs-2000 109 8 - - ADJ iajs-2000 109 9 prime prime ADJ iajs-2000 109 10 , , PUNCT iajs-2000 109 11 since since SCONJ iajs-2000 109 12 0 0 NUM iajs-2000 109 13 ψ ψ SYM iajs-2000 109 14 10 10 NUM iajs-2000 109 15 ψ ψ NOUN iajs-2000 109 16 ψ ψ SYM iajs-2000 109 17 10 10 NUM iajs-2000 109 18 10 10 NUM iajs-2000 109 19 ∈ ∈ PROPN iajs-2000 109 20 𝐾 𝐾 PROPN iajs-2000 109 21 , , PUNCT iajs-2000 109 22 implies imply VERB iajs-2000 109 23 that that SCONJ iajs-2000 109 24 ψ ψ ADP iajs-2000 109 25 10 10 NUM iajs-2000 109 26 10 10 NUM iajs-2000 109 27 ∈ ∈ PROPN iajs-2000 109 28 𝐾 𝐾 NOUN iajs-2000 109 29 " " PUNCT iajs-2000 109 30 . . PUNCT iajs-2000 110 1 proposition proposition NOUN iajs-2000 110 2 ( ( PUNCT iajs-2000 110 3 16 16 NUM iajs-2000 110 4 ) ) PUNCT iajs-2000 110 5 " " PUNCT iajs-2000 110 6 every every DET iajs-2000 110 7 we we PRON iajs-2000 110 8 - - PUNCT iajs-2000 110 9 semi semi ADJ iajs-2000 110 10 - - ADJ iajs-2000 110 11 prime prime ADJ iajs-2000 110 12 submodule submodule NOUN iajs-2000 110 13 of of ADP iajs-2000 110 14 an an DET iajs-2000 110 15 r r NOUN iajs-2000 110 16 - - PUNCT iajs-2000 110 17 module module NOUN iajs-2000 110 18 x x NOUN iajs-2000 110 19 " " PUNCT iajs-2000 110 20 " " PUNCT iajs-2000 110 21 is be AUX iajs-2000 110 22 a a DET iajs-2000 110 23 weakly weakly ADJ iajs-2000 110 24 semi semi ADJ iajs-2000 110 25 - - ADJ iajs-2000 110 26 prime prime ADJ iajs-2000 110 27 submodule submodule NOUN iajs-2000 110 28 of of ADP iajs-2000 110 29 x x NOUN iajs-2000 110 30 " " PUNCT iajs-2000 110 31 . . PUNCT iajs-2000 111 1 proof proof NOUN iajs-2000 111 2 " " PUNCT iajs-2000 111 3 let let VERB iajs-2000 111 4 k k PRON iajs-2000 111 5 be be AUX iajs-2000 111 6 a a DET iajs-2000 111 7 we we PRON iajs-2000 111 8 - - PUNCT iajs-2000 111 9 semi semi ADJ iajs-2000 111 10 - - ADJ iajs-2000 111 11 prime prime ADJ iajs-2000 111 12 submodule submodule NOUN iajs-2000 111 13 of of ADP iajs-2000 111 14 " " PUNCT iajs-2000 111 15 x x NOUN iajs-2000 111 16 , , PUNCT iajs-2000 111 17 " " PUNCT iajs-2000 111 18 and and CCONJ iajs-2000 111 19 " " PUNCT iajs-2000 111 20 0 0 NUM iajs-2000 112 1 π‘Ÿ π‘Ÿ NOUN iajs-2000 112 2 π‘₯ π‘₯ PRON iajs-2000 112 3 ∈ ∈ PROPN iajs-2000 112 4 𝐾 𝐾 PROPN iajs-2000 112 5 , , PUNCT iajs-2000 112 6 where where SCONJ iajs-2000 112 7 π‘Ÿ π‘Ÿ X iajs-2000 112 8 ∈ ∈ PROPN iajs-2000 112 9 𝑅 𝑅 PROPN iajs-2000 112 10 , , PUNCT iajs-2000 112 11 π‘₯ π‘₯ PRON iajs-2000 112 12 ∈ ∈ PROPN iajs-2000 112 13 𝐾 𝐾 PROPN iajs-2000 112 14 . . PUNCT iajs-2000 113 1 now now ADV iajs-2000 113 2 , , PUNCT iajs-2000 113 3 let let VERB iajs-2000 113 4 ψ ψ PRON iajs-2000 113 5 : : PUNCT iajs-2000 113 6 𝑋 𝑋 PROPN iajs-2000 113 7 ⟢ ⟢ NOUN iajs-2000 113 8 𝑋 𝑋 NOUN iajs-2000 113 9 defined define VERB iajs-2000 113 10 by by ADP iajs-2000 113 11 ψ ψ X iajs-2000 113 12 π‘₯ π‘₯ DET iajs-2000 113 13 π‘Ÿπ‘₯ π‘Ÿπ‘₯ NOUN iajs-2000 113 14 for for ADP iajs-2000 113 15 all all PRON iajs-2000 113 16 π‘₯ π‘₯ DET iajs-2000 113 17 ∈ ∈ PROPN iajs-2000 113 18 𝑋 𝑋 NOUN iajs-2000 113 19 , , PUNCT iajs-2000 113 20 clearly clearly ADV iajs-2000 113 21 ψ ψ ADP iajs-2000 113 22 ∈ ∈ PROPN iajs-2000 113 23 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 113 24 𝑋 𝑋 PROPN iajs-2000 113 25 " " PUNCT iajs-2000 113 26 . . PUNCT iajs-2000 114 1 " " PUNCT iajs-2000 114 2 now now ADV iajs-2000 114 3 , , PUNCT iajs-2000 114 4 0 0 NUM iajs-2000 114 5 π‘Ÿ π‘Ÿ X iajs-2000 114 6 π‘₯ π‘₯ PRON iajs-2000 114 7 ψ ψ X iajs-2000 114 8 π‘₯ π‘₯ PRON iajs-2000 114 9 ∈ ∈ PROPN iajs-2000 114 10 𝐾 𝐾 PROPN iajs-2000 114 11 , , PUNCT iajs-2000 114 12 but but CCONJ iajs-2000 114 13 " " PUNCT iajs-2000 114 14 k k X iajs-2000 114 15 is be AUX iajs-2000 114 16 a a DET iajs-2000 114 17 we we PRON iajs-2000 114 18 - - PUNCT iajs-2000 114 19 semi semi ADJ iajs-2000 114 20 - - ADJ iajs-2000 114 21 prime prime ADJ iajs-2000 114 22 submodule submodule NOUN iajs-2000 114 23 of of ADP iajs-2000 114 24 " " PUNCT iajs-2000 114 25 x x NOUN iajs-2000 114 26 , , PUNCT iajs-2000 114 27 " " PUNCT iajs-2000 114 28 implies imply VERB iajs-2000 114 29 that that SCONJ iajs-2000 114 30 ψ ψ ADP iajs-2000 114 31 π‘₯ π‘₯ X iajs-2000 114 32 π‘Ÿπ‘₯ π‘Ÿπ‘₯ ADP iajs-2000 114 33 ∈ ∈ PROPN iajs-2000 114 34 𝐾 𝐾 PROPN iajs-2000 114 35 " " PUNCT iajs-2000 114 36 . . PUNCT iajs-2000 115 1 " " PUNCT iajs-2000 115 2 thus thus ADV iajs-2000 115 3 k k X iajs-2000 115 4 is be AUX iajs-2000 115 5 a a DET iajs-2000 115 6 weakly weakly ADJ iajs-2000 115 7 semi semi ADJ iajs-2000 115 8 - - ADJ iajs-2000 115 9 prime prime ADJ iajs-2000 115 10 submodule submodule NOUN iajs-2000 115 11 of of ADP iajs-2000 115 12 x x NOUN iajs-2000 115 13 " " PUNCT iajs-2000 115 14 . . PUNCT iajs-2000 116 1 " " PUNCT iajs-2000 116 2 " " PUNCT iajs-2000 116 3 the the DET iajs-2000 116 4 converse converse NOUN iajs-2000 116 5 of of ADP iajs-2000 116 6 proposition proposition NOUN iajs-2000 116 7 " " PUNCT iajs-2000 116 8 " " PUNCT iajs-2000 116 9 ( ( PUNCT iajs-2000 116 10 3.4 3.4 NUM iajs-2000 116 11 ) ) PUNCT iajs-2000 116 12 is be AUX iajs-2000 116 13 not not PART iajs-2000 116 14 true true ADJ iajs-2000 116 15 in in ADP iajs-2000 116 16 general general ADJ iajs-2000 116 17 , , PUNCT iajs-2000 116 18 as as SCONJ iajs-2000 116 19 the the DET iajs-2000 116 20 following follow VERB iajs-2000 116 21 example example NOUN iajs-2000 116 22 shows show VERB iajs-2000 116 23 " " PUNCT iajs-2000 116 24 . . PUNCT iajs-2000 117 1 example example NOUN iajs-2000 117 2 ( ( PUNCT iajs-2000 117 3 17 17 NUM iajs-2000 117 4 ) ) PUNCT iajs-2000 117 5 " " PUNCT iajs-2000 117 6 " " PUNCT iajs-2000 117 7 let let VERB iajs-2000 117 8 𝑋 𝑋 PROPN iajs-2000 117 9 𝑍⨁𝑍 𝑍⨁𝑍 NOUN iajs-2000 117 10 , , PUNCT iajs-2000 117 11 r"=z r"=z VERB iajs-2000 117 12 , , PUNCT iajs-2000 117 13 𝐾 𝐾 PROPN iajs-2000 117 14 𝑍⨁10𝑍 𝑍⨁10𝑍 VERB iajs-2000 117 15 , , PUNCT iajs-2000 117 16 k k X iajs-2000 117 17 " " PUNCT iajs-2000 117 18 is be AUX iajs-2000 117 19 a a DET iajs-2000 117 20 weakly weakly ADJ iajs-2000 117 21 semi semi ADJ iajs-2000 117 22 - - ADJ iajs-2000 117 23 prime prime ADJ iajs-2000 117 24 submodule submodule NOUN iajs-2000 117 25 of of ADP iajs-2000 117 26 x x NOUN iajs-2000 117 27 " " PUNCT iajs-2000 117 28 but but CCONJ iajs-2000 117 29 not not PART iajs-2000 117 30 we we PRON iajs-2000 117 31 - - PUNCT iajs-2000 117 32 semi semi ADV iajs-2000 117 33 - - ADJ iajs-2000 117 34 prime prime ADJ iajs-2000 117 35 : : PUNCT iajs-2000 117 36 let let VERB iajs-2000 117 37 π‘Ÿ π‘Ÿ PRON iajs-2000 117 38 2 2 NUM iajs-2000 117 39 ∈ ∈ NOUN iajs-2000 117 40 𝑍 𝑍 NOUN iajs-2000 117 41 and and CCONJ iajs-2000 117 42 π‘₯ π‘₯ NOUN iajs-2000 117 43 3,5 3,5 NUM iajs-2000 117 44 ∈ ∈ NOUN iajs-2000 117 45 𝑋 𝑋 NOUN iajs-2000 117 46 , , PUNCT iajs-2000 117 47 then then ADV iajs-2000 117 48 0 0 NUM iajs-2000 117 49 2 2 NUM iajs-2000 117 50 3,5 3,5 NUM iajs-2000 117 51 12,20 12,20 NUM iajs-2000 117 52 ∈ ∈ PROPN iajs-2000 117 53 𝐾 𝐾 PROPN iajs-2000 117 54 , , PUNCT iajs-2000 117 55 implies imply VERB iajs-2000 117 56 that that SCONJ iajs-2000 117 57 2 2 NUM iajs-2000 117 58 3,5 3,5 NUM iajs-2000 117 59 6,10 6,10 NUM iajs-2000 117 60 ∈ ∈ NOUN iajs-2000 117 61 𝐾. 𝐾. NOUN iajs-2000 117 62 to to PART iajs-2000 117 63 show show VERB iajs-2000 117 64 that that SCONJ iajs-2000 117 65 k k PROPN iajs-2000 117 66 is be AUX iajs-2000 117 67 not not PART iajs-2000 117 68 we we PRON iajs-2000 117 69 - - PUNCT iajs-2000 117 70 semi semi ADV iajs-2000 117 71 - - ADJ iajs-2000 117 72 prime prime ADJ iajs-2000 117 73 : : PUNCT iajs-2000 117 74 let let VERB iajs-2000 117 75 ψ ψ PRON iajs-2000 117 76 : : PUNCT iajs-2000 117 77 𝑋 𝑋 PROPN iajs-2000 117 78 ⟢ ⟢ NOUN iajs-2000 117 79 𝑋 𝑋 PROPN iajs-2000 117 80 " " PUNCT iajs-2000 117 81 defined define VERB iajs-2000 117 82 by by ADP iajs-2000 117 83 ψ ψ X iajs-2000 117 84 π‘₯ π‘₯ PROPN iajs-2000 117 85 , , PUNCT iajs-2000 117 86 𝑦 𝑦 NOUN iajs-2000 117 87 𝑦 𝑦 NOUN iajs-2000 117 88 , , PUNCT iajs-2000 117 89 π‘₯ π‘₯ X iajs-2000 117 90 for for ADP iajs-2000 117 91 all all PRON iajs-2000 117 92 π‘₯ π‘₯ PROPN iajs-2000 117 93 , , PUNCT iajs-2000 117 94 𝑦 𝑦 PRON iajs-2000 117 95 ∈ ∈ PROPN iajs-2000 117 96 𝑍 𝑍 PROPN iajs-2000 117 97 " " PUNCT iajs-2000 117 98 . . PUNCT iajs-2000 118 1 clearly clearly ADV iajs-2000 118 2 ψ ψ X iajs-2000 118 3 ∈ ∈ PROPN iajs-2000 118 4 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 118 5 𝑋 𝑋 PROPN iajs-2000 118 6 . . PUNCT iajs-2000 119 1 now now ADV iajs-2000 119 2 , , PUNCT iajs-2000 119 3 take take VERB iajs-2000 119 4 ψ ψ X iajs-2000 119 5 0,5 0,5 NUM iajs-2000 119 6 5,0 5,0 NUM iajs-2000 119 7 βˆ‰ βˆ‰ X iajs-2000 119 8 𝐾 𝐾 PROPN iajs-2000 119 9 but but CCONJ iajs-2000 119 10 ψ ψ X iajs-2000 119 11 0,5 0,5 NUM iajs-2000 119 12 ψ ψ SYM iajs-2000 119 13 ψ ψ ADP iajs-2000 119 14 0,5 0,5 NUM iajs-2000 119 15 ψ ψ ADP iajs-2000 119 16 5,0 5,0 NUM iajs-2000 119 17 0,5 0,5 NUM iajs-2000 119 18 ∈ ∈ PROPN iajs-2000 119 19 𝐾 𝐾 PROPN iajs-2000 119 20 . . PUNCT iajs-2000 120 1 hence hence ADV iajs-2000 120 2 k k PROPN iajs-2000 120 3 is be AUX iajs-2000 120 4 not not PART iajs-2000 120 5 we we PRON iajs-2000 120 6 - - PUNCT iajs-2000 120 7 semiprime semiprime NOUN iajs-2000 120 8 submodule submodule NOUN iajs-2000 120 9 of of ADP iajs-2000 120 10 x x NOUN iajs-2000 120 11 " " PUNCT iajs-2000 120 12 . . PUNCT iajs-2000 121 1 proposition proposition NOUN iajs-2000 121 2 ( ( PUNCT iajs-2000 121 3 18 18 NUM iajs-2000 121 4 ) ) PUNCT iajs-2000 121 5 " " PUNCT iajs-2000 121 6 " " PUNCT iajs-2000 121 7 let let VERB iajs-2000 121 8 k k PRON iajs-2000 121 9 be be AUX iajs-2000 121 10 a a DET iajs-2000 121 11 submodule submodule NOUN iajs-2000 121 12 of of ADP iajs-2000 121 13 an an DET iajs-2000 121 14 r r NOUN iajs-2000 121 15 - - PUNCT iajs-2000 121 16 module module NOUN iajs-2000 121 17 " " PUNCT iajs-2000 121 18 x x NOUN iajs-2000 121 19 with with ADP iajs-2000 121 20 𝐾 𝐾 PROPN iajs-2000 121 21 ∩∝∈∧ ∩∝∈∧ PROPN iajs-2000 121 22 𝐿∝ 𝐿∝ NOUN iajs-2000 121 23 , , PUNCT iajs-2000 121 24 where where SCONJ iajs-2000 121 25 each each DET iajs-2000 121 26 𝐿∝ 𝐿∝ NOUN iajs-2000 121 27 " " PUNCT iajs-2000 121 28 is be AUX iajs-2000 121 29 a a DET iajs-2000 121 30 we we NOUN iajs-2000 121 31 - - PUNCT iajs-2000 121 32 prime prime NOUN iajs-2000 121 33 submodule