id sid tid token lemma pos iajs-2429 1 1 microsoft microsoft PROPN iajs-2429 1 2 word word NOUN iajs-2429 1 3 81 81 NUM iajs-2429 1 4 - - SYM iajs-2429 1 5 94 94 NUM iajs-2429 1 6     SPACE iajs-2429 1 7 81 81 NUM iajs-2429 1 8     SPACE iajs-2429 1 9 ibn ibn PROPN iajs-2429 1 10 al al PROPN iajs-2429 1 11 - - PUNCT iajs-2429 1 12 haitham haitham PROPN iajs-2429 1 13 jour jour X iajs-2429 1 14 . . PROPN iajs-2429 2 1 for for ADP iajs-2429 2 2 pure pure ADJ iajs-2429 2 3 & & CCONJ iajs-2429 2 4 appl appl PROPN iajs-2429 2 5 . . PUNCT iajs-2429 3 1 sci sci PROPN iajs-2429 3 2 . . PROPN iajs-2429 4 1 33 33 NUM iajs-2429 4 2 ( ( PUNCT iajs-2429 4 3 2 2 NUM iajs-2429 4 4 ) ) PUNCT iajs-2429 4 5 2020 2020 NUM iajs-2429 4 6             SPACE iajs-2429 4 7 on on ADP iajs-2429 4 8 semisecond semisecond ADJ iajs-2429 4 9 submodules submodule NOUN iajs-2429 4 10 zainab zainab PROPN iajs-2429 4 11 saadi saadi PROPN iajs-2429 4 12             SPACE iajs-2429 4 13 ghaleb ghaleb PROPN iajs-2429 4 14 ahmed ahmed PROPN iajs-2429 4 15       SPACE iajs-2429 4 16 abstract abstract ADJ iajs-2429 4 17 let let VERB iajs-2429 4 18 𝑀 𝑀 PROPN iajs-2429 4 19 be be AUX iajs-2429 4 20 a a DET iajs-2429 4 21 right right ADJ iajs-2429 4 22 module module NOUN iajs-2429 4 23 over over ADP iajs-2429 4 24 a a DET iajs-2429 4 25 ring ring NOUN iajs-2429 4 26 𝑅 𝑅 NOUN iajs-2429 4 27 with with ADP iajs-2429 4 28 identity identity NOUN iajs-2429 4 29 . . PUNCT iajs-2429 5 1 the the DET iajs-2429 5 2 semisecond semisecond ADJ iajs-2429 5 3 submodules submodule NOUN iajs-2429 5 4 are be AUX iajs-2429 5 5 studied study VERB iajs-2429 5 6 in in ADP iajs-2429 5 7 this this DET iajs-2429 5 8 paper paper NOUN iajs-2429 5 9 . . PUNCT iajs-2429 6 1 a a DET iajs-2429 6 2 nonzero nonzero PROPN iajs-2429 6 3 submodule submodule NOUN iajs-2429 6 4 𝑁 𝑁 PROPN iajs-2429 6 5 of of ADP iajs-2429 6 6 𝑀 𝑀 PROPN iajs-2429 6 7 is be AUX iajs-2429 6 8 called call VERB iajs-2429 6 9 semisecond semisecond ADJ iajs-2429 6 10 if if SCONJ iajs-2429 6 11 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 6 12 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 6 13 for for ADP iajs-2429 6 14 each each DET iajs-2429 6 15 𝑎 𝑎 PRON iajs-2429 6 16 ∈ ∈ NOUN iajs-2429 6 17 𝑅. 𝑅. NOUN iajs-2429 6 18 more more ADJ iajs-2429 6 19 information information NOUN iajs-2429 6 20 and and CCONJ iajs-2429 6 21 characterizations characterization NOUN iajs-2429 6 22 about about ADP iajs-2429 6 23 this this DET iajs-2429 6 24 concept concept NOUN iajs-2429 6 25 is be AUX iajs-2429 6 26 provided provide VERB iajs-2429 6 27 in in ADP iajs-2429 6 28 our our PRON iajs-2429 6 29 work work NOUN iajs-2429 6 30 . . PUNCT iajs-2429 7 1 keywords keyword NOUN iajs-2429 7 2 : : PUNCT iajs-2429 7 3 semisecond semisecond ADJ iajs-2429 7 4 submodules submodule NOUN iajs-2429 7 5 , , PUNCT iajs-2429 7 6 weak weak ADJ iajs-2429 7 7 semisecond semisecond ADJ iajs-2429 7 8 submodules submodule NOUN iajs-2429 7 9 , , PUNCT iajs-2429 7 10 𝑆-semisecond 𝑆-semisecond NOUN iajs-2429 7 11 submodules submodule NOUN iajs-2429 7 12 , , PUNCT iajs-2429 7 13 regular regular ADJ iajs-2429 7 14 modules module NOUN iajs-2429 7 15 . . PUNCT iajs-2429 8 1 1 1 X iajs-2429 8 2 . . X iajs-2429 8 3 introduction introduction NOUN iajs-2429 8 4 𝑅 𝑅 PROPN iajs-2429 8 5 is be AUX iajs-2429 8 6 indicated indicate VERB iajs-2429 8 7 a a DET iajs-2429 8 8 ring ring NOUN iajs-2429 8 9 with with ADP iajs-2429 8 10 identity identity NOUN iajs-2429 8 11 and and CCONJ iajs-2429 8 12 𝑀 𝑀 PROPN iajs-2429 8 13 is be AUX iajs-2429 8 14 viewed view VERB iajs-2429 8 15 as as ADP iajs-2429 8 16 a a DET iajs-2429 8 17 non non ADJ iajs-2429 8 18 - - ADJ iajs-2429 8 19 zero zero NUM iajs-2429 8 20 𝑆𝑅-bimodule 𝑆𝑅-bimodule NOUN iajs-2429 8 21 where where SCONJ iajs-2429 8 22 𝑆 𝑆 PROPN iajs-2429 8 23 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2429 8 24 𝑀 𝑀 PROPN iajs-2429 8 25 the the DET iajs-2429 8 26 endomorphism endomorphism PROPN iajs-2429 8 27 ring ring NOUN iajs-2429 8 28 of of ADP iajs-2429 8 29 𝑀. 