submodule NOUN iajs-2000 121 34 of of ADP iajs-2000 121 35 x. x. PROPN iajs-2000 122 1 " " PUNCT iajs-2000 122 2 then then ADV iajs-2000 122 3 k k PROPN iajs-2000 122 4 is be AUX iajs-2000 122 5 a a DET iajs-2000 122 6 we we PRON iajs-2000 122 7 - - PUNCT iajs-2000 122 8 semi semi ADJ iajs-2000 122 9 - - ADJ iajs-2000 122 10 prime prime ADJ iajs-2000 122 11 submodule submodule NOUN iajs-2000 122 12 of of ADP iajs-2000 122 13 x. x. PROPN iajs-2000 122 14 proof proof NOUN iajs-2000 122 15 " " PUNCT iajs-2000 122 16 " " PUNCT iajs-2000 122 17 suppose suppose VERB iajs-2000 122 18 that that SCONJ iajs-2000 122 19 " " PUNCT iajs-2000 122 20 0 0 NUM iajs-2000 122 21 ψ ψ NOUN iajs-2000 122 22 π‘₯ π‘₯ PRON iajs-2000 122 23 ∈ ∈ PROPN iajs-2000 122 24 𝐾 𝐾 PROPN iajs-2000 122 25 , , PUNCT iajs-2000 122 26 where where SCONJ iajs-2000 122 27 π‘₯ π‘₯ DET iajs-2000 122 28 ∈ ∈ PROPN iajs-2000 122 29 𝑋 𝑋 PROPN iajs-2000 122 30 , , PUNCT iajs-2000 122 31 ψ ψ ADP iajs-2000 122 32 ∈ ∈ PROPN iajs-2000 122 33 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 122 34 𝑋 𝑋 PROPN iajs-2000 122 35 , , PUNCT iajs-2000 122 36 then then ADV iajs-2000 122 37 0 0 NUM iajs-2000 122 38 ψ ψ NOUN iajs-2000 122 39 π‘₯ π‘₯ X iajs-2000 122 40 ∈ ∈ ADJ iajs-2000 122 41 𝐿∝ 𝐿∝ NOUN iajs-2000 122 42 for for ADP iajs-2000 122 43 each each DET iajs-2000 122 44 ∝ ∝ PROPN iajs-2000 122 45 ∈∧. ∈∧. PROPN iajs-2000 122 46 but but CCONJ iajs-2000 122 47 𝐿∝ 𝐿∝ PRON iajs-2000 122 48 " " PUNCT iajs-2000 122 49 is be AUX iajs-2000 122 50 a a DET iajs-2000 122 51 we we NOUN iajs-2000 122 52 - - PUNCT iajs-2000 122 53 prime prime NOUN iajs-2000 122 54 submodule submodule NOUN iajs-2000 122 55 of of ADP iajs-2000 122 56 x x PRON iajs-2000 122 57 , , PUNCT iajs-2000 122 58 hence hence ADV iajs-2000 122 59 by by ADP iajs-2000 122 60 proposition proposition NOUN iajs-2000 122 61 ( ( PUNCT iajs-2000 122 62 3.2 3.2 NUM iajs-2000 122 63 ) ) PUNCT iajs-2000 122 64 " " PUNCT iajs-2000 122 65 𝐿∝ 𝐿∝ X iajs-2000 122 66 is be AUX iajs-2000 122 67 a a DET iajs-2000 122 68 we we PRON iajs-2000 122 69 - - PUNCT iajs-2000 122 70 semi semi ADJ iajs-2000 122 71 mathematics mathematic NOUN iajs-2000 122 72 | | ADV iajs-2000 122 73 114 114 NUM iajs-2000 122 74 ibn ibn PROPN iajs-2000 122 75 al al PROPN iajs-2000 122 76 - - PUNCT iajs-2000 122 77 haitham haitham PROPN iajs-2000 122 78 jour jour X iajs-2000 122 79 . . PROPN iajs-2000 122 80 for for ADP iajs-2000 122 81 pure pure ADJ iajs-2000 122 82 & & CCONJ iajs-2000 122 83 appl appl PROPN iajs-2000 122 84 . . PUNCT iajs-2000 123 1 sci sci PROPN iajs-2000 123 2 . . PROPN iajs-2000 123 3 ihjpas ihjpa VERB iajs-2000 123 4 https://doi.org/10.30526/31.3.2000 https://doi.org/10.30526/31.3.2000 NUM iajs-2000 123 5 vol vol NOUN iajs-2000 123 6 . . PROPN iajs-2000 123 7 31 31 NUM iajs-2000 123 8 ( ( PUNCT iajs-2000 123 9 3 3 NUM iajs-2000 123 10 ) ) SYM iajs-2000 123 11 2018 2018 NUM iajs-2000 123 12 prime prime NOUN iajs-2000 123 13 . . PUNCT iajs-2000 124 1 " " PUNCT iajs-2000 124 2 thus thus ADV iajs-2000 124 3 ψ ψ X iajs-2000 124 4 π‘₯ π‘₯ X iajs-2000 124 5 ∈ ∈ ADJ iajs-2000 124 6 𝐿∝ 𝐿∝ NOUN iajs-2000 124 7 for for ADP iajs-2000 124 8 each each DET iajs-2000 124 9 ∝∈∧. ∝∈∧. NOUN iajs-2000 124 10 therefore therefore ADV iajs-2000 124 11 ψ ψ ADP iajs-2000 124 12 π‘₯ π‘₯ X iajs-2000 124 13 ∈ ∈ PROPN iajs-2000 124 14 ∩∝∈∧ ∩∝∈∧ NUM iajs-2000 124 15 𝐿∝. 𝐿∝. NOUN iajs-2000 124 16 hence hence ADV iajs-2000 124 17 k k PROPN iajs-2000 124 18 is be AUX iajs-2000 124 19 a a DET iajs-2000 124 20 we we PRON iajs-2000 124 21 - - PUNCT iajs-2000 124 22 semiprime semiprime NOUN iajs-2000 124 23 submodule submodule NOUN iajs-2000 124 24 of of ADP iajs-2000 124 25 x. x. NOUN iajs-2000 124 26 " " PUNCT iajs-2000 124 27 the the DET iajs-2000 124 28 following follow VERB iajs-2000 124 29 proposition proposition NOUN iajs-2000 124 30 shows show VERB iajs-2000 124 31 that that SCONJ iajs-2000 124 32 in in ADP iajs-2000 124 33 the the DET iajs-2000 124 34 class class NOUN iajs-2000 124 35 of of ADP iajs-2000 124 36 scalar scalar ADJ iajs-2000 124 37 modules module NOUN iajs-2000 124 38 , , PUNCT iajs-2000 124 39 weakly weakly ADJ iajs-2000 124 40 semi semi ADJ iajs-2000 124 41 - - ADJ iajs-2000 124 42 prime prime ADJ iajs-2000 124 43 submodule submodule NOUN iajs-2000 124 44 and and CCONJ iajs-2000 124 45 we we PRON iajs-2000 124 46 - - PUNCT iajs-2000 124 47 semi semi ADJ iajs-2000 124 48 - - ADJ iajs-2000 124 49 prime prime ADJ iajs-2000 124 50 submodules submodule NOUN iajs-2000 124 51 are be AUX iajs-2000 124 52 coinciding coincide VERB iajs-2000 124 53 . . PUNCT iajs-2000 125 1 proposition proposition NOUN iajs-2000 125 2 ( ( PUNCT iajs-2000 125 3 19 19 NUM iajs-2000 125 4 ) ) PUNCT iajs-2000 125 5 " " PUNCT iajs-2000 125 6 " " PUNCT iajs-2000 125 7 let let VERB iajs-2000 125 8 x x PRON iajs-2000 125 9 be be AUX iajs-2000 125 10 a a DET iajs-2000 125 11 scalar scalar ADJ iajs-2000 125 12 module module NOUN iajs-2000 125 13 , , PUNCT iajs-2000 125 14 and and CCONJ iajs-2000 125 15 l l NOUN iajs-2000 125 16 is be AUX iajs-2000 125 17 a a DET iajs-2000 125 18 proper proper ADJ iajs-2000 125 19 submodule submodule NOUN iajs-2000 125 20 of of ADP iajs-2000 125 21 x. x. NOUN iajs-2000 125 22 then then ADV iajs-2000 125 23 l l PROPN iajs-2000 125 24 is be AUX iajs-2000 125 25 a a DET iajs-2000 125 26 we we PRON iajs-2000 125 27 - - PUNCT iajs-2000 125 28 semi semi ADJ iajs-2000 125 29 - - ADJ iajs-2000 125 30 prime prime ADJ iajs-2000 125 31 submodule submodule NOUN iajs-2000 125 32 of of ADP iajs-2000 125 33 x x PRON iajs-2000 125 34 , , PUNCT iajs-2000 125 35 if if SCONJ iajs-2000 125 36 and and CCONJ iajs-2000 125 37 only only ADV iajs-2000 125 38 if if SCONJ iajs-2000 125 39 l l NOUN iajs-2000 125 40 is be AUX iajs-2000 125 41 a a DET iajs-2000 125 42 weakly weakly ADJ iajs-2000 125 43 semi semi ADJ iajs-2000 125 44 - - ADJ iajs-2000 125 45 prime prime ADJ iajs-2000 125 46 submodule submodule NOUN iajs-2000 125 47 of of ADP iajs-2000 125 48 x. x. PROPN iajs-2000 125 49 proof proof NOUN iajs-2000 125 50 " " PUNCT iajs-2000 125 51 ⟹ ⟹ NUM iajs-2000 125 52 " " PUNCT iajs-2000 125 53 follows follow VERB iajs-2000 125 54 from from ADP iajs-2000 125 55 proposition proposition NOUN iajs-2000 125 56 ( ( PUNCT iajs-2000 125 57 3.4 3.4 NUM iajs-2000 125 58 ) ) PUNCT iajs-2000 125 59 . . PUNCT iajs-2000 126 1 " " PUNCT iajs-2000 126 2 ⟸ ⟸ ADJ iajs-2000 126 3 suppose suppose VERB iajs-2000 126 4 that that SCONJ iajs-2000 126 5 l l NOUN iajs-2000 126 6 is be AUX iajs-2000 126 7 a a DET iajs-2000 126 8 weakly weakly ADJ iajs-2000 126 9 semi semi ADJ iajs-2000 126 10 - - ADJ iajs-2000 126 11 prime prime ADJ iajs-2000 126 12 submodule submodule NOUN iajs-2000 126 13 of of ADP iajs-2000 126 14 x x NOUN iajs-2000 126 15 " " PUNCT iajs-2000 126 16 , , PUNCT iajs-2000 126 17 and and CCONJ iajs-2000 126 18 0 0 NUM iajs-2000 126 19 πœ™ πœ™ DET iajs-2000 126 20 π‘₯ π‘₯ X iajs-2000 126 21 ∈ ∈ PROPN iajs-2000 126 22 𝐿 𝐿 PROPN iajs-2000 126 23 , , PUNCT iajs-2000 126 24 where where SCONJ iajs-2000 126 25 π‘₯ π‘₯ DET iajs-2000 126 26 ∈ ∈ PROPN iajs-2000 126 27 𝑋 𝑋 NOUN iajs-2000 126 28 and and CCONJ iajs-2000 126 29 πœ™ πœ™ PRON iajs-2000 126 30 ∈ ∈ PROPN iajs-2000 126 31 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 126 32 𝑋 𝑋 PROPN iajs-2000 126 33 . . PUNCT iajs-2000 127 1 since since SCONJ iajs-2000 127 2 x x PRON iajs-2000 127 3 is be AUX iajs-2000 127 4 a a DET iajs-2000 127 5 scalar scalar ADJ iajs-2000 127 6 module module NOUN iajs-2000 127 7 , , PUNCT iajs-2000 127 8 " " PUNCT iajs-2000 127 9 then then ADV iajs-2000 127 10 there there PRON iajs-2000 127 11 exists exist VERB iajs-2000 127 12 π‘Ÿ π‘Ÿ X iajs-2000 127 13 ∈ ∈ PROPN iajs-2000 127 14 𝑅 𝑅 PROPN iajs-2000 127 15 such such ADJ iajs-2000 127 16 that that SCONJ iajs-2000 127 17 πœ™ πœ™ PROPN iajs-2000 127 18 π‘₯ π‘₯ VERB iajs-2000 127 19 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PRON iajs-2000 127 20 for for ADP iajs-2000 127 21 each each DET iajs-2000 127 22 π‘₯ π‘₯ DET iajs-2000 127 23 ∈ ∈ PROPN iajs-2000 127 24 𝑋 𝑋 PROPN iajs-2000 127 25 " " PUNCT iajs-2000 127 26 . . PUNCT iajs-2000 128 1 " " PUNCT iajs-2000 128 2 now now ADV iajs-2000 128 3 " " PUNCT iajs-2000 128 4 , , PUNCT iajs-2000 128 5 0 0 NUM iajs-2000 128 6 πœ™ πœ™ DET iajs-2000 128 7 π‘₯ π‘₯ VERB iajs-2000 128 8 πœ™ πœ™ NOUN iajs-2000 128 9 πœ™ πœ™ VERB iajs-2000 128 10 π‘₯ π‘₯ INTJ iajs-2000 128 11 πœ™ πœ™ NOUN iajs-2000 128 12 π‘Ÿπ‘₯ π‘Ÿπ‘₯ ADP iajs-2000 128 13 π‘Ÿ π‘Ÿ NOUN iajs-2000 128 14 π‘₯ π‘₯ X iajs-2000 128 15 ∈ ∈ PROPN iajs-2000 128 16 𝐿. 𝐿. NOUN iajs-2000 128 17 but but CCONJ iajs-2000 128 18 l l NOUN iajs-2000 128 19 " " PUNCT iajs-2000 128 20 is be AUX iajs-2000 128 21 a a DET iajs-2000 128 22 weakly weakly ADJ iajs-2000 128 23 semi semi ADJ iajs-2000 128 24 - - ADJ iajs-2000 128 25 prime prime ADJ iajs-2000 128 26 submodule submodule NOUN iajs-2000 128 27 of of ADP iajs-2000 128 28 x x X iajs-2000 128 29 " " PUNCT iajs-2000 128 30 , , PUNCT iajs-2000 128 31 implies imply VERB iajs-2000 128 32 that that SCONJ iajs-2000 128 33 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2000 128 34 ∈ ∈ PROPN iajs-2000 128 35 𝐿. 𝐿. VERB iajs-2000 128 36 " " PUNCT iajs-2000 128 37 hence hence ADV iajs-2000 128 38 πœ™ πœ™ NOUN iajs-2000 128 39 π‘₯ π‘₯ X iajs-2000 128 40 ∈ ∈ PROPN iajs-2000 128 41 𝐿 𝐿 PROPN iajs-2000 128 42 " " PUNCT iajs-2000 128 43 . . PUNCT iajs-2000 129 1 " " PUNCT iajs-2000 129 2 thus thus ADV iajs-2000 129 3 l l NOUN iajs-2000 129 4 is be AUX iajs-2000 129 5 a a DET iajs-2000 129 6 we we PRON iajs-2000 129 7 - - PUNCT iajs-2000 129 8 semi semi ADJ iajs-2000 129 9 - - ADJ iajs-2000 129 10 prime prime ADJ iajs-2000 129 11 submodule submodule NOUN iajs-2000 129 12 of of ADP iajs-2000 129 13 x x NOUN iajs-2000 129 14 " " PUNCT iajs-2000 129 15 . . PUNCT iajs-2000 130 1 " " PUNCT iajs-2000 130 2 " " PUNCT iajs-2000 130 3 the the DET iajs-2000 130 4 following follow VERB iajs-2000 130 5 propositions proposition NOUN iajs-2000 130 6 are be AUX iajs-2000 130 7 characterizations characterization NOUN iajs-2000 130 8 of of ADP iajs-2000 130 9 we we PRON iajs-2000 130 10 - - PUNCT iajs-2000 130 11 semi semi ADJ iajs-2000 130 12 - - ADJ iajs-2000 130 13 prime prime ADJ iajs-2000 130 14 submodules submodule NOUN iajs-2000 130 15 " " PUNCT iajs-2000 130 16 . . PUNCT iajs-2000 131 1 proposition proposition NOUN iajs-2000 131 2 ( ( PUNCT iajs-2000 131 3 20 20 NUM iajs-2000 131 4 ) ) PUNCT iajs-2000 131 5 " " PUNCT iajs-2000 131 6 " " PUNCT iajs-2000 131 7 let let VERB iajs-2000 131 8 x x PRON iajs-2000 131 9 be be AUX iajs-2000 131 10 an an DET iajs-2000 131 11 r r NOUN iajs-2000 131 12 - - PUNCT iajs-2000 131 13 module module NOUN iajs-2000 131 14 , , PUNCT iajs-2000 131 15 and and CCONJ iajs-2000 131 16 " " PUNCT iajs-2000 131 17 l l NOUN iajs-2000 131 18 is be AUX iajs-2000 131 19 " " PUNCT iajs-2000 131 20 a a DET iajs-2000 131 21 proper proper ADJ iajs-2000 131 22 submodule submodule NOUN iajs-2000 131 23 of of ADP iajs-2000 131 24 x x NOUN iajs-2000 131 25 " " PUNCT iajs-2000 131 26 . . PUNCT iajs-2000 132 1 then then ADV iajs-2000 132 2 l l PROPN iajs-2000 132 3 is be AUX iajs-2000 132 4 a a DET iajs-2000 132 5 we we PRON iajs-2000 132 6 - - PUNCT iajs-2000 132 7 semi semi ADJ iajs-2000 132 8 - - ADJ iajs-2000 132 9 prime prime ADJ iajs-2000 132 10 submodule submodule NOUN iajs-2000 132 11 " " PUNCT iajs-2000 132 12 if if SCONJ iajs-2000 132 13 and and CCONJ iajs-2000 132 14 only only ADV iajs-2000 132 15 " " PUNCT iajs-2000 132 16 if if SCONJ iajs-2000 132 17 0 0 NUM iajs-2000 132 18 πœ™ πœ™ PROPN iajs-2000 132 19 𝐾 𝐾 PROPN iajs-2000 132 20 𝐿 𝐿 PROPN iajs-2000 132 21 , , PUNCT iajs-2000 132 22 where where SCONJ iajs-2000 132 23 k k PROPN iajs-2000 132 24 is be AUX iajs-2000 132 25 a a DET iajs-2000 132 26 submodule submodule NOUN iajs-2000 132 27 of of ADP iajs-2000 132 28 x x PUNCT iajs-2000 132 29 and and CCONJ iajs-2000 132 30 πœ™ πœ™ PROPN iajs-2000 132 31 ∈ ∈ PROPN iajs-2000 132 32 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 132 33 𝑋 𝑋 PROPN iajs-2000 132 34 , , PUNCT iajs-2000 132 35 implies imply VERB iajs-2000 132 36 that that SCONJ iajs-2000 132 37 πœ™ πœ™ PROPN iajs-2000 132 38 𝐾 𝐾 PROPN iajs-2000 132 39 𝐿 𝐿 PROPN iajs-2000 132 40 " " PUNCT iajs-2000 132 41 . . PUNCT iajs-2000 133 1 proof proof NOUN iajs-2000 133 2 " " PUNCT iajs-2000 133 3 ⟹ ⟹ NUM iajs-2000 133 4 assume assume VERB iajs-2000 133 5 that that SCONJ iajs-2000 133 6 0 0 NUM iajs-2000 133 7 πœ™ πœ™ PROPN iajs-2000 133 8 𝐾 𝐾 PROPN iajs-2000 133 9 𝐿 𝐿 PROPN iajs-2000 133 10 " " PUNCT iajs-2000 133 11 , , PUNCT iajs-2000 133 12 where where SCONJ iajs-2000 133 13 " " PUNCT iajs-2000 133 14 k k X iajs-2000 133 15 is be AUX iajs-2000 133 16 a a DET iajs-2000 133 17 submodule submodule NOUN iajs-2000 133 18 of of ADP iajs-2000 133 19 " " PUNCT iajs-2000 133 20 x x NOUN iajs-2000 133 21 , , PUNCT iajs-2000 133 22 πœ™ πœ™ PROPN iajs-2000 133 23 ∈ ∈ PROPN iajs-2000 133 24 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 133 25 𝑋 𝑋 PROPN iajs-2000 133 26 , , PUNCT iajs-2000 133 27 implies imply VERB iajs-2000 133 28 " " PUNCT iajs-2000 133 29 that that PRON iajs-2000 133 30 0 0 NUM iajs-2000 133 31 πœ™ πœ™ PRON iajs-2000 133 32 π‘₯ π‘₯ X iajs-2000 133 33 ∈ ∈ PROPN iajs-2000 133 34 𝐿 𝐿 PROPN iajs-2000 133 35 for for ADP iajs-2000 133 36 all all DET iajs-2000 133 37 π‘₯ π‘₯ DET iajs-2000 133 38 ∈ ∈ PROPN iajs-2000 133 39 𝐾 𝐾 PROPN iajs-2000 133 40 𝑋 𝑋 PROPN iajs-2000 133 41 " " PUNCT iajs-2000 133 42 . . PUNCT iajs-2000 134 1 " " PUNCT iajs-2000 134 2 since since SCONJ iajs-2000 134 3 l l NOUN iajs-2000 134 4 is be AUX iajs-2000 134 5 a a DET iajs-2000 134 6 we we PRON iajs-2000 134 7 - - PUNCT iajs-2000 134 8 semi semi ADJ iajs-2000 134 9 - - ADJ iajs-2000 134 10 prime prime ADJ iajs-2000 134 11 submodule submodule NOUN iajs-2000 134 12 of of ADP iajs-2000 134 13 x x PROPN iajs-2000 134 14 , , PUNCT iajs-2000 134 15 then then ADV iajs-2000 134 16 πœ™ πœ™ X iajs-2000 134 17 π‘₯ π‘₯ X iajs-2000 134 18 ∈ ∈ PROPN iajs-2000 134 19 𝐿 𝐿 PROPN iajs-2000 134 20 for for ADP iajs-2000 134 21 all all DET iajs-2000 134 22 π‘₯ π‘₯ DET iajs-2000 134 23 ∈ ∈ NOUN iajs-2000 134 24 𝑋. 𝑋. PROPN iajs-2000 134 25 thus thus ADV iajs-2000 134 26 πœ™ πœ™ PROPN iajs-2000 134 27 𝐾 𝐾 PROPN iajs-2000 134 28 𝐿. 𝐿. PROPN iajs-2000 134 29 " " PUNCT iajs-2000 134 30 ⟸ ⟸ ADJ iajs-2000 134 31 suppose suppose VERB iajs-2000 134 32 that that SCONJ iajs-2000 134 33 0 0 NUM iajs-2000 134 34 πœ™ πœ™ PRON iajs-2000 134 35 π‘₯ π‘₯ X iajs-2000 134 36 ∈ ∈ PROPN iajs-2000 134 37 𝐿 𝐿 PROPN iajs-2000 134 38 , , PUNCT iajs-2000 134 39 where where SCONJ iajs-2000 134 40 π‘₯ π‘₯ DET iajs-2000 134 41 ∈ ∈ PROPN iajs-2000 134 42 𝑋 𝑋 PROPN iajs-2000 134 43 , , PUNCT iajs-2000 134 44 and and CCONJ iajs-2000 134 45 πœ™ πœ™ PRON iajs-2000 134 46 ∈ ∈ PROPN iajs-2000 134 47 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 134 48 𝑋 𝑋 PROPN iajs-2000 134 49 , , PUNCT iajs-2000 134 50 then then ADV iajs-2000 134 51 by by ADP iajs-2000 134 52 hypothesis hypothesis NOUN iajs-2000 134 53 , , PUNCT iajs-2000 134 54 we we PRON iajs-2000 134 55 have have VERB iajs-2000 134 56 𝐾 𝐾 PROPN iajs-2000 134 57 π‘₯ π‘₯ PROPN iajs-2000 134 58 is be AUX iajs-2000 134 59 a a DET iajs-2000 134 60 submodule submodule NOUN iajs-2000 134 61 of of ADP iajs-2000 134 62 x x PROPN iajs-2000 134 63 , , PUNCT iajs-2000 134 64 and and CCONJ iajs-2000 134 65 0 0 NUM iajs-2000 134 66 πœ™ πœ™ NOUN iajs-2000 134 67 𝐾 𝐾 PROPN iajs-2000 134 68 ∈ ∈ PROPN iajs-2000 134 69 𝐿 𝐿 PROPN iajs-2000 134 70 , , PUNCT iajs-2000 134 71 implies imply VERB iajs-2000 134 72 that that SCONJ iajs-2000 134 73 πœ™ πœ™ PROPN iajs-2000 134 74 𝐾 𝐾 PROPN iajs-2000 134 75 𝐿 𝐿 PROPN iajs-2000 134 76 , , PUNCT iajs-2000 134 77 " " PUNCT iajs-2000 134 78 it it PRON iajs-2000 134 79 follows follow VERB iajs-2000 134 80 that that SCONJ iajs-2000 134 81 πœ™ πœ™ PROPN iajs-2000 134 82 π‘₯ π‘₯ X iajs-2000 134 83 ∈ ∈ PROPN iajs-2000 134 84 𝐿. 𝐿. VERB iajs-2000 134 85 " " PUNCT iajs-2000 134 86 hence hence ADV iajs-2000 134 87 l l NOUN iajs-2000 134 88 is be AUX iajs-2000 134 89 a a DET iajs-2000 134 90 we we PRON iajs-2000 134 91 - - PUNCT iajs-2000 134 92 semi semi ADJ iajs-2000 134 93 - - ADJ iajs-2000 134 94 prime prime ADJ iajs-2000 134 95 submodule submodule NOUN iajs-2000 134 96 of of ADP iajs-2000 134 97 x. x. NOUN iajs-2000 134 98 proposition proposition NOUN iajs-2000 134 99 ( ( PUNCT iajs-2000 134 100 21 21 NUM iajs-2000 134 101 ) ) PUNCT iajs-2000 134 102 " " PUNCT iajs-2000 134 103 " " PUNCT iajs-2000 134 104 let let VERB iajs-2000 134 105 x x PRON iajs-2000 134 106 be be AUX iajs-2000 134 107 an an DET iajs-2000 134 108 r r NOUN iajs-2000 134 109 - - PUNCT iajs-2000 134 110 module module NOUN iajs-2000 134 111 , , PUNCT iajs-2000 134 112 and and CCONJ iajs-2000 134 113 " " PUNCT iajs-2000 134 114 l l NOUN iajs-2000 134 115 is be AUX iajs-2000 134 116 " " PUNCT iajs-2000 134 117 a a DET iajs-2000 134 118 proper proper ADJ iajs-2000 134 119 submodule submodule NOUN iajs-2000 134 120 of of ADP iajs-2000 134 121 x x NOUN iajs-2000 134 122 " " PUNCT iajs-2000 134 123 . . PUNCT iajs-2000 135 1 " " PUNCT iajs-2000 135 2 then then ADV iajs-2000 135 3 l l NOUN iajs-2000 135 4 is be AUX iajs-2000 135 5 a a DET iajs-2000 135 6 we we PRON iajs-2000 135 7 - - PUNCT iajs-2000 135 8 semi semi ADJ iajs-2000 135 9 - - ADJ iajs-2000 135 10 prime prime ADJ iajs-2000 135 11 submodule submodule NOUN iajs-2000 135 12 of of ADP iajs-2000 135 13 x x PRON iajs-2000 135 14 , , PUNCT iajs-2000 135 15 if if SCONJ iajs-2000 135 16 and and CCONJ iajs-2000 135 17 only only ADV iajs-2000 135 18 if if SCONJ iajs-2000 135 19 " " PUNCT iajs-2000 135 20 , , PUNCT iajs-2000 135 21 " " PUNCT iajs-2000 135 22 wherever wherever SCONJ iajs-2000 135 23 0 0 NUM iajs-2000 135 24 πœ™ πœ™ PRON iajs-2000 135 25 π‘₯ π‘₯ X iajs-2000 135 26 ∈ ∈ PROPN iajs-2000 135 27 𝐿 𝐿 PROPN iajs-2000 135 28 , , PUNCT iajs-2000 135 29 π‘₯ π‘₯ PRON iajs-2000 135 30 ∈ ∈ PROPN iajs-2000 135 31 𝑋 𝑋 PROPN iajs-2000 135 32 , , PUNCT iajs-2000 135 33 πœ™ πœ™ PRON iajs-2000 135 34 ∈ ∈ PROPN iajs-2000 135 35 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 135 36 𝑋 𝑋 PROPN iajs-2000 135 37 , , PUNCT iajs-2000 135 38 and and CCONJ iajs-2000 135 39 for for ADP iajs-2000 135 40 𝑛 𝑛 DET iajs-2000 135 41 2 2 NUM iajs-2000 135 42 , , PUNCT iajs-2000 135 43 implies imply VERB iajs-2000 135 44 that that SCONJ iajs-2000 135 45 πœ™ πœ™ PROPN iajs-2000 135 46 π‘₯ π‘₯ X iajs-2000 135 47 ∈ ∈ PROPN iajs-2000 135 48 𝐿 𝐿 PROPN iajs-2000 135 49 " " PUNCT iajs-2000 135 50 . . PUNCT iajs-2000 136 1 proof proof NOUN iajs-2000 136 2 " " PUNCT iajs-2000 136 3 ⟹ ⟹ PROPN iajs-2000 136 4 follows follow VERB iajs-2000 136 5 by by ADP iajs-2000 136 6 inducation inducation NOUN iajs-2000 136 7 on on ADP iajs-2000 136 8 𝑛 𝑛 DET iajs-2000 136 9 ∈ ∈ PROPN iajs-2000 136 10 𝑍 𝑍 PROPN iajs-2000 136 11 " " PUNCT iajs-2000 136 12 . . PUNCT iajs-2000 136 13 " " PUNCT iajs-2000 137 1 ⟸ ⟸ ADV iajs-2000 137 2 direct direct ADJ iajs-2000 137 3 from from ADP iajs-2000 137 4 definition definition NOUN iajs-2000 137 5 of of ADP iajs-2000 137 6 we we PRON iajs-2000 137 7 - - PUNCT iajs-2000 137 8 semi semi ADJ iajs-2000 137 9 - - ADJ iajs-2000 137 10 prime prime ADJ iajs-2000 137 11 submodule submodule NOUN iajs-2000 137 12 " " PUNCT iajs-2000 137 13 . . PUNCT iajs-2000 138 1 " " PUNCT iajs-2000 138 2 in in ADP iajs-2000 138 3 the the DET iajs-2000 138 4 class class NOUN iajs-2000 138 5 of of ADP iajs-2000 138 6 scalar scalar ADJ iajs-2000 138 7 module module NOUN iajs-2000 138 8 , , PUNCT iajs-2000 138 9 we we PRON iajs-2000 138 10 get get VERB iajs-2000 138 11 the the DET iajs-2000 138 12 following follow VERB iajs-2000 138 13 characterizations characterization NOUN iajs-2000 138 14 of of ADP iajs-2000 138 15 we we PRON iajs-2000 138 16 - - PUNCT iajs-2000 138 17 semi semi ADJ iajs-2000 138 18 - - ADJ iajs-2000 138 19 prime prime ADJ iajs-2000 138 20 submodules submodule NOUN iajs-2000 138 21 . . PUNCT iajs-2000 139 1 proposition proposition NOUN iajs-2000 139 2 ( ( PUNCT iajs-2000 139 3 22 22 NUM iajs-2000 139 4 ) ) PUNCT iajs-2000 139 5 " " PUNCT iajs-2000 139 6 " " PUNCT iajs-2000 139 7 let let VERB iajs-2000 139 8 x x PRON iajs-2000 139 9 be be AUX iajs-2000 139 10 a a DET iajs-2000 139 11 scalar scalar ADJ iajs-2000 139 12 r r NOUN iajs-2000 139 13 - - PUNCT iajs-2000 139 14 module module NOUN iajs-2000 139 15 , , PUNCT iajs-2000 139 16 and and CCONJ iajs-2000 139 17 l l NOUN iajs-2000 139 18 be be AUX iajs-2000 139 19 a a DET iajs-2000 139 20 proper proper ADJ iajs-2000 139 21 submodule submodule NOUN iajs-2000 139 22 of of ADP iajs-2000 139 23 x. x. PROPN iajs-2000 139 24 then then ADV iajs-2000 139 25 the the DET iajs-2000 139 26 following follow VERB iajs-2000 139 27 statements statement NOUN iajs-2000 139 28 are be AUX iajs-2000 139 29 equivalent equivalent ADJ iajs-2000 139 30 : : PUNCT iajs-2000 139 31 " " PUNCT iajs-2000 139 32 1 1 X iajs-2000 139 33 . . PUNCT iajs-2000 140 1 " " PUNCT iajs-2000 140 2 l l NOUN iajs-2000 140 3 is be AUX iajs-2000 140 4 a a DET iajs-2000 140 5 we we PRON iajs-2000 140 6 - - PUNCT iajs-2000 140 7 semi semi ADJ iajs-2000 140 8 - - ADJ iajs-2000 140 9 prime prime ADJ iajs-2000 140 10 submodule submodule NOUN iajs-2000 140 11 of of ADP iajs-2000 140 12 x x NOUN iajs-2000 140 13 " " PUNCT iajs-2000 140 14 . . PUNCT iajs-2000 141 1 2 2 X iajs-2000 141 2 . . PUNCT iajs-2000 141 3 " " PUNCT iajs-2000 141 4 𝐿 𝐿 NOUN iajs-2000 141 5 : : PUNCT iajs-2000 141 6 π‘Ÿ π‘Ÿ NOUN iajs-2000 141 7 0 0 NUM iajs-2000 141 8 : : PUNCT iajs-2000 141 9 π‘Ÿ π‘Ÿ NOUN iajs-2000 141 10 βˆͺ βˆͺ ADP iajs-2000 141 11 𝐿 𝐿 NOUN iajs-2000 141 12 : : PUNCT iajs-2000 141 13 π‘Ÿ π‘Ÿ NOUN iajs-2000 141 14 for for ADP iajs-2000 141 15 non non ADJ iajs-2000 141 16 - - ADJ iajs-2000 141 17 zero zero ADJ iajs-2000 141 18 r r NOUN iajs-2000 141 19 in in ADP iajs-2000 141 20 r r NOUN iajs-2000 141 21 " " PUNCT iajs-2000 141 22 . . PUNCT iajs-2000 142 1 3 3 X iajs-2000 142 2 . . X iajs-2000 142 3 𝐿 𝐿 NOUN iajs-2000 142 4 : : PUNCT iajs-2000 142 5 π‘Ÿ π‘Ÿ NOUN iajs-2000 142 6 0 0 NUM iajs-2000 142 7 : : PUNCT iajs-2000 142 8 π‘Ÿ π‘Ÿ NOUN iajs-2000 142 9 or or CCONJ iajs-2000 142 10 0 0 NUM iajs-2000 142 11 : : PUNCT iajs-2000 142 12 π‘Ÿ π‘Ÿ X iajs-2000 142 13 𝐿 𝐿 NOUN iajs-2000 142 14 : : PUNCT iajs-2000 142 15 π‘Ÿ π‘Ÿ NOUN iajs-2000 142 16 for for ADP iajs-2000 142 17 non non ADJ iajs-2000 142 18 - - ADJ iajs-2000 142 19 zero zero ADJ iajs-2000 142 20 r r NOUN iajs-2000 142 21 in in ADP iajs-2000 142 22 r r NOUN iajs-2000 142 23 " " PUNCT iajs-2000 142 24 . . PUNCT iajs-2000 143 1 mathematics mathematic NOUN iajs-2000 143 2 | | ADV iajs-2000 143 3 115 115 NUM iajs-2000 143 4 ibn ibn PROPN iajs-2000 143 5 al al PROPN iajs-2000 143 6 - - PUNCT iajs-2000 143 7 haitham haitham PROPN iajs-2000 143 8 jour jour X iajs-2000 143 9 . . PROPN iajs-2000 143 10 for for ADP iajs-2000 143 11 pure pure ADJ iajs-2000 143 12 & & CCONJ iajs-2000 143 13 appl appl PROPN iajs-2000 143 14 . . PUNCT iajs-2000 144 1 sci sci PROPN iajs-2000 144 2 . . PROPN iajs-2000 144 3 ihjpas ihjpa VERB iajs-2000 144 4 https://doi.org/10.30526/31.3.2000 https://doi.org/10.30526/31.3.2000 NUM iajs-2000 144 5 vol vol NOUN iajs-2000 144 6 . . PROPN iajs-2000 144 7 31 31 NUM iajs-2000 144 8 ( ( PUNCT iajs-2000 144 9 3 3 NUM iajs-2000 144 10 ) ) SYM iajs-2000 144 11 2018 2018 NUM iajs-2000 144 12 proof proof NOUN iajs-2000 144 13 " " PUNCT iajs-2000 144 14 1 1 NUM iajs-2000 144 15 ⟹ ⟹ NUM iajs-2000 144 16 2 2 NUM iajs-2000 144 17 since since SCONJ iajs-2000 144 18 l l NOUN iajs-2000 144 19 is be AUX iajs-2000 144 20 a a DET iajs-2000 144 21 we we PRON iajs-2000 144 22 - - PUNCT iajs-2000 144 23 semi semi ADJ iajs-2000 144 24 - - ADJ iajs-2000 144 25 prime prime ADJ iajs-2000 144 26 submodule submodule NOUN iajs-2000 144 27 of of ADP iajs-2000 144 28 x x PRON iajs-2000 144 29 , , PUNCT iajs-2000 144 30 " " PUNCT iajs-2000 144 31 then then ADV iajs-2000 144 32 by by ADP iajs-2000 144 33 proposition proposition NOUN iajs-2000 144 34 ( ( PUNCT iajs-2000 144 35 3.4 3.4 NUM iajs-2000 144 36 ) ) PUNCT iajs-2000 144 37 l l NOUN iajs-2000 144 38 is be AUX iajs-2000 144 39 a a DET iajs-2000 144 40 weakly weakly ADJ iajs-2000 144 41 semi semi ADJ iajs-2000 144 42 - - ADJ iajs-2000 144 43 prime prime ADJ iajs-2000 144 44 submodule submodule NOUN iajs-2000 144 45 of of ADP iajs-2000 144 46 x. x. NOUN iajs-2000 144 47 now now ADV iajs-2000 144 48 , , PUNCT iajs-2000 144 49 let let VERB iajs-2000 144 50 π‘₯ π‘₯ PRON iajs-2000 144 51 ∈ ∈ PROPN iajs-2000 144 52 𝐿 𝐿 PROPN iajs-2000 144 53 : : PUNCT iajs-2000 144 54 π‘Ÿ π‘Ÿ NOUN iajs-2000 144 55 , , PUNCT iajs-2000 144 56 implies imply VERB iajs-2000 144 57 that that SCONJ iajs-2000 144 58 π‘Ÿ π‘Ÿ X iajs-2000 144 59 π‘₯ π‘₯ PRON iajs-2000 144 60 ∈ ∈ PROPN iajs-2000 144 61 𝐿 𝐿 PROPN iajs-2000 144 62 , , PUNCT iajs-2000 144 63 either either ADV iajs-2000 144 64 0 0 NUM iajs-2000 144 65 π‘Ÿ π‘Ÿ NOUN iajs-2000 144 66 π‘₯ π‘₯ DET iajs-2000 144 67 ∈ ∈ PROPN iajs-2000 144 68 𝐿 𝐿 PROPN iajs-2000 144 69 or or CCONJ iajs-2000 144 70 π‘Ÿ π‘Ÿ PRON iajs-2000 144 71 π‘₯ π‘₯ PROPN iajs-2000 145 1 0 0 X iajs-2000 145 2 . . PUNCT iajs-2000 146 1 if if SCONJ iajs-2000 146 2 0 0 NUM iajs-2000 146 3 π‘Ÿ π‘Ÿ NOUN iajs-2000 146 4 π‘₯ π‘₯ PRON iajs-2000 146 5 ∈ ∈ PROPN iajs-2000 146 6 𝐿 𝐿 PROPN iajs-2000 146 7 , , PUNCT iajs-2000 146 8 implies imply VERB iajs-2000 146 9 that that SCONJ iajs-2000 146 10 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2000 146 11 ∈ ∈ PROPN iajs-2000 146 12 𝐿 𝐿 PROPN iajs-2000 146 13 , , PUNCT iajs-2000 146 14 hence hence ADV iajs-2000 146 15 π‘₯ π‘₯ PRON iajs-2000 146 16 ∈ ∈ PROPN iajs-2000 146 17 𝐿 𝐿 PROPN iajs-2000 146 18 : : PUNCT iajs-2000 146 19 π‘Ÿ π‘Ÿ NOUN iajs-2000 146 20 . . PUNCT iajs-2000 147 1 if if SCONJ iajs-2000 147 2 π‘Ÿ π‘Ÿ PRON iajs-2000 147 3 π‘₯ π‘₯ X iajs-2000 147 4 0 0 NUM iajs-2000 147 5 , , PUNCT iajs-2000 147 6 implies imply VERB iajs-2000 147 7 that that SCONJ iajs-2000 147 8 π‘₯ π‘₯ PROPN iajs-2000 147 9 ∈ ∈ NOUN iajs-2000 147 10 0 0 PUNCT iajs-2000 147 11 : : PUNCT iajs-2000 147 12 π‘Ÿ π‘Ÿ NOUN iajs-2000 147 13 , , PUNCT iajs-2000 147 14 hence hence ADV iajs-2000 147 15 , , PUNCT iajs-2000 147 16 we we PRON iajs-2000 147 17 get get VERB iajs-2000 147 18 𝐿 𝐿 PROPN iajs-2000 147 19 : : PUNCT iajs-2000 147 20 π‘Ÿ π‘Ÿ X iajs-2000 147 21 𝐿 𝐿 NOUN iajs-2000 147 22 : : PUNCT iajs-2000 147 23 π‘Ÿ π‘Ÿ NOUN iajs-2000 147 24 βˆͺ βˆͺ ADJ iajs-2000 147 25 0 0 NUM iajs-2000 147 26 : : PUNCT iajs-2000 147 27 π‘Ÿ π‘Ÿ X iajs-2000 147 28 . . PUNCT iajs-2000 148 1 clearly clearly ADV iajs-2000 148 2 we we PRON iajs-2000 148 3 have have VERB iajs-2000 148 4 by by ADP iajs-2000 148 5 lemma lemma PROPN iajs-2000 148 6 ( ( PUNCT iajs-2000 148 7 2.7 2.7 NUM iajs-2000 148 8 ) ) PUNCT iajs-2000 148 9 , , PUNCT iajs-2000 148 10 𝐿 𝐿 PROPN iajs-2000 148 11 : : PUNCT iajs-2000 148 12 π‘Ÿ π‘Ÿ X iajs-2000 148 13 𝐿 𝐿 NOUN iajs-2000 148 14 : : PUNCT iajs-2000 148 15 π‘Ÿ π‘Ÿ NOUN iajs-2000 148 16 , , PUNCT iajs-2000 148 17 and and CCONJ iajs-2000 148 18 0 0 NUM iajs-2000 148 19 : : PUNCT iajs-2000 148 20 π‘Ÿ π‘Ÿ X iajs-2000 148 21 𝐿 𝐿 NOUN iajs-2000 148 22 : : PUNCT iajs-2000 148 23 π‘Ÿ π‘Ÿ NOUN iajs-2000 148 24 , , PUNCT iajs-2000 148 25 hence hence ADV iajs-2000 148 26 𝐿 𝐿 PROPN iajs-2000 148 27 : : PUNCT iajs-2000 148 28 π‘Ÿ π‘Ÿ NOUN iajs-2000 148 29 βˆͺ βˆͺ ADJ iajs-2000 148 30 0 0 NUM iajs-2000 148 31 : : PUNCT iajs-2000 148 32 π‘Ÿ π‘Ÿ X iajs-2000 148 33 𝐿 𝐿 NOUN iajs-2000 148 34 : : PUNCT iajs-2000 148 35 π‘Ÿ π‘Ÿ NOUN iajs-2000 148 36 . . PUNCT iajs-2000 149 1 thus thus ADV iajs-2000 149 2 the the DET iajs-2000 149 3 equality equality NOUN iajs-2000 149 4 holds hold VERB iajs-2000 149 5 " " PUNCT iajs-2000 149 6 . . PUNCT iajs-2000 149 7 " " PUNCT iajs-2000 150 1 2 2 NUM iajs-2000 150 2 ⟹ ⟹ NUM iajs-2000 150 3 3 3 NUM iajs-2000 150 4 direct direct ADJ iajs-2000 150 5 " " PUNCT iajs-2000 150 6 . . PUNCT iajs-2000 150 7 " " PUNCT iajs-2000 151 1 3 3 NUM iajs-2000 151 2 ⟹ ⟹ NUM iajs-2000 151 3 1 1 NUM iajs-2000 151 4 to to PART iajs-2000 151 5 prove prove VERB iajs-2000 151 6 first first ADJ iajs-2000 151 7 l l NOUN iajs-2000 151 8 is be AUX iajs-2000 151 9 a a DET iajs-2000 151 10 weakly weakly ADJ iajs-2000 151 11 semi semi ADJ iajs-2000 151 12 - - ADJ iajs-2000 151 13 prime prime ADJ iajs-2000 151 14 submodule submodule NOUN iajs-2000 151 15 of of ADP iajs-2000 151 16 x x NOUN iajs-2000 151 17 " " PUNCT iajs-2000 151 18 . . PUNCT iajs-2000 152 1 " " PUNCT iajs-2000 152 2 suppose suppose VERB iajs-2000 152 3 that that SCONJ iajs-2000 152 4 " " PUNCT iajs-2000 152 5 0 0 NUM iajs-2000 152 6 π‘Ÿ π‘Ÿ NOUN iajs-2000 152 7 π‘₯ π‘₯ PRON iajs-2000 152 8 ∈ ∈ PROPN iajs-2000 152 9 𝐿 𝐿 PROPN iajs-2000 152 10 , , PUNCT iajs-2000 152 11 where where SCONJ iajs-2000 152 12 π‘₯ π‘₯ DET iajs-2000 152 13 ∈ ∈ PROPN iajs-2000 152 14 𝑋 𝑋 PROPN iajs-2000 152 15 , , PUNCT iajs-2000 152 16 π‘Ÿ π‘Ÿ X iajs-2000 152 17 ∈ ∈ PROPN iajs-2000 152 18 𝑅 𝑅 PROPN iajs-2000 152 19 , , PUNCT iajs-2000 152 20 implies imply VERB iajs-2000 152 21 that that SCONJ iajs-2000 152 22 π‘₯ π‘₯ PROPN iajs-2000 152 23 ∈ ∈ PROPN iajs-2000 152 24 𝐿 𝐿 PROPN iajs-2000 152 25 : : PUNCT iajs-2000 152 26 π‘Ÿ π‘Ÿ NOUN iajs-2000 152 27 and and CCONJ iajs-2000 152 28 π‘₯ π‘₯ X iajs-2000 152 29 βˆ‰ βˆ‰ PROPN iajs-2000 152 30 0 0 NUM iajs-2000 152 31 : : PUNCT iajs-2000 152 32 π‘Ÿ π‘Ÿ X iajs-2000 152 33 . . PUNCT iajs-2000 152 34 thus thus ADV iajs-2000 152 35 by by ADP iajs-2000 152 36 hypothesis hypothesis NOUN iajs-2000 152 37 , , PUNCT iajs-2000 152 38 we we PRON iajs-2000 152 39 get get VERB iajs-2000 152 40 π‘₯ π‘₯ DET iajs-2000 152 41 ∈ ∈ PROPN iajs-2000 152 42 𝐿 𝐿 PROPN iajs-2000 152 43 : : PUNCT iajs-2000 152 44 π‘Ÿ π‘Ÿ NOUN iajs-2000 152 45 , , PUNCT iajs-2000 152 46 implies imply VERB iajs-2000 152 47 that that SCONJ iajs-2000 152 48 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2000 152 49 ∈ ∈ PROPN iajs-2000 152 50 𝐿 𝐿 PROPN iajs-2000 152 51 , , PUNCT iajs-2000 152 52 hence hence ADV iajs-2000 152 53 l l NOUN iajs-2000 152 54 " " PUNCT iajs-2000 152 55 is be AUX iajs-2000 152 56 a a DET iajs-2000 152 57 weakly weakly ADJ iajs-2000 152 58 semi semi ADJ iajs-2000 152 59 - - ADJ iajs-2000 152 60 prime prime ADJ iajs-2000 152 61 submodule submodule NOUN iajs-2000 152 62 of of ADP iajs-2000 152 63 x x NOUN iajs-2000 152 64 " " PUNCT iajs-2000 152 65 . . PUNCT iajs-2000 153 1 " " PUNCT iajs-2000 153 2 thus thus ADV iajs-2000 153 3 by by ADP iajs-2000 153 4 proposition proposition NOUN iajs-2000 153 5 ( ( PUNCT iajs-2000 153 6 3.7 3.7 NUM iajs-2000 153 7 ) ) PUNCT iajs-2000 153 8 , , PUNCT iajs-2000 153 9 we we PRON iajs-2000 153 10 have have VERB iajs-2000 153 11 l l NOUN iajs-2000 153 12 is be AUX iajs-2000 153 13 a a DET iajs-2000 153 14 we we PRON iajs-2000 153 15 - - PUNCT iajs-2000 153 16 semi semi ADJ iajs-2000 153 17 - - ADJ iajs-2000 153 18 prime prime ADJ iajs-2000 153 19 submodule submodule NOUN iajs-2000 153 20 of of ADP iajs-2000 153 21 x x NOUN iajs-2000 153 22 " " PUNCT iajs-2000 153 23 . . PUNCT iajs-2000 154 1 recall recall VERB iajs-2000 154 2 that that SCONJ iajs-2000 154 3 " " PUNCT iajs-2000 154 4 an an DET iajs-2000 154 5 element element NOUN iajs-2000 154 6 x x PUNCT iajs-2000 154 7 in in ADP iajs-2000 154 8 r r NOUN iajs-2000 154 9 - - PUNCT iajs-2000 154 10 module module NOUN iajs-2000 154 11 x x PRON iajs-2000 154 12 is be AUX iajs-2000 154 13 called call VERB iajs-2000 154 14 " " PUNCT iajs-2000 154 15 torsion torsion NOUN iajs-2000 154 16 if if SCONJ iajs-2000 154 17 0 0 NUM iajs-2000 154 18 π‘Žπ‘›π‘› π‘Žπ‘›π‘› VERB iajs-2000 154 19 π‘₯ π‘₯ NOUN iajs-2000 154 20 π‘Ÿ π‘Ÿ DET iajs-2000 154 21 ∈ ∈ PROPN iajs-2000 154 22 𝑅 𝑅 PROPN iajs-2000 154 23 ∢ ∢ NOUN iajs-2000 154 24 π‘Ÿπ‘₯ π‘Ÿπ‘₯ ADP iajs-2000 154 25 0 0 NUM iajs-2000 154 26 . . PUNCT iajs-2000 155 1 the the DET iajs-2000 155 2 set set NOUN iajs-2000 155 3 of of ADP iajs-2000 155 4 all all DET iajs-2000 155 5 torsion torsion NOUN iajs-2000 155 6 elements element NOUN iajs-2000 155 7 denoted denote VERB iajs-2000 155 8 by by ADP iajs-2000 155 9 t(x t(x PROPN iajs-2000 155 10 ) ) PUNCT iajs-2000 155 11 , , PUNCT iajs-2000 155 12 which which PRON iajs-2000 155 13 is be AUX iajs-2000 155 14 a a DET iajs-2000 155 15 submodule submodule NOUN iajs-2000 155 16 of of ADP iajs-2000 155 17 x. x. NOUN iajs-2000 155 18 if if SCONJ iajs-2000 155 19 t(x)=(0 t(x)=(0 NUM iajs-2000 155 20 ) ) PUNCT iajs-2000 155 21 , , PUNCT iajs-2000 155 22 then then ADV iajs-2000 155 23 x x PUNCT iajs-2000 155 24 is be AUX iajs-2000 155 25 called call VERB iajs-2000 155 26 torsion torsion NOUN iajs-2000 155 27 free free ADJ iajs-2000 155 28 [ [ X iajs-2000 155 29 3 3 NUM iajs-2000 155 30 ] ] PUNCT iajs-2000 155 31 . . PUNCT iajs-2000 156 1 proposition proposition NOUN iajs-2000 156 2 ( ( PUNCT iajs-2000 156 3 23 23 NUM iajs-2000 156 4 ) ) PUNCT iajs-2000 156 5 " " PUNCT iajs-2000 156 6 let let VERB iajs-2000 156 7 x x PRON iajs-2000 156 8 is be AUX iajs-2000 156 9 a a DET iajs-2000 156 10 torsion torsion NOUN iajs-2000 156 11 free free ADJ iajs-2000 156 12 scalar scalar ADJ iajs-2000 156 13 r r NOUN iajs-2000 156 14 - - PUNCT iajs-2000 156 15 module module NOUN iajs-2000 156 16 , , PUNCT iajs-2000 156 17 and and CCONJ iajs-2000 156 18 l l NOUN iajs-2000 156 19 be be AUX iajs-2000 156 20 a a DET iajs-2000 156 21 proper proper ADJ iajs-2000 156 22 submodule submodule NOUN iajs-2000 156 23 of of ADP iajs-2000 156 24 x x PROPN iajs-2000 156 25 , , PUNCT iajs-2000 156 26 such such ADJ iajs-2000 156 27 that that SCONJ iajs-2000 156 28 l l NOUN iajs-2000 156 29 is be AUX iajs-2000 156 30 a a DET iajs-2000 156 31 we we PRON iajs-2000 156 32 - - PUNCT iajs-2000 156 33 semi semi ADJ iajs-2000 156 34 - - ADJ iajs-2000 156 35 prime prime ADJ iajs-2000 156 36 submodule submodule NOUN iajs-2000 156 37 of of ADP iajs-2000 156 38 x. x. PROPN iajs-2000 156 39 then then ADV iajs-2000 156 40 𝐿 𝐿 PROPN iajs-2000 156 41 : : PUNCT iajs-2000 156 42 𝐼 𝐼 PROPN iajs-2000 156 43 is be AUX iajs-2000 156 44 a a DET iajs-2000 156 45 we we PRON iajs-2000 156 46 - - PUNCT iajs-2000 156 47 semi-"prime semi-"prime NOUN iajs-2000 156 48 submodule submodule NOUN iajs-2000 156 49 of of ADP iajs-2000 156 50 x x PUNCT iajs-2000 156 51 for for ADP iajs-2000 156 52 any any DET iajs-2000 156 53 non non ADJ iajs-2000 156 54 - - ADJ iajs-2000 156 55 zero zero NUM iajs-2000 156 56 ideal ideal NOUN iajs-2000 156 57 i i PRON iajs-2000 156 58 of of ADP iajs-2000 156 59 r r NOUN iajs-2000 156 60 " " PUNCT iajs-2000 156 61 . . PUNCT iajs-2000 157 1 proof proof NOUN iajs-2000 157 2 " " PUNCT iajs-2000 157 3 since since SCONJ iajs-2000 157 4 l l NOUN iajs-2000 157 5 is be AUX iajs-2000 157 6 a a DET iajs-2000 157 7 we we PRON iajs-2000 157 8 - - PUNCT iajs-2000 157 9 semi semi ADJ iajs-2000 157 10 - - ADJ iajs-2000 157 11 prime prime ADJ iajs-2000 157 12 submodule submodule NOUN iajs-2000 157 13 of of ADP iajs-2000 157 14 x x NOUN iajs-2000 157 15 " " PUNCT iajs-2000 157 16 , , PUNCT iajs-2000 157 17 " " PUNCT iajs-2000 157 18 then then ADV iajs-2000 157 19 by by ADP iajs-2000 157 20 proposition proposition NOUN iajs-2000 157 21 ( ( PUNCT iajs-2000 157 22 3.4 3.4 NUM iajs-2000 157 23 ) ) PUNCT iajs-2000 157 24 l l NOUN iajs-2000 157 25 is be AUX iajs-2000 157 26 a a DET iajs-2000 157 27 weakly weakly ADJ iajs-2000 157 28 semi semi ADJ iajs-2000 157 29 - - ADJ iajs-2000 157 30 prime prime ADJ iajs-2000 157 31 submodule submodule NOUN iajs-2000 157 32 of of ADP iajs-2000 157 33 x x NOUN iajs-2000 157 34 " " PUNCT iajs-2000 157 35 . . PUNCT iajs-2000 158 1 " " PUNCT iajs-2000 158 2 thus thus ADV iajs-2000 158 3 by by ADP iajs-2000 158 4 [ [ X iajs-2000 158 5 2 2 NUM iajs-2000 158 6 , , PUNCT iajs-2000 158 7 prop.27 prop.27 PROPN iajs-2000 158 8 ] ] X iajs-2000 158 9 we we PRON iajs-2000 158 10 get get VERB iajs-2000 158 11 𝐿 𝐿 NOUN iajs-2000 158 12 : : PUNCT iajs-2000 158 13 𝐼 𝐼 PROPN iajs-2000 158 14 is be AUX iajs-2000 158 15 a a DET iajs-2000 158 16 weakly weakly ADJ iajs-2000 158 17 semi semi ADJ iajs-2000 158 18 - - ADJ iajs-2000 158 19 prime prime ADJ iajs-2000 158 20 submodule submodule NOUN iajs-2000 158 21 of of ADP iajs-2000 158 22 x. x. NOUN iajs-2000 158 23 " " PUNCT iajs-2000 158 24 but but CCONJ iajs-2000 158 25 x x X iajs-2000 158 26 is be AUX iajs-2000 158 27 a a DET iajs-2000 158 28 scalar scalar ADJ iajs-2000 158 29 module module NOUN iajs-2000 158 30 , , PUNCT iajs-2000 158 31 hence hence ADV iajs-2000 158 32 by by ADP iajs-2000 158 33 proposition proposition NOUN iajs-2000 158 34 ( ( PUNCT iajs-2000 158 35 3.7 3.7 NUM iajs-2000 158 36 ) ) PUNCT iajs-2000 158 37 , , PUNCT iajs-2000 158 38 we we PRON iajs-2000 158 39 have have VERB iajs-2000 158 40 𝐿 𝐿 NOUN iajs-2000 158 41 : : PUNCT iajs-2000 158 42 𝐼 𝐼 PROPN iajs-2000 158 43 is be AUX iajs-2000 158 44 a a DET iajs-2000 158 45 we we PRON iajs-2000 158 46 - - PUNCT iajs-2000 158 47 semi semi ADJ iajs-2000 158 48 - - ADJ iajs-2000 158 49 prime prime ADJ iajs-2000 158 50 submodule submodule NOUN iajs-2000 158 51 of of ADP iajs-2000 158 52 x x NOUN iajs-2000 158 53 " " PUNCT iajs-2000 158 54 . . PUNCT iajs-2000 159 1 proposition proposition NOUN iajs-2000 159 2 ( ( PUNCT iajs-2000 159 3 24 24 NUM iajs-2000 159 4 ) ) PUNCT iajs-2000 159 5 let let VERB iajs-2000 159 6 πœ™ πœ™ NOUN iajs-2000 159 7 : : PUNCT iajs-2000 159 8 𝑋 𝑋 NOUN iajs-2000 159 9 ⟢ ⟢ NOUN iajs-2000 159 10 𝑋′ 𝑋′ NOUN iajs-2000 159 11 be be AUX iajs-2000 159 12 an an DET iajs-2000 159 13 r r NOUN iajs-2000 159 14 - - PUNCT iajs-2000 159 15 epimorphism epimorphism NOUN iajs-2000 159 16 , , PUNCT iajs-2000 159 17 and and CCONJ iajs-2000 159 18 " " PUNCT iajs-2000 159 19 l l NOUN iajs-2000 159 20 is be AUX iajs-2000 159 21 a a DET iajs-2000 159 22 we we PRON iajs-2000 159 23 - - PUNCT iajs-2000 159 24 semi semi ADJ iajs-2000 159 25 - - ADJ iajs-2000 159 26 prime prime ADJ iajs-2000 159 27 submodule submodule NOUN iajs-2000 159 28 of of ADP iajs-2000 159 29 x x PUNCT iajs-2000 159 30 with with ADP iajs-2000 159 31 πΎπ‘’π‘Ÿπœ™ πΎπ‘’π‘Ÿπœ™ PROPN iajs-2000 159 32 𝐿. 𝐿. PROPN iajs-2000 159 33 " " PUNCT iajs-2000 159 34 then then ADV iajs-2000 159 35 πœ™ πœ™ PROPN iajs-2000 159 36 𝐿 𝐿 PROPN iajs-2000 159 37 is be AUX iajs-2000 159 38 a a DET iajs-2000 159 39 we we PRON iajs-2000 159 40 - - PUNCT iajs-2000 159 41 semi semi ADJ iajs-2000 159 42 - - ADJ iajs-2000 159 43 prime prime ADJ iajs-2000 159 44 submodule submodule NOUN iajs-2000 159 45 of of ADP iajs-2000 159 46 x x NOUN iajs-2000 159 47 ' ' PROPN iajs-2000 159 48 , , PUNCT iajs-2000 159 49 where where SCONJ iajs-2000 159 50 x x X iajs-2000 159 51 ' ' PUNCT iajs-2000 159 52 is be AUX iajs-2000 159 53 an an DET iajs-2000 159 54 x x ADJ iajs-2000 159 55 - - ADJ iajs-2000 159 56 projective projective ADJ iajs-2000 159 57 rmodule rmodule NOUN iajs-2000 159 58 . . PUNCT iajs-2000 160 1 proof proof NOUN iajs-2000 160 2 clearly clearly ADV iajs-2000 160 3 πœ™ πœ™ PROPN iajs-2000 160 4 𝐿 𝐿 PROPN iajs-2000 160 5 is be AUX iajs-2000 160 6 a a DET iajs-2000 160 7 proper proper ADJ iajs-2000 160 8 submodule submodule NOUN iajs-2000 160 9 of of ADP iajs-2000 160 10 x x NOUN iajs-2000 160 11 ' ' PUNCT iajs-2000 160 12 . . PUNCT iajs-2000 160 13 assume assume VERB iajs-2000 160 14 that that SCONJ iajs-2000 160 15 0 0 NUM iajs-2000 161 1 𝑓 𝑓 X iajs-2000 161 2 π‘₯β€² π‘₯β€² NUM iajs-2000 161 3 ∈ ∈ PROPN iajs-2000 161 4 πœ™ πœ™ PRON iajs-2000 161 5 𝐿 𝐿 PROPN iajs-2000 161 6 where where SCONJ iajs-2000 161 7 π‘₯β€² π‘₯β€² PUNCT iajs-2000 161 8 ∈ ∈ PROPN iajs-2000 161 9 𝑋′ 𝑋′ NOUN iajs-2000 161 10 , , PUNCT iajs-2000 161 11 and and CCONJ iajs-2000 161 12 𝑓 𝑓 DET iajs-2000 161 13 ∈ ∈ PROPN iajs-2000 161 14 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 161 15 𝑋′ 𝑋′ NOUN iajs-2000 161 16 , , PUNCT iajs-2000 161 17 we we PRON iajs-2000 161 18 prove prove VERB iajs-2000 161 19 that that SCONJ iajs-2000 161 20 𝑓 𝑓 X iajs-2000 161 21 π‘₯β€² π‘₯β€² PUNCT iajs-2000 161 22 ∈ ∈ PROPN iajs-2000 161 23 πœ™ πœ™ PRON iajs-2000 161 24 𝐿 𝐿 PROPN iajs-2000 161 25 , , PUNCT iajs-2000 161 26 since since SCONJ iajs-2000 161 27 πœ™ πœ™ PRON iajs-2000 161 28 is be AUX iajs-2000 161 29 an an DET iajs-2000 161 30 epimorphism epimorphism NOUN iajs-2000 161 31 , , PUNCT iajs-2000 161 32 and and CCONJ iajs-2000 161 33 π‘₯β€² π‘₯β€² PUNCT iajs-2000 161 34 ∈ ∈ PROPN iajs-2000 161 35 𝑋′ 𝑋′ NOUN iajs-2000 161 36 , , PUNCT iajs-2000 161 37 then then ADV iajs-2000 161 38 there there PRON iajs-2000 161 39 exists exist VERB iajs-2000 161 40 π‘₯ π‘₯ DET iajs-2000 161 41 ∈ ∈ PROPN iajs-2000 161 42 𝑋 𝑋 NOUN iajs-2000 161 43 such such ADJ iajs-2000 161 44 that that SCONJ iajs-2000 161 45 πœ™ πœ™ PROPN iajs-2000 162 1 π‘₯ π‘₯ X iajs-2000 162 2 π‘₯β€² π‘₯β€² PUNCT iajs-2000 162 3 . . PUNCT iajs-2000 163 1 " " PUNCT iajs-2000 163 2 consider consider VERB