𝑀. PROPN iajs-2429 8 30 we we PRON iajs-2429 8 31 use use VERB iajs-2429 8 32 the the DET iajs-2429 8 33 notation notation NOUN iajs-2429 8 34 ʻʻ ʻʻ ADP iajs-2429 8 35 ⊆ ⊆ NUM iajs-2429 8 36 ʼʼ ʼʼ NOUN iajs-2429 8 37 to to PART iajs-2429 8 38 denote denote VERB iajs-2429 8 39 inclusion inclusion NOUN iajs-2429 8 40 . . PUNCT iajs-2429 9 1 a a DET iajs-2429 9 2 non non ADJ iajs-2429 9 3 - - ADJ iajs-2429 9 4 zero zero NUM iajs-2429 9 5 submodule submodule NOUN iajs-2429 9 6 𝑁 𝑁 PROPN iajs-2429 9 7 of of ADP iajs-2429 9 8 𝑀 𝑀 PROPN iajs-2429 9 9 is be AUX iajs-2429 9 10 said say VERB iajs-2429 9 11 to to PART iajs-2429 9 12 be be AUX iajs-2429 9 13 a a DET iajs-2429 9 14 second second ADJ iajs-2429 9 15 submodule submodule NOUN iajs-2429 9 16 if if SCONJ iajs-2429 9 17 for for ADP iajs-2429 9 18 any any DET iajs-2429 9 19 𝑎 𝑎 PROPN iajs-2429 9 20 ∈ ∈ PROPN iajs-2429 9 21 𝑅 𝑅 PROPN iajs-2429 9 22 , , PUNCT iajs-2429 9 23 the the DET iajs-2429 9 24 endomorphism endomorphism NOUN iajs-2429 9 25 𝑓 𝑓 X iajs-2429 9 26 : : PUNCT iajs-2429 9 27 𝑁 𝑁 PROPN iajs-2429 9 28 → → SYM iajs-2429 9 29 𝑁 𝑁 PROPN iajs-2429 9 30 defined define VERB iajs-2429 9 31 by by ADP iajs-2429 9 32 𝑓 𝑓 DET iajs-2429 9 33 𝑛 𝑛 DET iajs-2429 9 34 𝑛𝑎 𝑛𝑎 PROPN iajs-2429 9 35 for for ADP iajs-2429 9 36 each each DET iajs-2429 9 37 𝑛 𝑛 PRON iajs-2429 9 38 ∈ ∈ PROPN iajs-2429 9 39 𝑁 𝑁 PROPN iajs-2429 9 40 , , PUNCT iajs-2429 9 41 is be AUX iajs-2429 9 42 either either CCONJ iajs-2429 9 43 surjective surjective ADJ iajs-2429 9 44 or or CCONJ iajs-2429 9 45 zero zero NUM iajs-2429 9 46 ( ( PUNCT iajs-2429 9 47 that that PRON iajs-2429 9 48 is be AUX iajs-2429 9 49 𝐼𝑚𝑓 𝐼𝑚𝑓 PROPN iajs-2429 9 50 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 9 51 𝑁 𝑁 PROPN iajs-2429 9 52 or or CCONJ iajs-2429 9 53 𝐼𝑚𝑓 𝐼𝑚𝑓 PROPN iajs-2429 9 54 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 9 55 0 0 NUM iajs-2429 9 56 ) ) PUNCT iajs-2429 10 1 [ [ X iajs-2429 10 2 1 1 NUM iajs-2429 10 3 ] ] PUNCT iajs-2429 10 4 . . PUNCT iajs-2429 11 1 equivalently equivalently ADV iajs-2429 11 2 0 0 PUNCT iajs-2429 12 1 𝑁 𝑁 NOUN iajs-2429 12 2 is be AUX iajs-2429 12 3 a a DET iajs-2429 12 4 second second ADJ iajs-2429 12 5 submodule submodule NOUN iajs-2429 12 6 of of ADP iajs-2429 12 7 𝑀 𝑀 PROPN iajs-2429 12 8 if if SCONJ iajs-2429 12 9 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 12 10 𝑁 𝑁 PROPN iajs-2429 12 11 or or CCONJ iajs-2429 12 12 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 12 13 0 0 NUM iajs-2429 12 14 for for ADP iajs-2429 12 15 every every DET iajs-2429 12 16 ideal ideal ADJ iajs-2429 12 17 𝐼 𝐼 PROPN iajs-2429 12 18 of of ADP iajs-2429 12 19 𝑅 𝑅 PROPN iajs-2429 12 20 [ [ PROPN iajs-2429 12 21 1 1 NUM iajs-2429 12 22 ] ] PUNCT iajs-2429 12 23 . . PUNCT iajs-2429 13 1 in in ADP iajs-2429 13 2 that that DET iajs-2429 13 3 situation situation NOUN iajs-2429 13 4 , , PUNCT iajs-2429 13 5 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 13 6 𝑁 𝑁 PROPN iajs-2429 13 7 is be AUX iajs-2429 13 8 a a DET iajs-2429 13 9 prime prime ADJ iajs-2429 13 10 ideal ideal NOUN iajs-2429 13 11 of of ADP iajs-2429 13 12 𝑅[1 𝑅[1 PROPN iajs-2429 13 13 ] ] X iajs-2429 13 14 . . PUNCT iajs-2429 14 1 a a DET iajs-2429 14 2 non non ADJ iajs-2429 14 3 - - ADJ iajs-2429 14 4 zero zero NUM iajs-2429 14 5 module module NOUN iajs-2429 14 6 𝑀 𝑀 PROPN iajs-2429 14 7 is be AUX iajs-2429 14 8 second second ADJ iajs-2429 14 9 ( ( PUNCT iajs-2429 14 10 or or CCONJ iajs-2429 14 11 coprime coprime NOUN iajs-2429 14 12 ) ) PUNCT iajs-2429 14 13 if if SCONJ iajs-2429 14 14 𝑀 𝑀 PROPN iajs-2429 14 15 is be AUX iajs-2429 14 16 a a DET iajs-2429 14 17 second second ADJ iajs-2429 14 18 submodule submodule NOUN iajs-2429 14 19 of of ADP iajs-2429 14 20 itself itself PRON iajs-2429 14 21 [ [ X iajs-2429 14 22 1 1 NUM iajs-2429 14 23 ] ] PUNCT iajs-2429 14 24 . . PUNCT iajs-2429 15 1 as as ADP iajs-2429 15 2 a a DET iajs-2429 15 3 new new ADJ iajs-2429 15 4 type type NOUN iajs-2429 15 5 of of ADP iajs-2429 15 6 second second ADJ iajs-2429 15 7 submodules submodule NOUN iajs-2429 15 8 , , PUNCT iajs-2429 15 9 the the DET iajs-2429 15 10 concept concept NOUN iajs-2429 15 11 of of ADP iajs-2429 15 12 weakly weakly ADJ iajs-2429 15 13 second second ADJ iajs-2429 15 14 submodules submodule NOUN iajs-2429 15 15 is be AUX iajs-2429 15 16 presented present VERB iajs-2429 15 17 in in ADP iajs-2429 15 18 [ [ X iajs-2429 15 19 2 2 NUM iajs-2429 15 20 ] ] PUNCT iajs-2429 15 21 . . PUNCT iajs-2429 16 1 a a DET iajs-2429 16 2 non non ADJ iajs-2429 16 3 - - ADJ iajs-2429 16 4 zero zero NUM iajs-2429 16 5 submodule submodule NOUN iajs-2429 16 6 𝑁 𝑁 PROPN iajs-2429 16 7 of of ADP iajs-2429 16 8 𝑀 𝑀 PROPN iajs-2429 16 9 is be AUX iajs-2429 16 10 weakly weakly ADJ iajs-2429 16 11 second second ADJ iajs-2429 16 12 submodule submodule NOUN iajs-2429 16 13 whenever whenever SCONJ iajs-2429 16 14 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 16 15 ⊆ ⊆ NUM iajs-2429 16 16 𝐾 𝐾 PROPN iajs-2429 16 17 where where SCONJ iajs-2429 16 18 𝑎 𝑎 X iajs-2429 16 19 , , PUNCT iajs-2429 16 20 𝑏 𝑏 PROPN iajs-2429 16 21 ∈ ∈ PROPN iajs-2429 16 22 𝑅 𝑅 PROPN iajs-2429 16 23 and and CCONJ iajs-2429 16 24 𝐾 𝐾 PROPN iajs-2429 16 25 a a DET iajs-2429 16 26 submodule submodule NOUN iajs-2429 16 27 of of ADP iajs-2429 16 28 𝑀 𝑀 PROPN iajs-2429 16 29 implies imply VERB iajs-2429 16 30 either either CCONJ iajs-2429 16 31 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 16 32 ⊆ ⊆ PROPN iajs-2429 16 33 𝐾 𝐾 PROPN iajs-2429 16 34 or or CCONJ iajs-2429 16 35 𝑁𝑏 𝑁𝑏 PROPN iajs-2429 16 36 ⊆ ⊆ NUM iajs-2429 16 37 𝐾 𝐾 PROPN iajs-2429 17 1 [ [ X iajs-2429 17 2 2 2 NUM iajs-2429 17 3 ] ] PUNCT iajs-2429 17 4 . . PUNCT iajs-2429 18 1 equivalently equivalently ADV iajs-2429 18 2 , , PUNCT iajs-2429 18 3 a a DET iajs-2429 18 4 non non ADJ iajs-2429 18 5 - - ADJ iajs-2429 18 6 zero zero NUM iajs-2429 18 7 submodule submodule NOUN iajs-2429 18 8 𝑁 𝑁 PROPN iajs-2429 18 9 of of ADP iajs-2429 18 10 𝑀 𝑀 PROPN iajs-2429 18 11 is be AUX iajs-2429 18 12 called call VERB iajs-2429 18 13 weakly weakly ADV iajs-2429 18 14 second second ADV iajs-2429 18 15 if if SCONJ iajs-2429 18 16 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 18 17 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 18 18 or or CCONJ iajs-2429 18 19 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 18 20 𝑁𝑏 𝑁𝑏 PROPN iajs-2429 18 21 for for ADP iajs-2429 18 22 every every DET iajs-2429 18 23 𝑎 𝑎 NOUN iajs-2429 18 24 , , PUNCT iajs-2429 18 25 𝑏 𝑏 PROPN iajs-2429 18 26 ∈ ∈ PROPN iajs-2429 18 27 𝑅 𝑅 PROPN iajs-2429 18 28 [ [ NOUN iajs-2429 18 29 2 2 NUM iajs-2429 18 30 ] ] PUNCT iajs-2429 18 31 . . PUNCT iajs-2429 19 1 more more ADJ iajs-2429 19 2 characterizations characterization NOUN iajs-2429 19 3 of of ADP iajs-2429 19 4 the the DET iajs-2429 19 5 weakly weakly ADJ iajs-2429 19 6 second second ADJ iajs-2429 19 7 concept concept NOUN iajs-2429 19 8 are be AUX iajs-2429 19 9 provided provide VERB iajs-2429 19 10 in in ADP iajs-2429 19 11 [ [ X iajs-2429 19 12 3 3 NUM iajs-2429 19 13 ] ] PUNCT iajs-2429 19 14 . . PUNCT iajs-2429 20 1 in in ADP iajs-2429 20 2 fact fact NOUN iajs-2429 20 3 this this DET iajs-2429 20 4 idea idea NOUN iajs-2429 20 5 as as ADP iajs-2429 20 6 a a DET iajs-2429 20 7 dual dual ADJ iajs-2429 20 8 notion notion NOUN iajs-2429 20 9 of of ADP iajs-2429 20 10 the the DET iajs-2429 20 11 concept concept NOUN iajs-2429 20 12 weakly weakly ADJ iajs-2429 20 13 prime prime NOUN iajs-2429 20 14 ( ( PUNCT iajs-2429 20 15 sometimes sometimes ADV iajs-2429 20 16 is be AUX iajs-2429 20 17 called call VERB iajs-2429 20 18 classical classical ADJ iajs-2429 20 19 prime prime ADJ iajs-2429 20 20 ) ) PUNCT iajs-2429 20 21 submodules submodule NOUN iajs-2429 20 22 . . PUNCT iajs-2429 21 1 a a DET iajs-2429 21 2 proper proper ADJ iajs-2429 21 3 submodule submodule NOUN iajs-2429 21 4 𝑁 𝑁 PROPN iajs-2429 21 5 of of ADP iajs-2429 21 6 𝑀 𝑀 PROPN iajs-2429 21 7 is be AUX iajs-2429 21 8 wekly wekly ADV iajs-2429 21 9 prime prime NOUN iajs-2429 21 10 whenever whenever SCONJ iajs-2429 21 11 𝐾𝑎𝑏 𝐾𝑎𝑏 PROPN iajs-2429 21 12 ⊆ ⊆ NUM iajs-2429 21 13 𝑁 𝑁 PROPN iajs-2429 21 14 where where SCONJ iajs-2429 21 15 𝑎 𝑎 NOUN iajs-2429 21 16 , , PUNCT iajs-2429 21 17 𝑏 𝑏 PROPN iajs-2429 21 18 ∈ ∈ PROPN iajs-2429 21 19 𝑅 𝑅 PROPN iajs-2429 21 20 and and CCONJ iajs-2429 21 21 𝐾 𝐾 PROPN iajs-2429 21 22 a a DET iajs-2429 21 23 submodule submodule NOUN iajs-2429 21 24 of of ADP iajs-2429 21 25 𝑀 𝑀 PROPN iajs-2429 21 26 implies imply VERB iajs-2429 21 27 either either CCONJ iajs-2429 21 28 𝐾𝑎 𝐾𝑎 PROPN iajs-2429 21 29 ⊆ ⊆ NUM iajs-2429 21 30 𝑁 𝑁 PROPN iajs-2429 21 31 or or CCONJ iajs-2429 21 32 𝐾𝑏 𝐾𝑏 ADP iajs-2429 21 33 ⊆ ⊆ NUM iajs-2429 21 34 𝑁 𝑁 PROPN iajs-2429 21 35 [ [ X iajs-2429 21 36 4 4 NUM iajs-2429 21 37 ] ] PUNCT iajs-2429 21 38 . . PUNCT iajs-2429 22 1 in in ADP iajs-2429 22 2 [ [ X iajs-2429 22 3 5 5 NUM iajs-2429 22 4 ] ] PUNCT iajs-2429 22 5 . . PUNCT iajs-2429 23 1 we we PRON iajs-2429 23 2 define define VERB iajs-2429 23 3 the the DET iajs-2429 23 4 idea idea NOUN iajs-2429 23 5 of of ADP iajs-2429 23 6 weakly weakly ADV iajs-2429 23 7 secondary secondary ADJ iajs-2429 23 8 as as ADP iajs-2429 23 9 a a DET iajs-2429 23 10 generalization generalization NOUN iajs-2429 23 11 of of ADP iajs-2429 23 12 weakly weakly ADJ iajs-2429 23 13 second second ADJ iajs-2429 23 14 concept concept NOUN iajs-2429 23 15 and and CCONJ iajs-2429 23 16 the the DET iajs-2429 23 17 same same ADJ iajs-2429 23 18 time time NOUN iajs-2429 23 19 , , PUNCT iajs-2429 23 20 it it PRON iajs-2429 23 21 is be AUX iajs-2429 23 22 a a DET iajs-2429 23 23 new new ADJ iajs-2429 23 24 class class NOUN iajs-2429 23 25 of of ADP iajs-2429 23 26 secondary secondary ADJ iajs-2429 23 27 submodules submodule NOUN iajs-2429 23 28 and and CCONJ iajs-2429 23 29 a a DET iajs-2429 23 30 dual dual ADJ iajs-2429 23 31 notion notion NOUN iajs-2429 23 32 of of ADP iajs-2429 23 33 classical classical ADJ iajs-2429 23 34 primary primary ADJ iajs-2429 23 35 submodules submodule NOUN iajs-2429 23 36 respectively respectively ADV iajs-2429 23 37 . . PUNCT iajs-2429 24 1 a a DET iajs-2429 24 2 nonzero nonzero PROPN iajs-2429 24 3 submodule submodule NOUN iajs-2429 24 4 𝑁 𝑁 PROPN iajs-2429 24 5 of of ADP iajs-2429 24 6 𝑀 𝑀 PROPN iajs-2429 24 7 is be AUX iajs-2429 24 8 weakly weakly ADV iajs-2429 24 9 secondary secondary ADJ iajs-2429 24 10 submodule submodule NOUN iajs-2429 24 11 if if SCONJ iajs-2429 24 12 ibn ibn PROPN iajs-2429 24 13 al al PROPN iajs-2429 24 14 haitham haitham PROPN iajs-2429 24 15 journal journal PROPN iajs-2429 24 16 for for ADP iajs-2429 24 17 pure pure ADJ iajs-2429 24 18 and and CCONJ iajs-2429 24 19 applied apply VERB iajs-2429 24 20 science science NOUN iajs-2429 24 21 journal journal PROPN iajs-2429 24 22 homepage homepage NOUN iajs-2429 24 23 : : PUNCT iajs-2429 24 24 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2429 24 25 doi doi NOUN iajs-2429 24 26 : : PUNCT iajs-2429 24 27 10.30526/33.2.2429 10.30526/33.2.2429 PROPN iajs-2429 24 28 zoozaih89@gmail.com zoozaih89@gmail.com PROPN iajs-2429 24 29 ghaleb.a.h@ihcoedu.uobaghdad.edu.iq ghaleb.a.h@ihcoedu.uobaghdad.edu.iq PROPN iajs-2429 24 30 article article NOUN iajs-2429 24 31 history history NOUN iajs-2429 24 32 : : PUNCT iajs-2429 24 33 received receive VERB iajs-2429 24 34 2 2 NUM iajs-2429 24 35 june june PROPN iajs-2429 24 36 2019 2019 NUM iajs-2429 24 37 , , PUNCT iajs-2429 24 38 accepted accept VERB iajs-2429 24 39 15 15 NUM iajs-2429 24 40 july july PROPN iajs-2429 24 41 2019 2019 NUM iajs-2429 24 42 , , PUNCT iajs-2429 24 43 published publish VERB iajs-2429 24 44 april april PROPN iajs-2429 24 45 2020 2020 NUM iajs-2429 24 46 . . PUNCT iajs-2429 25 1 department department NOUN iajs-2429 25 2 of of ADP iajs-2429 25 3 mathematics mathematics PROPN iajs-2429 25 4 , , PUNCT iajs-2429 25 5 college college NOUN iajs-2429 25 6 of of ADP iajs-2429 25 7 science science NOUN iajs-2429 25 8 , , PUNCT iajs-2429 25 9 university university NOUN iajs-2429 25 10 of of ADP iajs-2429 25 11 mustansiriyah mustansiriyah NOUN iajs-2429 25 12     SPACE iajs-2429 25 13 82 82 NUM iajs-2429 25 14     SPACE iajs-2429 25 15 ibn ibn PROPN iajs-2429 25 16 al al PROPN iajs-2429 25 17 - - PUNCT iajs-2429 25 18 haitham haitham PROPN iajs-2429 25 19 jour jour X iajs-2429 25 20 . . PROPN iajs-2429 25 21 for for ADP iajs-2429 25 22 pure pure ADJ iajs-2429 25 23 & & CCONJ iajs-2429 25 24 appl appl PROPN iajs-2429 25 25 . . PUNCT iajs-2429 26 1 sci sci PROPN iajs-2429 26 2 . . PROPN iajs-2429 27 1 33 33 NUM iajs-2429 27 2 ( ( PUNCT iajs-2429 27 3 2 2 NUM iajs-2429 27 4 ) ) PUNCT iajs-2429 27 5 2020 2020 NUM iajs-2429 28 1 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 28 2 ⊆ ⊆ NUM iajs-2429 28 3 𝐾 𝐾 PROPN iajs-2429 28 4 where where SCONJ iajs-2429 28 5 𝑎 𝑎 X iajs-2429 28 6 , , PUNCT iajs-2429 28 7 𝑏 𝑏 PROPN iajs-2429 28 8 ∈ ∈ PROPN iajs-2429 28 9 𝑅 𝑅 PROPN iajs-2429 28 10 and and CCONJ iajs-2429 28 11 𝐾 𝐾 PROPN iajs-2429 28 12 is be AUX iajs-2429 28 13 a a DET iajs-2429 28 14 submodule submodule NOUN iajs-2429 28 15 of of ADP iajs-2429 28 16 𝑀 𝑀 PROPN iajs-2429 28 17 implies imply VERB iajs-2429 28 18 either either CCONJ iajs-2429 28 19 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 28 20 ⊆ ⊆ PROPN iajs-2429 28 21 𝐾 𝐾 PROPN iajs-2429 28 22 or or CCONJ iajs-2429 28 23 𝑁𝑏 𝑁𝑏 PROPN iajs-2429 28 24 ⊆ ⊆ NUM iajs-2429 28 25 𝐾 𝐾 PROPN iajs-2429 28 26 for for ADP iajs-2429 28 27 some some DET iajs-2429 28 28 positive positive ADJ iajs-2429 28 29 integer integer NOUN iajs-2429 28 30 𝑡. 