iajs-2000 163 3 the the DET iajs-2000 163 4 following follow VERB iajs-2000 163 5 diagram diagram NOUN iajs-2000 163 6 since since SCONJ iajs-2000 163 7 x x PRON iajs-2000 163 8 ' ' PUNCT iajs-2000 163 9 is be AUX iajs-2000 163 10 xprojective xprojective ADJ iajs-2000 163 11 " " PUNCT iajs-2000 163 12 , , PUNCT iajs-2000 163 13 then then ADV iajs-2000 163 14 there there PRON iajs-2000 163 15 exists exist VERB iajs-2000 163 16 a a DET iajs-2000 163 17 homomorphism homomorphism NOUN iajs-2000 163 18 h h NOUN iajs-2000 164 1 such such ADJ iajs-2000 164 2 that that SCONJ iajs-2000 164 3 πœ™oh πœ™oh PROPN iajs-2000 164 4 f f X iajs-2000 164 5 . . PUNCT iajs-2000 165 1 now now ADV iajs-2000 165 2 , , PUNCT iajs-2000 165 3 0 0 NUM iajs-2000 165 4 𝑓 𝑓 X iajs-2000 165 5 π‘₯β€² π‘₯β€² NUM iajs-2000 165 6 𝑓 𝑓 DET iajs-2000 165 7 𝑓 𝑓 DET iajs-2000 165 8 π‘₯β€² π‘₯β€² PUNCT iajs-2000 165 9 ∈ ∈ PROPN iajs-2000 165 10 πœ™ πœ™ PRON iajs-2000 165 11 𝐿 𝐿 PROPN iajs-2000 165 12 , , PUNCT iajs-2000 165 13 " " PUNCT iajs-2000 165 14 implies imply VERB iajs-2000 165 15 that that SCONJ iajs-2000 165 16 0 0 NUM iajs-2000 165 17 πœ™ πœ™ PRON iajs-2000 165 18 ∘ ∘ PROPN iajs-2000 165 19 h h PROPN iajs-2000 165 20 ∘ ∘ PROPN iajs-2000 165 21 πœ™ πœ™ PROPN iajs-2000 165 22 ∘ ∘ PROPN iajs-2000 165 23 h h NOUN iajs-2000 165 24 π‘₯β€² π‘₯β€² NOUN iajs-2000 165 25 ∈ ∈ PROPN iajs-2000 165 26 πœ™ πœ™ NOUN iajs-2000 165 27 𝐿 𝐿 PROPN iajs-2000 165 28 , , PUNCT iajs-2000 165 29 and and CCONJ iajs-2000 166 1 hence hence ADV iajs-2000 166 2 0 0 NUM iajs-2000 166 3 πœ™ πœ™ PROPN iajs-2000 166 4 h h NOUN iajs-2000 166 5 ∘ ∘ NOUN iajs-2000 166 6 πœ™ πœ™ X iajs-2000 166 7 π‘₯ π‘₯ X iajs-2000 166 8 ∈ ∈ PROPN iajs-2000 166 9 πœ™ πœ™ X iajs-2000 166 10 𝐿 𝐿 PROPN iajs-2000 166 11 . . PUNCT iajs-2000 167 1 but but CCONJ iajs-2000 167 2 πΎπ‘’π‘Ÿπœ™ πΎπ‘’π‘Ÿπœ™ PROPN iajs-2000 167 3 𝐿 𝐿 PROPN iajs-2000 167 4 , , PUNCT iajs-2000 167 5 " " PUNCT iajs-2000 167 6 " " PUNCT iajs-2000 167 7 then then ADV iajs-2000 168 1 0 0 NUM iajs-2000 168 2 h h NOUN iajs-2000 168 3 ∘ ∘ NOUN iajs-2000 168 4 πœ™ πœ™ VERB iajs-2000 168 5 π‘₯ π‘₯ X iajs-2000 168 6 ∈ ∈ PROPN iajs-2000 168 7 𝐿. 𝐿. VERB iajs-2000 168 8 since since SCONJ iajs-2000 168 9 l l PROPN iajs-2000 168 10 is be AUX iajs-2000 168 11 a a DET iajs-2000 168 12 we we PRON iajs-2000 168 13 - - PUNCT iajs-2000 168 14 semi semi ADJ iajs-2000 168 15 - - ADJ iajs-2000 168 16 prime prime ADJ iajs-2000 168 17 submodule submodule NOUN iajs-2000 168 18 of of ADP iajs-2000 168 19 x x PROPN iajs-2000 168 20 , , PUNCT iajs-2000 168 21 then then ADV iajs-2000 168 22 πœ™ πœ™ PROPN iajs-2000 168 23 ∘ ∘ PROPN iajs-2000 168 24 h h PROPN iajs-2000 168 25 π‘₯ π‘₯ PROPN iajs-2000 168 26 , , PUNCT iajs-2000 168 27 implies imply VERB iajs-2000 168 28 that that SCONJ iajs-2000 169 1 πœ™ πœ™ PROPN iajs-2000 169 2 h h NOUN iajs-2000 169 3 ∘ ∘ VERB iajs-2000 169 4 πœ™ πœ™ X iajs-2000 169 5 π‘₯ π‘₯ X iajs-2000 169 6 ∈ ∈ NOUN iajs-2000 169 7 πœ™ πœ™ PRON iajs-2000 169 8 𝐿 𝐿 PROPN iajs-2000 169 9 hence hence ADV iajs-2000 169 10 πœ™ πœ™ PROPN iajs-2000 169 11 ∘ ∘ X iajs-2000 169 12 h h NOUN iajs-2000 169 13 πœ™ πœ™ NOUN iajs-2000 169 14 π‘₯ π‘₯ X iajs-2000 169 15 ∈ ∈ PROPN iajs-2000 169 16 πœ™ πœ™ DET iajs-2000 169 17 𝐿 𝐿 PROPN iajs-2000 169 18 implies imply VERB iajs-2000 169 19 that that SCONJ iajs-2000 169 20 𝑓 𝑓 X iajs-2000 169 21 π‘₯β€² π‘₯β€² PUNCT iajs-2000 169 22 ∈ ∈ PROPN iajs-2000 169 23 πœ™ πœ™ PROPN iajs-2000 169 24 𝐿 𝐿 PROPN iajs-2000 169 25 . . PUNCT iajs-2000 170 1 therefore therefore ADV iajs-2000 170 2 πœ™ πœ™ PROPN iajs-2000 170 3 𝐿 𝐿 PROPN iajs-2000 170 4 is be AUX iajs-2000 170 5 a a DET iajs-2000 170 6 we we PRON iajs-2000 170 7 - - PUNCT iajs-2000 170 8 semi semi ADJ iajs-2000 170 9 - - ADJ iajs-2000 170 10 prime prime ADJ iajs-2000 170 11 submodule submodule NOUN iajs-2000 170 12 of of ADP iajs-2000 170 13 x x NOUN iajs-2000 170 14 ' ' PUNCT iajs-2000 170 15 . . PUNCT iajs-2000 171 1 " " PUNCT iajs-2000 171 2 as as ADP iajs-2000 171 3 a a DET iajs-2000 171 4 direct direct ADJ iajs-2000 171 5 consequence consequence NOUN iajs-2000 171 6 of of ADP iajs-2000 171 7 proposition proposition NOUN iajs-2000 171 8 ( ( PUNCT iajs-2000 171 9 3.12 3.12 NUM iajs-2000 171 10 ) ) PUNCT iajs-2000 171 11 we we PRON iajs-2000 171 12 get get VERB iajs-2000 171 13 the the DET iajs-2000 171 14 following follow VERB iajs-2000 171 15 corollary corollary NOUN iajs-2000 171 16 . . PUNCT iajs-2000 172 1 corollary corollary ADJ iajs-2000 172 2 ( ( PUNCT iajs-2000 172 3 25 25 NUM iajs-2000 172 4 ) ) PUNCT iajs-2000 172 5 " " PUNCT iajs-2000 172 6 " " PUNCT iajs-2000 172 7 let let VERB iajs-2000 172 8 l l NOUN iajs-2000 172 9 and and CCONJ iajs-2000 172 10 k k PROPN iajs-2000 172 11 be be AUX iajs-2000 172 12 a a DET iajs-2000 172 13 submodule submodule NOUN iajs-2000 172 14 of of ADP iajs-2000 172 15 an an DET iajs-2000 172 16 r r NOUN iajs-2000 172 17 - - PUNCT iajs-2000 172 18 module module NOUN iajs-2000 172 19 x x PUNCT iajs-2000 172 20 with with ADP iajs-2000 172 21 𝐾 𝐾 PROPN iajs-2000 172 22 𝐿 𝐿 PROPN iajs-2000 172 23 , , PUNCT iajs-2000 172 24 " " PUNCT iajs-2000 172 25 and and CCONJ iajs-2000 172 26 l l NOUN iajs-2000 172 27 is be AUX iajs-2000 172 28 a a DET iajs-2000 172 29 we we PRON iajs-2000 172 30 - - PUNCT iajs-2000 172 31 semi semi ADJ iajs-2000 172 32 - - ADJ iajs-2000 172 33 prime prime ADJ iajs-2000 172 34 submodule submodule NOUN iajs-2000 172 35 of of ADP iajs-2000 172 36 x. x. PROPN iajs-2000 173 1 " " PUNCT iajs-2000 173 2 then then ADV iajs-2000 173 3 is be AUX iajs-2000 173 4 a a DET iajs-2000 173 5 we we PRON iajs-2000 173 6 - - PUNCT iajs-2000 173 7 semi semi ADJ iajs-2000 173 8 - - ADJ iajs-2000 173 9 prime prime ADJ iajs-2000 173 10 submodule submodule NOUN iajs-2000 173 11 of of ADP iajs-2000 173 12 , , PUNCT iajs-2000 173 13 where where SCONJ iajs-2000 173 14 is be AUX iajs-2000 173 15 an an DET iajs-2000 173 16 x x ADJ iajs-2000 173 17 - - ADJ iajs-2000 173 18 projective projective ADJ iajs-2000 173 19 rmodule rmodule NOUN iajs-2000 173 20 . . PUNCT iajs-2000 174 1 mathematics mathematic NOUN iajs-2000 174 2 | | ADV iajs-2000 174 3 116 116 NUM iajs-2000 174 4 ibn ibn PROPN iajs-2000 174 5 al al PROPN iajs-2000 174 6 - - PUNCT iajs-2000 174 7 haitham haitham PROPN iajs-2000 174 8 jour jour X iajs-2000 174 9 . . PROPN iajs-2000 175 1 for for ADP iajs-2000 175 2 pure pure ADJ iajs-2000 175 3 & & CCONJ iajs-2000 175 4 appl appl PROPN iajs-2000 175 5 . . PUNCT iajs-2000 176 1 sci sci PROPN iajs-2000 176 2 . . PROPN iajs-2000 176 3 ihjpas ihjpa VERB iajs-2000 176 4 https://doi.org/10.30526/31.3.2000 https://doi.org/10.30526/31.3.2000 NUM iajs-2000 176 5 vol vol NOUN iajs-2000 176 6 . . PROPN iajs-2000 176 7 31 31 NUM iajs-2000 176 8 ( ( PUNCT iajs-2000 176 9 3 3 NUM iajs-2000 176 10 ) ) PUNCT iajs-2000 176 11 2018 2018 NUM iajs-2000 176 12 " " PUNCT iajs-2000 176 13 recall recall VERB iajs-2000 176 14 that that SCONJ iajs-2000 176 15 an an DET iajs-2000 176 16 r r NOUN iajs-2000 176 17 - - PUNCT iajs-2000 176 18 module module NOUN iajs-2000 176 19 x x PUNCT iajs-2000 176 20 is be AUX iajs-2000 176 21 multiplication multiplication NOUN iajs-2000 176 22 if if SCONJ iajs-2000 176 23 every every DET iajs-2000 176 24 submodule submodule NOUN iajs-2000 176 25 k k PROPN iajs-2000 176 26 of of ADP iajs-2000 176 27 x x PROPN iajs-2000 176 28 is be AUX iajs-2000 176 29 of of ADP iajs-2000 176 30 the the DET iajs-2000 176 31 form form NOUN iajs-2000 176 32 k k NOUN iajs-2000 176 33 = = NOUN iajs-2000 176 34 ix ix ADJ iajs-2000 176 35 for for ADP iajs-2000 176 36 some some DET iajs-2000 176 37 ideal ideal ADJ iajs-2000 177 1 i i PRON iajs-2000 177 2 of of ADP iajs-2000 177 3 r r NOUN iajs-2000 177 4 [ [ X iajs-2000 177 5 7 7 NUM iajs-2000 177 6 ] ] PUNCT iajs-2000 177 7 . . PUNCT iajs-2000 178 1 proposition proposition NOUN iajs-2000 178 2 ( ( PUNCT iajs-2000 178 3 26 26 NUM iajs-2000 178 4 ) ) PUNCT iajs-2000 178 5 " " PUNCT iajs-2000 178 6 " " PUNCT iajs-2000 178 7 let let VERB iajs-2000 178 8 x x PRON iajs-2000 178 9 be be AUX iajs-2000 178 10 a a DET iajs-2000 178 11 multiplication multiplication NOUN iajs-2000 178 12 r r NOUN iajs-2000 178 13 - - PUNCT iajs-2000 178 14 module module NOUN iajs-2000 178 15 and and CCONJ iajs-2000 178 16 l l NOUN iajs-2000 178 17 is be AUX iajs-2000 178 18 a a DET iajs-2000 178 19 weakly weakly ADJ iajs-2000 178 20 semi semi ADJ iajs-2000 178 21 - - ADJ iajs-2000 178 22 prime prime ADJ iajs-2000 178 23 submodule submodule NOUN iajs-2000 178 24 of of ADP iajs-2000 178 25 x x NOUN iajs-2000 178 26 " " PUNCT iajs-2000 178 27 , , PUNCT iajs-2000 178 28 " " PUNCT iajs-2000 178 29 then then ADV iajs-2000 178 30 l l PROPN iajs-2000 178 31 is be AUX iajs-2000 178 32 a a DET iajs-2000 178 33 we we PRON iajs-2000 178 34 - - PUNCT iajs-2000 178 35 semi semi ADJ iajs-2000 178 36 - - ADJ iajs-2000 178 37 prime prime ADJ iajs-2000 178 38 submodule submodule NOUN iajs-2000 178 39 of of ADP iajs-2000 178 40 x x NOUN iajs-2000 178 41 " " PUNCT iajs-2000 178 42 . . PUNCT iajs-2000 179 1 proof proof NOUN iajs-2000 179 2 " " PUNCT iajs-2000 179 3 " " PUNCT iajs-2000 179 4 suppose suppose VERB iajs-2000 179 5 that that SCONJ iajs-2000 179 6 0 0 NUM iajs-2000 179 7 𝑓 𝑓 PRON iajs-2000 179 8 π‘₯ π‘₯ PRON iajs-2000 179 9 ∈ ∈ PROPN iajs-2000 179 10 𝐿 𝐿 PROPN iajs-2000 179 11 , , PUNCT iajs-2000 179 12 where where SCONJ iajs-2000 179 13 π‘₯ π‘₯ DET iajs-2000 179 14 ∈ ∈ PROPN iajs-2000 179 15 𝑋 𝑋 PROPN iajs-2000 179 16 , , PUNCT iajs-2000 179 17 𝑓 𝑓 DET iajs-2000 179 18 ∈ ∈ PROPN iajs-2000 179 19 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2000 179 20 𝑋 𝑋 NOUN iajs-2000 179 21 ."since ."since PUNCT iajs-2000 179 22 x x PUNCT iajs-2000 179 23 is be AUX iajs-2000 179 24 a a DET iajs-2000 179 25 multiplication multiplication NOUN iajs-2000 179 26 , , PUNCT iajs-2000 179 27 then then ADV iajs-2000 179 28 by by ADP iajs-2000 179 29 [ [ X iajs-2000 179 30 8 8 NUM iajs-2000 179 31 , , PUNCT iajs-2000 179 32 coro.1.2 coro.1.2 PROPN iajs-2000 179 33 ] ] PUNCT iajs-2000 179 34 there there PRON iajs-2000 179 35 exists exist VERB iajs-2000 179 36 𝑠 𝑠 PROPN iajs-2000 179 37 ∈ ∈ PROPN iajs-2000 179 38 𝑅 𝑅 PROPN iajs-2000 179 39 such such ADJ iajs-2000 179 40 that that SCONJ iajs-2000 179 41 𝑓 𝑓 DET iajs-2000 179 42 π‘₯ π‘₯ X iajs-2000 179 43 𝑠π‘₯ 𝑠π‘₯ NOUN iajs-2000 179 44 for for ADP iajs-2000 179 45 all all PRON iajs-2000 179 46 π‘₯ π‘₯ DET iajs-2000 179 47 ∈ ∈ PROPN iajs-2000 179 48 𝑋 𝑋 PROPN iajs-2000 179 49 " " PUNCT iajs-2000 179 50 . . PUNCT iajs-2000 180 1 " " PUNCT iajs-2000 180 2 hence hence ADV iajs-2000 180 3 0 0 PUNCT iajs-2000 180 4 𝑓 𝑓 PRON iajs-2000 180 5 𝑓 𝑓 PRON iajs-2000 180 6 π‘₯ π‘₯ X iajs-2000 180 7 𝑠 𝑠 INTJ iajs-2000 180 8 π‘₯ π‘₯ X iajs-2000 180 9 ∈ ∈ PROPN iajs-2000 180 10 𝐿. 