𝑡. VERB iajs-2429 28 31 a a DET iajs-2429 28 32 nonzero nonzero PROPN iajs-2429 28 33 submodule submodule NOUN iajs-2429 28 34 𝑁 𝑁 PROPN iajs-2429 28 35 is be AUX iajs-2429 28 36 a a DET iajs-2429 28 37 secondary secondary ADJ iajs-2429 28 38 submodule submodule NOUN iajs-2429 28 39 of of ADP iajs-2429 28 40 𝑀 𝑀 PROPN iajs-2429 28 41 if if SCONJ iajs-2429 28 42 for for ADP iajs-2429 28 43 any any DET iajs-2429 28 44 𝑎 𝑎 PROPN iajs-2429 28 45 ∈ ∈ PROPN iajs-2429 28 46 𝑅 𝑅 PROPN iajs-2429 28 47 , , PUNCT iajs-2429 28 48 the the DET iajs-2429 28 49 endomorphism endomorphism NOUN iajs-2429 28 50 𝑓 𝑓 X iajs-2429 28 51 : : PUNCT iajs-2429 28 52 𝑁 𝑁 PROPN iajs-2429 28 53 → → SYM iajs-2429 28 54 𝑁 𝑁 PROPN iajs-2429 28 55 defined define VERB iajs-2429 28 56 by by ADP iajs-2429 28 57 𝑓 𝑓 DET iajs-2429 28 58 𝑛 𝑛 DET iajs-2429 28 59 𝑛𝑎 𝑛𝑎 PROPN iajs-2429 28 60 for for ADP iajs-2429 28 61 each each DET iajs-2429 28 62 𝑛 𝑛 PRON iajs-2429 28 63 ∈ ∈ PROPN iajs-2429 28 64 𝑁 𝑁 PROPN iajs-2429 28 65 , , PUNCT iajs-2429 28 66 is be AUX iajs-2429 28 67 either either CCONJ iajs-2429 28 68 surjective surjective ADJ iajs-2429 28 69 or or CCONJ iajs-2429 28 70 nilpotent nilpotent ADJ iajs-2429 28 71 ( ( PUNCT iajs-2429 28 72 that that PRON iajs-2429 28 73 is be AUX iajs-2429 28 74 𝐼𝑚𝑓 𝐼𝑚𝑓 PROPN iajs-2429 28 75 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 28 76 𝑁 𝑁 PROPN iajs-2429 28 77 or or CCONJ iajs-2429 28 78 𝐼𝑚𝑓 𝐼𝑚𝑓 PROPN iajs-2429 28 79 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 28 80 0 0 NUM iajs-2429 28 81 for for ADP iajs-2429 28 82 some some DET iajs-2429 28 83 positive positive ADJ iajs-2429 28 84 integer integer NOUN iajs-2429 28 85 𝑡 𝑡 PROPN iajs-2429 28 86 ) ) PUNCT iajs-2429 29 1 [ [ X iajs-2429 29 2 1 1 NUM iajs-2429 29 3 ] ] PUNCT iajs-2429 29 4 . . PUNCT iajs-2429 30 1 equivalently equivalently ADV iajs-2429 30 2 , , PUNCT iajs-2429 30 3 0 0 PUNCT iajs-2429 31 1 𝑁 𝑁 PROPN iajs-2429 31 2 is be AUX iajs-2429 31 3 a a DET iajs-2429 31 4 secondary secondary ADJ iajs-2429 31 5 submodule submodule NOUN iajs-2429 31 6 of of ADP iajs-2429 31 7 𝑀 𝑀 PROPN iajs-2429 31 8 if if SCONJ iajs-2429 31 9 for for ADP iajs-2429 31 10 every every DET iajs-2429 31 11 ideal ideal ADJ iajs-2429 31 12 𝐼 𝐼 PROPN iajs-2429 31 13 of of ADP iajs-2429 31 14 𝑅 𝑅 PROPN iajs-2429 31 15 , , PUNCT iajs-2429 31 16 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 31 17 𝑁 𝑁 PROPN iajs-2429 31 18 or or CCONJ iajs-2429 31 19 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 31 20 0 0 NUM iajs-2429 31 21 for for ADP iajs-2429 31 22 some some DET iajs-2429 31 23 positive positive ADJ iajs-2429 31 24 integer integer NOUN iajs-2429 31 25 𝑡 𝑡 PROPN iajs-2429 31 26 [ [ X iajs-2429 31 27 1 1 NUM iajs-2429 31 28 ] ] PUNCT iajs-2429 31 29 . . PUNCT iajs-2429 32 1 in in ADP iajs-2429 32 2 this this DET iajs-2429 32 3 case case NOUN iajs-2429 32 4 , , PUNCT iajs-2429 32 5 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 32 6 𝑁 𝑁 PROPN iajs-2429 32 7 is be AUX iajs-2429 32 8 a a DET iajs-2429 32 9 primary primary ADJ iajs-2429 32 10 ideal ideal NOUN iajs-2429 32 11 of of ADP iajs-2429 32 12 𝑅 𝑅 PROPN iajs-2429 32 13 ( ( PUNCT iajs-2429 32 14 that that PRON iajs-2429 32 15 is is ADV iajs-2429 32 16 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 32 17 𝑁 𝑁 PROPN iajs-2429 32 18 is be AUX iajs-2429 32 19 a a DET iajs-2429 32 20 prime prime ADJ iajs-2429 32 21 ideal ideal NOUN iajs-2429 32 22 of of ADP iajs-2429 32 23 𝑅 𝑅 NOUN iajs-2429 32 24 ) ) PUNCT iajs-2429 33 1 [ [ X iajs-2429 33 2 1 1 NUM iajs-2429 33 3 ] ] PUNCT iajs-2429 33 4 . . PUNCT iajs-2429 34 1 a a DET iajs-2429 34 2 proper proper ADJ iajs-2429 34 3 submodule submodule NOUN iajs-2429 34 4 𝐾 𝐾 PROPN iajs-2429 34 5 of of ADP iajs-2429 34 6 𝑀 𝑀 PROPN iajs-2429 34 7 is be AUX iajs-2429 34 8 classical classical ADJ iajs-2429 34 9 primary primary NOUN iajs-2429 34 10 if if SCONJ iajs-2429 34 11 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 34 12 ⊆ ⊆ NUM iajs-2429 34 13 𝐾 𝐾 PROPN iajs-2429 34 14 where where SCONJ iajs-2429 34 15 𝑎 𝑎 X iajs-2429 34 16 , , PUNCT iajs-2429 34 17 𝑏 𝑏 PROPN iajs-2429 34 18 ∈ ∈ PROPN iajs-2429 34 19 𝑅 𝑅 PROPN iajs-2429 34 20 and and CCONJ iajs-2429 34 21 𝑁 𝑁 PROPN iajs-2429 34 22 is be AUX iajs-2429 34 23 a a DET iajs-2429 34 24 submodule submodule NOUN iajs-2429 34 25 of of ADP iajs-2429 34 26 𝑀 𝑀 PROPN iajs-2429 34 27 then then ADV iajs-2429 34 28 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 34 29 ⊆ ⊆ PROPN iajs-2429 34 30 𝐾 𝐾 PROPN iajs-2429 34 31 or or CCONJ iajs-2429 34 32 𝑁𝑏 𝑁𝑏 PROPN iajs-2429 34 33 ⊆ ⊆ NUM iajs-2429 34 34 𝐾 𝐾 PROPN iajs-2429 34 35 for for ADP iajs-2429 34 36 some some DET iajs-2429 34 37 positive positive ADJ iajs-2429 34 38 integer integer NOUN iajs-2429 34 39 𝑡 𝑡 PROPN iajs-2429 35 1 [ [ X iajs-2429 35 2 6 6 NUM iajs-2429 35 3 ] ] PUNCT iajs-2429 35 4 . . PUNCT iajs-2429 36 1 a a DET iajs-2429 36 2 proper proper ADJ iajs-2429 36 3 submodule submodule NOUN iajs-2429 36 4 𝐾 𝐾 PROPN iajs-2429 36 5 of of ADP iajs-2429 36 6 𝑀 𝑀 PROPN iajs-2429 36 7 is be AUX iajs-2429 36 8 called call VERB iajs-2429 36 9 completely completely ADV iajs-2429 36 10 irreducible irreducible ADJ iajs-2429 36 11 when when SCONJ iajs-2429 36 12 𝐾 𝐾 PROPN iajs-2429 36 13 ⋂ ⋂ PROPN iajs-2429 36 14 𝐻∈∧ 𝐻∈∧ PROPN iajs-2429 36 15 where where SCONJ iajs-2429 36 16 𝐻 𝐻 PROPN iajs-2429 36 17 ∈∧ ∈∧ NOUN iajs-2429 36 18 is be AUX iajs-2429 36 19 a a DET iajs-2429 36 20 family family NOUN iajs-2429 36 21 of of ADP iajs-2429 36 22 submodules submodule NOUN iajs-2429 36 23 of of ADP iajs-2429 36 24 𝑀 𝑀 PROPN iajs-2429 36 25 implies imply VERB iajs-2429 36 26 that that SCONJ iajs-2429 36 27 𝐾 𝐾 PROPN iajs-2429 36 28 𝐻 𝐻 PROPN iajs-2429 36 29 for for ADP iajs-2429 36 30 some some DET iajs-2429 36 31 𝑖 𝑖 SYM iajs-2429 36 32 ∈∧ ∈∧ NOUN iajs-2429 36 33 [ [ X iajs-2429 36 34 2 2 NUM iajs-2429 36 35 ] ] PUNCT iajs-2429 36 36 . . PUNCT iajs-2429 37 1 it it PRON iajs-2429 37 2 is be AUX iajs-2429 37 3 not not PART iajs-2429 37 4 hard hard ADJ iajs-2429 37 5 to to PART iajs-2429 37 6 see see VERB iajs-2429 37 7 that that SCONJ iajs-2429 37 8 every every DET iajs-2429 37 9 submodule submodule NOUN iajs-2429 37 10 is be AUX iajs-2429 37 11 an an DET iajs-2429 37 12 intersection intersection NOUN iajs-2429 37 13 of of ADP iajs-2429 37 14 completely completely ADV iajs-2429 37 15 irreducible irreducible ADJ iajs-2429 37 16 submodules submodule NOUN iajs-2429 37 17 of of ADP iajs-2429 37 18 𝑀 𝑀 PROPN iajs-2429 37 19 consequently consequently ADV iajs-2429 37 20 the the DET iajs-2429 37 21 intersection intersection NOUN iajs-2429 37 22 of of ADP iajs-2429 37 23 all all DET iajs-2429 37 24 completely completely ADV iajs-2429 37 25 irreducible irreducible ADJ iajs-2429 37 26 submodules submodule NOUN iajs-2429 37 27 of of ADP iajs-2429 37 28 𝑀 𝑀 PROPN iajs-2429 37 29 is be AUX iajs-2429 37 30 zero zero NUM iajs-2429 37 31 . . PUNCT iajs-2429 38 1 𝑁 𝑁 PROPN iajs-2429 38 2 is be AUX iajs-2429 38 3 called call VERB iajs-2429 38 4 simple simple ADJ iajs-2429 38 5 ( ( PUNCT iajs-2429 38 6 sometimes sometimes ADV iajs-2429 38 7 minimal minimal ADJ iajs-2429 38 8 ) ) PUNCT iajs-2429 38 9 submodule submodule NOUN iajs-2429 38 10 of of ADP iajs-2429 38 11 a a DET iajs-2429 38 12 module module NOUN iajs-2429 38 13 𝑀 𝑀 NOUN iajs-2429 38 14 if if SCONJ iajs-2429 38 15 𝑁 𝑁 PROPN iajs-2429 38 16 0 0 NUM iajs-2429 38 17 and and CCONJ iajs-2429 38 18 for for ADP iajs-2429 38 19 each each DET iajs-2429 38 20 submodule submodule NOUN iajs-2429 38 21 𝐿 𝐿 PROPN iajs-2429 38 22 of of ADP iajs-2429 38 23 𝑀 𝑀 PROPN iajs-2429 38 24 and and CCONJ iajs-2429 38 25 𝑁 𝑁 PROPN iajs-2429 38 26 contains contain VERB iajs-2429 38 27 𝐿 𝐿 PROPN iajs-2429 38 28 properly properly ADV iajs-2429 38 29 implies imply VERB iajs-2429 38 30 𝐿 𝐿 PROPN iajs-2429 38 31 0 0 PUNCT iajs-2429 39 1 [ [ X iajs-2429 39 2 7 7 NUM iajs-2429 39 3 ] ] PUNCT iajs-2429 39 4 . . PUNCT iajs-2429 40 1 𝑀 𝑀 PROPN iajs-2429 40 2 is be AUX iajs-2429 40 3 coquasi coquasi NOUN iajs-2429 40 4 - - PUNCT iajs-2429 40 5 dedekind dedekind NOUN iajs-2429 40 6 if if SCONJ iajs-2429 40 7 all all DET iajs-2429 40 8 nonzero nonzero NOUN iajs-2429 40 9 endomorphism endomorphism PROPN iajs-2429 40 10 of of ADP iajs-2429 40 11 𝑀 𝑀 PROPN iajs-2429 40 12 is be AUX iajs-2429 40 13 epimorphism epimorphism NOUN iajs-2429 40 14 ( ( PUNCT iajs-2429 40 15 in in ADP iajs-2429 40 16 other other ADJ iajs-2429 40 17 word word NOUN iajs-2429 40 18 , , PUNCT iajs-2429 40 19 𝑓 𝑓 DET iajs-2429 40 20 𝑀 𝑀 PROPN iajs-2429 40 21 𝑀 𝑀 PROPN iajs-2429 40 22 for for ADP iajs-2429 40 23 every every DET iajs-2429 40 24 0 0 NUM iajs-2429 40 25 𝑓 𝑓 PRON iajs-2429 40 26 ∈ ∈ PROPN iajs-2429 40 27 𝑆 𝑆 PROPN iajs-2429 40 28 ) ) PUNCT iajs-2429 41 1 [ [ X iajs-2429 41 2 8 8 NUM iajs-2429 41 3 ] ] PUNCT iajs-2429 41 4 . . PUNCT iajs-2429 42 1 let let VERB iajs-2429 42 2 𝑅 𝑅 NOUN iajs-2429 42 3 be be AUX iajs-2429 42 4 a a DET iajs-2429 42 5 commutative commutative ADJ iajs-2429 42 6 integral integral ADJ iajs-2429 42 7 domain domain NOUN iajs-2429 42 8 , , PUNCT iajs-2429 42 9 𝑀 𝑀 PROPN iajs-2429 42 10 is be AUX iajs-2429 42 11 called call VERB iajs-2429 42 12 divisible divisible ADJ iajs-2429 42 13 module module NOUN iajs-2429 42 14 over over ADP iajs-2429 42 15 𝑅 𝑅 PROPN iajs-2429 42 16 if if SCONJ iajs-2429 42 17 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 42 18 𝑀 𝑀 PROPN iajs-2429 42 19 for for ADP iajs-2429 42 20 each each DET iajs-2429 42 21 0 0 NUM iajs-2429 42 22 𝑎 𝑎 PRON iajs-2429 42 23 ∈ ∈ PROPN iajs-2429 42 24 𝑅 𝑅 PROPN iajs-2429 42 25 [ [ NOUN iajs-2429 42 26 7 7 NUM iajs-2429 42 27 ] ] PUNCT iajs-2429 42 28 . . PUNCT iajs-2429 43 1 a a DET iajs-2429 43 2 proper proper ADJ iajs-2429 43 3 submodule submodule NOUN iajs-2429 43 4 𝑁 𝑁 PROPN iajs-2429 43 5 is be AUX iajs-2429 43 6 maximal maximal ADJ iajs-2429 43 7 if if SCONJ iajs-2429 43 8 it it PRON iajs-2429 43 9 is be AUX iajs-2429 43 10 not not PART iajs-2429 43 11 properly properly ADV iajs-2429 43 12 contained contain VERB iajs-2429 43 13 in in ADP iajs-2429 43 14 any any DET iajs-2429 43 15 proper proper ADJ iajs-2429 43 16 submodule submodule NOUN iajs-2429 43 17 of of ADP iajs-2429 43 18 𝑀 𝑀 PROPN iajs-2429 43 19 [ [ X iajs-2429 43 20 7 7 NUM iajs-2429 43 21 ] ] PUNCT iajs-2429 43 22 . . PUNCT iajs-2429 44 1 a a DET iajs-2429 44 2 proper proper ADJ iajs-2429 44 3 submodule submodule NOUN iajs-2429 44 4 𝑁 𝑁 PROPN iajs-2429 44 5 is be AUX iajs-2429 44 6 called call VERB iajs-2429 44 7 prime prime ADJ iajs-2429 44 8 if if SCONJ iajs-2429 44 9 𝑚𝑟 𝑚𝑟 PROPN iajs-2429 44 10 ∈ ∈ PROPN iajs-2429 44 11 𝑁 𝑁 PROPN iajs-2429 44 12 implies imply VERB iajs-2429 44 13 𝑚 𝑚 ADP iajs-2429 44 14 ∈ ∈ NOUN iajs-2429 44 15 𝑁 𝑁 PROPN iajs-2429 44 16 or or CCONJ iajs-2429 44 17 𝑀𝑟 𝑀𝑟 PROPN iajs-2429 44 18 ⊆ ⊆ NUM iajs-2429 44 19 𝑁 𝑁 PROPN iajs-2429 45 1 [ [ NOUN iajs-2429 45 2 9 9 NUM iajs-2429 45 3 ] ] PUNCT iajs-2429 45 4 . . PUNCT iajs-2429 46 1 𝑀 𝑀 PROPN iajs-2429 46 2 is be AUX iajs-2429 46 3 called call VERB iajs-2429 46 4 a a DET iajs-2429 46 5 prime prime ADJ iajs-2429 46 6 module module NOUN iajs-2429 46 7 if if SCONJ iajs-2429 46 8 the the DET iajs-2429 46 9 zero zero NUM iajs-2429 46 10 submodule submodule NOUN iajs-2429 46 11 is be AUX iajs-2429 46 12 prime prime ADJ iajs-2429 46 13 . . PUNCT iajs-2429 47 1 a a DET iajs-2429 47 2 proper proper ADJ iajs-2429 47 3 ideal ideal NOUN iajs-2429 47 4 𝐼 𝐼 PROPN iajs-2429 47 5 is be AUX iajs-2429 47 6 prime prime ADJ iajs-2429 47 7 if if SCONJ iajs-2429 47 8 𝑎𝑏 𝑎𝑏 PROPN iajs-2429 47 9 ∈ ∈ PROPN iajs-2429 47 10 𝐼 𝐼 ADP iajs-2429 47 11 where where SCONJ iajs-2429 47 12 𝑎 𝑎 X iajs-2429 47 13 , , PUNCT iajs-2429 47 14 𝑏 𝑏 PROPN iajs-2429 47 15 ∈ ∈ PROPN iajs-2429 47 16 𝑅 𝑅 PROPN iajs-2429 47 17 implies imply VERB iajs-2429 47 18 𝑎 𝑎 DET iajs-2429 47 19 ∈ ∈ ADJ iajs-2429 47 20 𝐼 𝐼 NOUN iajs-2429 47 21 or or CCONJ iajs-2429 47 22 𝑏 𝑏 PRON iajs-2429 47 23 ∈ ∈ NOUN iajs-2429 47 24 𝐼 𝐼 ADP iajs-2429 47 25 [ [ X iajs-2429 47 26 10 10 NUM iajs-2429 47 27 ] ] PUNCT iajs-2429 47 28 . . PUNCT iajs-2429 48 1 equivalently equivalently ADV iajs-2429 48 2 , , PUNCT iajs-2429 48 3 a a DET iajs-2429 48 4 proper proper ADJ iajs-2429 48 5 ideal ideal NOUN iajs-2429 48 6 𝐼 𝐼 PROPN iajs-2429 48 7 is be AUX iajs-2429 48 8 prime prime ADJ iajs-2429 48 9 if if SCONJ iajs-2429 48 10 𝐴𝐵 𝐴𝐵 PROPN iajs-2429 48 11 ⊆ ⊆ NUM iajs-2429 48 12 𝐼 𝐼 ADP iajs-2429 48 13 where where SCONJ iajs-2429 48 14 𝐴 𝐴 PROPN iajs-2429 48 15 and and CCONJ iajs-2429 48 16 𝐵 𝐵 NOUN iajs-2429 48 17 are be AUX iajs-2429 48 18 ideals ideal NOUN iajs-2429 48 19 of of ADP iajs-2429 48 20 𝑅 𝑅 PROPN iajs-2429 48 21 implies imply VERB iajs-2429 48 22 𝐴 𝐴 PROPN iajs-2429 48 23 ⊆ ⊆ NUM iajs-2429 48 24 𝐼 𝐼 PROPN iajs-2429 48 25 or or CCONJ iajs-2429 48 26 𝐵 𝐵 NOUN iajs-2429 48 27 ⊆ ⊆ NUM iajs-2429 48 28 𝐼 𝐼 PROPN iajs-2429 48 29 [ [ X iajs-2429 48 30 10 10 NUM iajs-2429 48 31 ] ] PUNCT iajs-2429 48 32 . . PUNCT iajs-2429 49 1 a a DET iajs-2429 49 2 ring ring NOUN iajs-2429 49 3 in in ADP iajs-2429 49 4 which which PRON iajs-2429 49 5 every every DET iajs-2429 49 6 ideal ideal ADJ iajs-2429 49 7 prime prime NOUN iajs-2429 49 8 is be AUX iajs-2429 49 9 called call VERB iajs-2429 49 10 fully fully ADV iajs-2429 49 11 prime prime ADJ iajs-2429 49 12 [ [ X iajs-2429 49 13 11 11 NUM iajs-2429 49 14 ] ] PUNCT iajs-2429 49 15 . . PUNCT iajs-2429 50 1 equivalently equivalently ADV iajs-2429 50 2 , , PUNCT iajs-2429 50 3 a a DET iajs-2429 50 4 ring ring NOUN iajs-2429 50 5 𝑅 𝑅 PROPN iajs-2429 50 6 is be AUX iajs-2429 50 7 fully fully ADV iajs-2429 50 8 prime prime ADJ iajs-2429 50 9 if if SCONJ iajs-2429 51 1 and and CCONJ iajs-2429 51 2 only only ADV iajs-2429 51 3 if if SCONJ iajs-2429 51 4 it it PRON iajs-2429 51 5 is be AUX iajs-2429 51 6 fully fully ADV iajs-2429 51 7 