𝐿. PROPN iajs-2000 180 11 but but CCONJ iajs-2000 180 12 l l NOUN iajs-2000 180 13 is be AUX iajs-2000 180 14 a a DET iajs-2000 180 15 weakly weakly ADJ iajs-2000 180 16 semi semi ADJ iajs-2000 180 17 - - ADJ iajs-2000 180 18 prime prime ADJ iajs-2000 180 19 , , PUNCT iajs-2000 180 20 implies imply VERB iajs-2000 180 21 that that SCONJ iajs-2000 180 22 𝑠π‘₯ 𝑠π‘₯ AUX iajs-2000 180 23 ∈ ∈ PROPN iajs-2000 180 24 𝐿. 𝐿. VERB iajs-2000 180 25 thus thus ADV iajs-2000 180 26 𝑓 𝑓 PRON iajs-2000 180 27 π‘₯ π‘₯ PRON iajs-2000 180 28 ∈ ∈ PROPN iajs-2000 180 29 𝐿 𝐿 PROPN iajs-2000 180 30 , , PUNCT iajs-2000 180 31 so so SCONJ iajs-2000 180 32 l l NOUN iajs-2000 180 33 is be AUX iajs-2000 180 34 a a DET iajs-2000 180 35 wesemi wesemi NOUN iajs-2000 180 36 - - PUNCT iajs-2000 180 37 prime prime NOUN iajs-2000 180 38 submodule submodule NOUN iajs-2000 180 39 of of ADP iajs-2000 180 40 x. x. NOUN iajs-2000 181 1 " " PUNCT iajs-2000 181 2 it it PRON iajs-2000 181 3 is be AUX iajs-2000 181 4 well well ADV iajs-2000 181 5 - - PUNCT iajs-2000 181 6 known know VERB iajs-2000 181 7 every every DET iajs-2000 181 8 cyclic cyclic ADJ iajs-2000 181 9 r r NOUN iajs-2000 181 10 - - PUNCT iajs-2000 181 11 module module NOUN iajs-2000 181 12 is be AUX iajs-2000 181 13 a a DET iajs-2000 181 14 multiplication multiplication NOUN iajs-2000 181 15 [ [ X iajs-2000 181 16 7 7 NUM iajs-2000 181 17 ] ] PUNCT iajs-2000 181 18 , , PUNCT iajs-2000 181 19 we we PRON iajs-2000 181 20 get get VERB iajs-2000 181 21 the the DET iajs-2000 181 22 following follow VERB iajs-2000 181 23 result result NOUN iajs-2000 181 24 . . PUNCT iajs-2000 182 1 corollary corollary ADJ iajs-2000 182 2 ( ( PUNCT iajs-2000 182 3 27 27 NUM iajs-2000 182 4 ) ) PUNCT iajs-2000 182 5 " " PUNCT iajs-2000 182 6 " " PUNCT iajs-2000 182 7 let let VERB iajs-2000 182 8 x x PRON iajs-2000 182 9 be be AUX iajs-2000 182 10 a a DET iajs-2000 182 11 cyclic cyclic ADJ iajs-2000 182 12 r r NOUN iajs-2000 182 13 - - PUNCT iajs-2000 182 14 module module NOUN iajs-2000 182 15 , , PUNCT iajs-2000 182 16 and and CCONJ iajs-2000 182 17 l l NOUN iajs-2000 182 18 is be AUX iajs-2000 182 19 a a DET iajs-2000 182 20 proper proper ADJ iajs-2000 182 21 submodule submodule NOUN iajs-2000 182 22 of of ADP iajs-2000 182 23 x. x. NOUN iajs-2000 182 24 " " PUNCT iajs-2000 182 25 then then ADV iajs-2000 182 26 l l PROPN iajs-2000 182 27 is be AUX iajs-2000 182 28 a a DET iajs-2000 182 29 we we PRON iajs-2000 182 30 - - PUNCT iajs-2000 182 31 semiprime semiprime NOUN iajs-2000 182 32 submodule submodule NOUN iajs-2000 183 1 if if SCONJ iajs-2000 183 2 and and CCONJ iajs-2000 183 3 only only ADV iajs-2000 183 4 if if SCONJ iajs-2000 183 5 l l NOUN iajs-2000 183 6 is be AUX iajs-2000 183 7 a a DET iajs-2000 183 8 weakly weakly ADJ iajs-2000 183 9 semi semi ADJ iajs-2000 183 10 - - ADJ iajs-2000 183 11 prime prime ADJ iajs-2000 183 12 . . PUNCT iajs-2000 184 1 we we PRON iajs-2000 184 2 end end VERB iajs-2000 184 3 this this DET iajs-2000 184 4 section section NOUN iajs-2000 184 5 by by ADP iajs-2000 184 6 the the DET iajs-2000 184 7 following follow VERB iajs-2000 184 8 result result NOUN iajs-2000 184 9 . . PUNCT iajs-2000 185 1 proposition proposition NOUN iajs-2000 185 2 ( ( PUNCT iajs-2000 185 3 28 28 NUM iajs-2000 185 4 ) ) PUNCT iajs-2000 185 5 " " PUNCT iajs-2000 185 6 " " PUNCT iajs-2000 185 7 let let VERB iajs-2000 185 8 x x PRON iajs-2000 185 9 be be AUX iajs-2000 185 10 a a DET iajs-2000 185 11 faithful faithful ADJ iajs-2000 185 12 multiplication multiplication NOUN iajs-2000 185 13 r r NOUN iajs-2000 185 14 - - PUNCT iajs-2000 185 15 module module NOUN iajs-2000 185 16 , , PUNCT iajs-2000 185 17 and and CCONJ iajs-2000 185 18 l l NOUN iajs-2000 185 19 is be AUX iajs-2000 185 20 a a DET iajs-2000 185 21 proper proper ADJ iajs-2000 185 22 submodule submodule NOUN iajs-2000 185 23 of of ADP iajs-2000 185 24 x. x. NOUN iajs-2000 185 25 " " PUNCT iajs-2000 185 26 then then ADV iajs-2000 185 27 l l PROPN iajs-2000 185 28 is be AUX iajs-2000 185 29 a a DET iajs-2000 185 30 we we PRON iajs-2000 185 31 - - PUNCT iajs-2000 185 32 semi semi ADJ iajs-2000 185 33 - - ADJ iajs-2000 185 34 prime prime ADJ iajs-2000 185 35 submodule submodule NOUN iajs-2000 185 36 of of ADP iajs-2000 185 37 x x SYM iajs-2000 185 38 if if SCONJ iajs-2000 185 39 and and CCONJ iajs-2000 185 40 only only ADV iajs-2000 185 41 if if SCONJ iajs-2000 185 42 𝐿 𝐿 PROPN iajs-2000 185 43 : : PUNCT iajs-2000 185 44 𝑋 𝑋 NOUN iajs-2000 185 45 is be AUX iajs-2000 185 46 a a DET iajs-2000 185 47 we we PRON iajs-2000 185 48 - - PUNCT iajs-2000 185 49 semi semi ADJ iajs-2000 185 50 - - ADJ iajs-2000 185 51 prime prime ADJ iajs-2000 185 52 ideal ideal NOUN iajs-2000 185 53 of of ADP iajs-2000 185 54 r r NOUN iajs-2000 185 55 " " PUNCT iajs-2000 185 56 . . PUNCT iajs-2000 186 1 proof proof NOUN iajs-2000 186 2 " " PUNCT iajs-2000 186 3 ⟹ ⟹ NUM iajs-2000 186 4 since since SCONJ iajs-2000 186 5 l l NOUN iajs-2000 186 6 is be AUX iajs-2000 186 7 a a DET iajs-2000 186 8 we we PRON iajs-2000 186 9 - - PUNCT iajs-2000 186 10 semi semi ADJ iajs-2000 186 11 - - ADJ iajs-2000 186 12 prime prime ADJ iajs-2000 186 13 submodule submodule NOUN iajs-2000 186 14 of of ADP iajs-2000 186 15 x x PRON iajs-2000 186 16 , , PUNCT iajs-2000 186 17 " " PUNCT iajs-2000 186 18 then then ADV iajs-2000 186 19 by by ADP iajs-2000 186 20 proposition proposition NOUN iajs-2000 186 21 ( ( PUNCT iajs-2000 186 22 3.4 3.4 NUM iajs-2000 186 23 ) ) PUNCT iajs-2000 186 24 l l NOUN iajs-2000 186 25 is be AUX iajs-2000 186 26 a a DET iajs-2000 186 27 weakly weakly ADJ iajs-2000 186 28 semi semi ADJ iajs-2000 186 29 - - ADJ iajs-2000 186 30 prime prime ADJ iajs-2000 186 31 submodule submodule NOUN iajs-2000 186 32 of of ADP iajs-2000 186 33 x x NOUN iajs-2000 186 34 " " PUNCT iajs-2000 186 35 . . PUNCT iajs-2000 187 1 " " PUNCT iajs-2000 187 2 hence hence ADV iajs-2000 187 3 by by ADP iajs-2000 187 4 [ [ X iajs-2000 187 5 2 2 NUM iajs-2000 187 6 , , PUNCT iajs-2000 187 7 prop.29 prop.29 NOUN iajs-2000 187 8 ] ] PUNCT iajs-2000 187 9 , , PUNCT iajs-2000 187 10 we we PRON iajs-2000 187 11 have have VERB iajs-2000 187 12 𝐿 𝐿 PROPN iajs-2000 187 13 : : PUNCT iajs-2000 187 14 𝑋 𝑋 NOUN iajs-2000 187 15 is be AUX iajs-2000 187 16 a a DET iajs-2000 187 17 weakly weakly ADJ iajs-2000 187 18 semi semi ADJ iajs-2000 187 19 - - ADJ iajs-2000 187 20 prime prime ADJ iajs-2000 187 21 ideal ideal NOUN iajs-2000 187 22 of of ADP iajs-2000 187 23 r. r. PROPN iajs-2000 187 24 " " PUNCT iajs-2000 187 25 therefore therefore ADV iajs-2000 187 26 𝐿 𝐿 PROPN iajs-2000 187 27 : : PUNCT iajs-2000 187 28 𝑋 𝑋 NOUN iajs-2000 187 29 is be AUX iajs-2000 187 30 a a DET iajs-2000 187 31 weakly weakly ADJ iajs-2000 187 32 semi semi ADJ iajs-2000 187 33 - - ADJ iajs-2000 187 34 prime prime ADJ iajs-2000 187 35 as as ADP iajs-2000 187 36 r r NOUN iajs-2000 187 37 - - PUNCT iajs-2000 187 38 submodule submodule NOUN iajs-2000 187 39 of of ADP iajs-2000 187 40 r r NOUN iajs-2000 187 41 - - PUNCT iajs-2000 187 42 module module NOUN iajs-2000 187 43 r r NOUN iajs-2000 187 44 " " PUNCT iajs-2000 187 45 . . PUNCT iajs-2000 188 1 " " PUNCT iajs-2000 188 2 but but CCONJ iajs-2000 188 3 r r NOUN iajs-2000 188 4 is be AUX iajs-2000 188 5 cyclic cyclic ADJ iajs-2000 188 6 r r NOUN iajs-2000 188 7 - - PUNCT iajs-2000 188 8 module module NOUN iajs-2000 188 9 , , PUNCT iajs-2000 188 10 implies imply VERB iajs-2000 188 11 that that SCONJ iajs-2000 188 12 by by ADP iajs-2000 188 13 corollary corollary ADJ iajs-2000 188 14 ( ( PUNCT iajs-2000 188 15 27 27 NUM iajs-2000 188 16 ) ) PUNCT iajs-2000 188 17 𝐿 𝐿 NOUN iajs-2000 188 18 : : PUNCT iajs-2000 188 19 𝑋 𝑋 NOUN iajs-2000 188 20 is be AUX iajs-2000 188 21 a a DET iajs-2000 188 22 we we PRON iajs-2000 188 23 - - PUNCT iajs-2000 188 24 semi semi ADJ iajs-2000 188 25 - - ADJ iajs-2000 188 26 prime prime ADJ iajs-2000 188 27 r r NOUN iajs-2000 188 28 - - PUNCT iajs-2000 188 29 submodule submodule NOUN iajs-2000 188 30 of of ADP iajs-2000 188 31 r r NOUN iajs-2000 188 32 - - PUNCT iajs-2000 188 33 module module NOUN iajs-2000 188 34 r. r. NOUN iajs-2000 188 35 hence hence ADV iajs-2000 188 36 𝐿 𝐿 PROPN iajs-2000 188 37 : : PUNCT iajs-2000 188 38 𝑋 𝑋 NOUN iajs-2000 188 39 is be AUX iajs-2000 188 40 a a DET iajs-2000 188 41 we we PRON iajs-2000 188 42 - - PUNCT iajs-2000 188 43 semi semi ADJ iajs-2000 188 44 - - ADJ iajs-2000 188 45 prime prime ADJ iajs-2000 188 46 ideal ideal NOUN iajs-2000 188 47 of of ADP iajs-2000 188 48 r. r. PROPN iajs-2000 188 49 " " PUNCT iajs-2000 188 50 ⟸ ⟸ ADJ iajs-2000 188 51 since since SCONJ iajs-2000 188 52 𝐿 𝐿 PROPN iajs-2000 188 53 : : PUNCT iajs-2000 188 54 𝑋 𝑋 NOUN iajs-2000 188 55 is be AUX iajs-2000 188 56 a a DET iajs-2000 188 57 we we PRON iajs-2000 188 58 - - PUNCT iajs-2000 188 59 semi semi ADJ iajs-2000 188 60 - - ADJ iajs-2000 188 61 prime prime ADJ iajs-2000 188 62 ideal ideal NOUN iajs-2000 188 63 of of ADP iajs-2000 188 64 r r NOUN iajs-2000 188 65 , , PUNCT iajs-2000 188 66 " " PUNCT iajs-2000 188 67 implies imply VERB iajs-2000 188 68 that that SCONJ iajs-2000 188 69 𝐿 𝐿 PROPN iajs-2000 188 70 : : PUNCT iajs-2000 188 71 𝑋 𝑋 NOUN iajs-2000 188 72 is be AUX iajs-2000 188 73 a a DET iajs-2000 188 74 weakly weakly ADJ iajs-2000 188 75 semiprime semiprime NOUN iajs-2000 188 76 ideal ideal NOUN iajs-2000 188 77 of of ADP iajs-2000 188 78 r r NOUN iajs-2000 188 79 " " PUNCT iajs-2000 188 80 . . PUNCT iajs-2000 189 1 " " PUNCT iajs-2000 189 2 hence hence ADV iajs-2000 189 3 by by ADP iajs-2000 189 4 [ [ X iajs-2000 189 5 2 2 NUM iajs-2000 189 6 , , PUNCT iajs-2000 189 7 theo.30 theo.30 NOUN iajs-2000 189 8 ] ] PUNCT iajs-2000 190 1 we we PRON iajs-2000 190 2 have have VERB iajs-2000 190 3 l l NOUN iajs-2000 190 4 is be AUX iajs-2000 190 5 a a DET iajs-2000 190 6 weakly weakly ADJ iajs-2000 190 7 semi semi ADJ iajs-2000 190 8 - - ADJ iajs-2000 190 9 prime prime ADJ iajs-2000 190 10 submodule submodule NOUN iajs-2000 190 11 of of ADP iajs-2000 190 12 x. x. PROPN iajs-2000 191 1 but but CCONJ iajs-2000 191 2 x x X iajs-2000 191 3 is be AUX iajs-2000 191 4 a a DET iajs-2000 191 5 multiplication multiplication NOUN iajs-2000 191 6 , , PUNCT iajs-2000 191 7 then then ADV iajs-2000 191 8 by by ADP iajs-2000 191 9 proposition proposition NOUN iajs-2000 191 10 ( ( PUNCT iajs-2000 191 11 26 26 NUM iajs-2000 191 12 ) ) PUNCT iajs-2000 191 13 l l NOUN iajs-2000 191 14 is be AUX iajs-2000 191 15 a a DET iajs-2000 191 16 we we PRON iajs-2000 191 17 - - PUNCT iajs-2000 191 18 semi semi ADJ iajs-2000 191 19 - - ADJ iajs-2000 191 20 prime prime ADJ iajs-2000 191 21 submodule submodule NOUN iajs-2000 191 22 of of ADP iajs-2000 191 23 x. x. NOUN iajs-2000 191 24 " " PUNCT iajs-2000 191 25 " " PUNCT iajs-2000 191 26 as as ADP iajs-2000 191 27 a a DET iajs-2000 191 28 direct direct ADJ iajs-2000 191 29 consequence consequence NOUN iajs-2000 191 30 of of ADP iajs-2000 191 31 proposition proposition NOUN iajs-2000 191 32 ( ( PUNCT iajs-2000 191 33 27 27 NUM iajs-2000 191 34 ) ) PUNCT iajs-2000 191 35 , , PUNCT iajs-2000 191 36 we we PRON iajs-2000 191 37 get get VERB iajs-2000 191 38 the the DET iajs-2000 191 39 following follow VERB iajs-2000 191 40 result result NOUN iajs-2000 191 41 " " PUNCT iajs-2000 191 42 . . PUNCT iajs-2000 192 1 corollary corollary ADJ iajs-2000 192 2 ( ( PUNCT iajs-2000 192 3 3.17 3.17 NUM iajs-2000 192 4 ) ) PUNCT iajs-2000 192 5 " " PUNCT iajs-2000 192 6 " " PUNCT iajs-2000 192 7 let let VERB iajs-2000 192 8 x x PRON iajs-2000 192 9 be be AUX iajs-2000 192 10 a a DET iajs-2000 192 11 faithful faithful ADJ iajs-2000 192 12 cyclic cyclic ADJ iajs-2000 192 13 r r NOUN iajs-2000 192 14 - - PUNCT iajs-2000 192 15 module module NOUN iajs-2000 192 16 " " PUNCT iajs-2000 192 17 , , PUNCT iajs-2000 192 18 " " PUNCT iajs-2000 192 19 and and CCONJ iajs-2000 192 20 l l NOUN iajs-2000 192 21 is be AUX iajs-2000 192 22 a a DET iajs-2000 192 23 proper proper ADJ iajs-2000 192 24 submodule submodule NOUN iajs-2000 192 25 of of ADP iajs-2000 192 26 x. x. NOUN iajs-2000 193 1 " " PUNCT iajs-2000 193 2 then then ADV iajs-2000 193 3 l l PROPN iajs-2000 193 4 is be AUX iajs-2000 193 5 a a DET iajs-2000 193 6 wesemi wesemi NOUN iajs-2000 193 7 - - PUNCT iajs-2000 193 8 prime prime NOUN iajs-2000 193 9 submodule submodule NOUN iajs-2000 193 10 of of ADP iajs-2000 193 11 x x SYM iajs-2000 193 12 if if SCONJ iajs-2000 193 13 and and CCONJ iajs-2000 193 14 only only ADV iajs-2000 193 15 if if SCONJ iajs-2000 193 16 𝐿 𝐿 PROPN iajs-2000 193 17 : : PUNCT iajs-2000 193 18 𝑋 𝑋 NOUN iajs-2000 193 19 is be AUX iajs-2000 193 20 a a DET iajs-2000 193 21 we we PRON iajs-2000 193 22 - - PUNCT iajs-2000 193 23 semi semi ADJ iajs-2000 193 24 - - ADJ iajs-2000 193 25 prime prime ADJ iajs-2000 193 26 ideal ideal NOUN iajs-2000 193 27 of of ADP iajs-2000 193 28 r. r. PROPN iajs-2000 193 29 references reference NOUN iajs-2000 193 30 1 1 NUM iajs-2000 193 31 . . PUNCT iajs-2000 194 1 hadi hadi PROPN iajs-2000 194 2 , , PUNCT iajs-2000 194 3 a.m. a.m. ADV iajs-2000 194 4 on on ADP iajs-2000 194 5 weakly weakly ADJ iajs-2000 194 6 prime prime ADJ iajs-2000 194 7 submodules submodule NOUN iajs-2000 194 8 . . PUNCT iajs-2000 195 1 ibn ibn PROPN iajs-2000 195 2 al al PROPN iajs-2000 195 3 - - PUNCT iajs-2000 195 4 haitham haitham PROPN iajs-2000 195 5 j. j. PROPN iajs-2000 195 6 for for ADP iajs-2000 195 7 pure pure ADJ iajs-2000 195 8 and and CCONJ iajs-2000 195 9 appl appl NOUN iajs-2000 195 10 . . PUNCT iajs-2000 196 1 sci sci PROPN iajs-2000 196 2 . . PROPN iajs-2000 196 3 2009 2009 NUM iajs-2000 196 4 , , PUNCT iajs-2000 196 5 22 22 NUM iajs-2000 196 6 , , PUNCT iajs-2000 196 7 3 3 NUM iajs-2000 196 8 , , PUNCT iajs-2000 196 9 62 62 NUM iajs-2000 196 10 - - SYM iajs-2000 196 11 69 69 NUM iajs-2000 196 12 . . PUNCT iajs-2000 197 1 2 2 X iajs-2000 197 2 . . X iajs-2000 197 3 farzalipour farzalipour PROPN iajs-2000 197 4 , , PUNCT iajs-2000 197 5 f. f. PROPN iajs-2000 197 6 on on ADP iajs-2000 197 7 almost almost ADV iajs-2000 197 8 semi semi ADJ iajs-2000 197 9 - - ADJ iajs-2000 197 10 prime prime ADJ iajs-2000 197 11 submodules submodule NOUN iajs-2000 197 12 . . PUNCT iajs-2000 198 1 hindowi hindowi PROPN iajs-2000 198 2 publishing publish VERB iajs-2000 198 3 corporation corporation NOUN iajs-2000 198 4 algebra algebra PROPN iajs-2000 198 5 . . PUNCT iajs-2000 199 1 2014 2014 NUM iajs-2000 199 2 , , PUNCT iajs-2000 199 3 3 3 NUM iajs-2000 199 4 , , PUNCT iajs-2000 199 5 231 231 NUM iajs-2000 199 6 - - SYM iajs-2000 199 7 237 237 NUM iajs-2000 199 8 . . PUNCT iajs-2000 200 1 3 3 X iajs-2000 200 2 . . X iajs-2000 200 3 kash kash PROPN iajs-2000 200 4 , , PUNCT iajs-2000 200 5 f. f. PROPN iajs-2000 200 6 modules module NOUN iajs-2000 200 7 and and CCONJ iajs-2000 200 8 rings ring NOUN iajs-2000 200 9 . . PUNCT iajs-2000 201 1 academic academic PROPN iajs-2000 201 2 press press PROPN iajs-2000 201 3 inc inc PROPN iajs-2000 201 4 . . PROPN iajs-2000 201 5 london london PROPN iajs-2000 201 6 . . PUNCT iajs-2000 202 1 1982 1982 NUM iajs-2000 202 2 . . PUNCT iajs-2000 203 1 4 4 X iajs-2000 203 2 . . X iajs-2000 203 3 ozcan ozcan PROPN iajs-2000 203 4 . . PUNCT iajs-2000 204 1 c. c. PROPN iajs-2000 204 2 a. a. PROPN iajs-2000 204 3 ; ; PUNCT iajs-2000 204 4 haranc haranc PROPN iajs-2000 204 5 a. a. NOUN iajs-2000 204 6 ; ; PUNCT iajs-2000 204 7 f. f. PROPN iajs-2000 204 8 smith smith PROPN iajs-2000 204 9 . . PUNCT iajs-2000 205 1 p. p. NOUN iajs-2000 205 2 duo duo NOUN iajs-2000 205 3 modules module NOUN iajs-2000 205 4 . . PUNCT iajs-2000 206 1 clasgow clasgow PROPN iajs-2000 206 2 math math NOUN iajs-2000 206 3 . . PUNCT iajs-2000 207 1 journal journal PROPN iajs-2000 207 2 trust trust PROPN iajs-2000 207 3 . . PUNCT iajs-2000 208 1 2006 2006 NUM iajs-2000 208 2 , , PUNCT iajs-2000 208 3 48 48 NUM iajs-2000 208 4 , , PUNCT iajs-2000 208 5 533 533 NUM iajs-2000 208 6 - - SYM iajs-2000 208 7 545 545 NUM iajs-2000 208 8 . . PUNCT iajs-2000 209 1 5 5 NUM iajs-2000 209 2 . . X iajs-2000 209 3 azum azum PROPN iajs-2000 209 4 , , PUNCT iajs-2000 209 5 g. g. PROPN iajs-2000 209 6 f. f. PROPN iajs-2000 209 7 ; ; PUNCT iajs-2000 209 8 mbuntum mbuntum NOUN iajs-2000 209 9 ; ; PUNCT iajs-2000 209 10 varadarajan varadarajan ADJ iajs-2000 209 11 , , PUNCT iajs-2000 209 12 k. k. PROPN iajs-2000 209 13 on on ADP iajs-2000 209 14 m m PROPN iajs-2000 209 15 - - PUNCT iajs-2000 209 16 projective projective ADJ iajs-2000 209 17 and and CCONJ iajs-2000 209 18 m m ADJ iajs-2000 209 19 - - ADJ iajs-2000 209 20 injective injective ADJ iajs-2000 209 21 modules module NOUN iajs-2000 209 22 . . PUNCT iajs-2000 210 1 pacific pacific PROPN iajs-2000 210 2 journal journal PROPN iajs-2000 210 3 of of ADP iajs-2000 210 4 math math NOUN iajs-2000 210 5 . . PUNCT iajs-2000 211 1 1967 1967 NUM iajs-2000 211 2 , , PUNCT iajs-2000 211 3 59 59 NUM iajs-2000 211 4 , , PUNCT iajs-2000 211 5 1 1 NUM iajs-2000 211 6 , , PUNCT iajs-2000 211 7 9 9 NUM iajs-2000 211 8 - - SYM iajs-2000 211 9 16 16 NUM iajs-2000 211 10 . . PUNCT iajs-2000 212 1 mathematics mathematic NOUN iajs-2000 212 2 | | ADV iajs-2000 212 3 117 117 NUM iajs-2000 212 4 ibn ibn PROPN iajs-2000 212 5 al al PROPN iajs-2000 212 6 - - PUNCT iajs-2000 212 7 haitham haitham PROPN iajs-2000 212 8 jour jour X iajs-2000 212 9 . . PROPN iajs-2000 213 1 for for ADP iajs-2000 213 2 pure pure ADJ iajs-2000 213 3 & & CCONJ iajs-2000 213 4 appl appl PROPN iajs-2000 213 5 . . PUNCT iajs-2000 214 1 sci sci PROPN iajs-2000 214 2 . . PROPN iajs-2000 214 3 ihjpas ihjpa VERB iajs-2000 214 4 https://doi.org/10.30526/31.3.2000 https://doi.org/10.30526/31.3.2000 NUM iajs-2000 214 5 vol vol NOUN iajs-2000 214 6 . . PROPN iajs-2000 214 7 31 31 NUM iajs-2000 214 8 ( ( PUNCT iajs-2000 214 9 3 3 NUM iajs-2000 214 10 ) ) PUNCT iajs-2000 214 11 2018 2018 NUM iajs-2000 214 12 6 6 NUM iajs-2000 214 13 . . PUNCT iajs-2000 215 1 shihap shihap PROPN iajs-2000 215 2 , , PUNCT iajs-2000 215 3 n.b n.b PROPN iajs-2000 215 4 . . PROPN iajs-2000 215 5 scalar scalar ADJ iajs-2000 215 6 reflexine reflexine NOUN iajs-2000 215 7 modules module NOUN iajs-2000 215 8 . . PUNCT iajs-2000 216 1 ph.d ph.d PROPN iajs-2000 216 2 . . PUNCT iajs-2000 217 1 thesis thesis NOUN iajs-2000 217 2 , , PUNCT iajs-2000 217 3 university university NOUN iajs-2000 217 4 of of ADP iajs-2000 217 5 baghdad baghdad PROPN iajs-2000 217 6 . . PUNCT iajs-2000 218 1 2004 2004 NUM iajs-2000 218 2 7 7 NUM iajs-2000 218 3 . . X iajs-2000 219 1 barnard barnard PROPN iajs-2000 219 2 , , PUNCT iajs-2000 219 3 a. a. NOUN iajs-2000 219 4 multiplication multiplication NOUN iajs-2000 219 5 modules module NOUN iajs-2000 219 6 , , PUNCT iajs-2000 219 7 j. j. PROPN iajs-2000 219 8 of of ADP iajs-2000 219 9 algebra algebra PROPN iajs-2000 219 10 , , PUNCT iajs-2000 219 11 1981 1981 NUM iajs-2000 219 12 , , PUNCT iajs-2000 219 13 71 71 NUM iajs-2000 219 14 , , PUNCT iajs-2000 219 15 174 174 NUM iajs-2000 219 16 - - SYM iajs-2000 219 17 178 178 NUM iajs-2000 219 18 . . PUNCT iajs-2000 219 19 8 8 NUM iajs-2000 219 20 . . PUNCT iajs-2000 220 1 naoum naoum PROPN iajs-2000 220 2 , , PUNCT iajs-2000 220 3 g. g. PROPN iajs-2000 220 4 a. a. PROPN iajs-2000 220 5 on on ADP iajs-2000 220 6 the the DET iajs-2000 220 7 ring ring NOUN iajs-2000 220 8 of of ADP iajs-2000 220 9 endomorphisms endomorphism NOUN iajs-2000 220 10 of of ADP iajs-2000 220 11 a a DET iajs-2000 220 12 multiplication multiplication NOUN iajs-2000 220 13 modules module NOUN iajs-2000 220 14 . . PUNCT iajs-2000 221 1 mathematics mathematic NOUN iajs-2000 221 2 hungarica hungarica PROPN iajs-2000 221 3 . . PUNCT iajs-2000 222 1 1994 1994 NUM iajs-2000 222 2 , , PUNCT iajs-2000 222 3 29 29 NUM iajs-2000 222 4 , , PUNCT iajs-2000 222 5 3 3 NUM iajs-2000 222 6 , , PUNCT iajs-2000 222 7 277 277 NUM iajs-2000 222 8 - - SYM iajs-2000 222 9 284 284 NUM iajs-2000 222 10 . . PUNCT