idempotent idempotent ADJ iajs-2429 51 8 ( ( PUNCT iajs-2429 51 9 a a DET iajs-2429 51 10 ring ring NOUN iajs-2429 51 11 in in ADP iajs-2429 51 12 which which PRON iajs-2429 51 13 every every DET iajs-2429 51 14 ideal ideal NOUN iajs-2429 51 15 is be AUX iajs-2429 51 16 an an DET iajs-2429 51 17 idempotent idempotent NOUN iajs-2429 51 18 that that PRON iajs-2429 51 19 is be AUX iajs-2429 51 20 𝐼 𝐼 ADP iajs-2429 51 21 𝐼 𝐼 PROPN iajs-2429 51 22 for for ADP iajs-2429 51 23 each each DET iajs-2429 51 24 ideal ideal NOUN iajs-2429 51 25 𝐼 𝐼 PROPN iajs-2429 51 26 of of ADP iajs-2429 51 27 ) ) PUNCT iajs-2429 51 28 and and CCONJ iajs-2429 51 29 the the DET iajs-2429 51 30 set set NOUN iajs-2429 51 31 of of ADP iajs-2429 51 32 ideals ideal NOUN iajs-2429 51 33 of of ADP iajs-2429 51 34 𝑅 𝑅 PROPN iajs-2429 51 35 is be AUX iajs-2429 51 36 totally totally ADV iajs-2429 51 37 ordered order VERB iajs-2429 51 38 under under ADP iajs-2429 51 39 inclusion inclusion NOUN iajs-2429 51 40 [ [ X iajs-2429 51 41 11 11 NUM iajs-2429 51 42 ] ] PUNCT iajs-2429 51 43 . . PUNCT iajs-2429 52 1 a a DET iajs-2429 52 2 proper proper ADJ iajs-2429 52 3 submodule submodule NOUN iajs-2429 52 4 𝑁 𝑁 PROPN iajs-2429 52 5 is be AUX iajs-2429 52 6 called call VERB iajs-2429 52 7 primary primary ADJ iajs-2429 52 8 if if SCONJ iajs-2429 52 9 𝑚𝑟 𝑚𝑟 PROPN iajs-2429 52 10 ∈ ∈ PROPN iajs-2429 52 11 𝑁 𝑁 PROPN iajs-2429 52 12 implies imply VERB iajs-2429 52 13 𝑚 𝑚 ADP iajs-2429 52 14 ∈ ∈ NOUN iajs-2429 52 15 𝑁 𝑁 PROPN iajs-2429 52 16 or or CCONJ iajs-2429 52 17 𝑀𝑟 𝑀𝑟 PROPN iajs-2429 52 18 ⊆ ⊆ NUM iajs-2429 52 19 𝑁 𝑁 PROPN iajs-2429 52 20 for for ADP iajs-2429 52 21 some some DET iajs-2429 52 22 positive positive ADJ iajs-2429 52 23 integer integer NOUN iajs-2429 52 24 𝑡 𝑡 PROPN iajs-2429 52 25 [ [ X iajs-2429 52 26 6 6 NUM iajs-2429 52 27 ] ] PUNCT iajs-2429 52 28 . . PUNCT iajs-2429 53 1 𝑀 𝑀 PROPN iajs-2429 53 2 is be AUX iajs-2429 53 3 called call VERB iajs-2429 53 4 a a DET iajs-2429 53 5 primary primary ADJ iajs-2429 53 6 module module NOUN iajs-2429 53 7 if if SCONJ iajs-2429 53 8 the the DET iajs-2429 53 9 zero zero NUM iajs-2429 53 10 submodule submodule NOUN iajs-2429 53 11 is be AUX iajs-2429 53 12 primary primary ADJ iajs-2429 53 13 . . PUNCT iajs-2429 54 1 a a DET iajs-2429 54 2 proper proper ADJ iajs-2429 54 3 ideal ideal NOUN iajs-2429 54 4 𝐼 𝐼 PROPN iajs-2429 54 5 is be AUX iajs-2429 54 6 primary primary ADJ iajs-2429 54 7 if if SCONJ iajs-2429 54 8 𝑎𝑏 𝑎𝑏 PROPN iajs-2429 54 9 ∈ ∈ PROPN iajs-2429 54 10 𝐼 𝐼 ADP iajs-2429 54 11 where where SCONJ iajs-2429 54 12 𝑎 𝑎 X iajs-2429 54 13 , , PUNCT iajs-2429 54 14 𝑏 𝑏 PROPN iajs-2429 54 15 ∈ ∈ PROPN iajs-2429 54 16 𝑅 𝑅 PROPN iajs-2429 54 17 implies imply VERB iajs-2429 54 18 𝑎 𝑎 DET iajs-2429 54 19 ∈ ∈ ADJ iajs-2429 54 20 𝐼 𝐼 NOUN iajs-2429 54 21 or or CCONJ iajs-2429 54 22 𝑏 𝑏 DET iajs-2429 54 23 ∈ ∈ NOUN iajs-2429 54 24 𝐼 𝐼 ADP iajs-2429 54 25 for for ADP iajs-2429 54 26 some some DET iajs-2429 54 27 positive positive ADJ iajs-2429 54 28 integer integer NOUN iajs-2429 54 29 𝑡 𝑡 PROPN iajs-2429 55 1 [ [ X iajs-2429 55 2 6 6 NUM iajs-2429 55 3 ] ] PUNCT iajs-2429 55 4 . . PUNCT iajs-2429 55 5 0 0 NUM iajs-2429 56 1 𝑀 𝑀 PROPN iajs-2429 56 2 is be AUX iajs-2429 56 3 called call VERB iajs-2429 56 4 an an DET iajs-2429 56 5 𝑆-second 𝑆-second PROPN iajs-2429 56 6 module module NOUN iajs-2429 56 7 if if SCONJ iajs-2429 56 8 for for SCONJ iajs-2429 56 9 every every DET iajs-2429 56 10 𝑓 𝑓 PROPN iajs-2429 56 11 ∈ ∈ PROPN iajs-2429 56 12 𝑆 𝑆 PROPN iajs-2429 56 13 implies imply VERB iajs-2429 56 14 𝑓 𝑓 DET iajs-2429 56 15 𝑀 𝑀 PROPN iajs-2429 56 16 𝑀 𝑀 PROPN iajs-2429 56 17 or or CCONJ iajs-2429 56 18 𝑓 𝑓 DET iajs-2429 56 19 𝑀 𝑀 PROPN iajs-2429 56 20 0 0 PUNCT iajs-2429 57 1 [ [ X iajs-2429 57 2 12 12 NUM iajs-2429 57 3 ] ] PUNCT iajs-2429 57 4 . . PUNCT iajs-2429 57 5 0 0 NUM iajs-2429 58 1 𝑀 𝑀 PROPN iajs-2429 58 2 is be AUX iajs-2429 58 3 called call VERB iajs-2429 58 4 an an DET iajs-2429 58 5 s s NOUN iajs-2429 58 6 - - PUNCT iajs-2429 58 7 weakly weakly ADJ iajs-2429 58 8 second second ADJ iajs-2429 58 9 module module NOUN iajs-2429 58 10 whenever whenever SCONJ iajs-2429 58 11 𝑓𝑔 𝑓𝑔 VERB iajs-2429 58 12 𝑀 𝑀 PROPN iajs-2429 58 13 ⊆ ⊆ PROPN iajs-2429 58 14 𝐾 𝐾 PROPN iajs-2429 58 15 , , PUNCT iajs-2429 58 16 where where SCONJ iajs-2429 58 17 𝑓 𝑓 X iajs-2429 58 18 , , PUNCT iajs-2429 58 19 𝑔 𝑔 PROPN iajs-2429 58 20 ∈ ∈ PROPN iajs-2429 58 21 𝑆 𝑆 PROPN iajs-2429 58 22 and and CCONJ iajs-2429 58 23 𝐾 𝐾 PROPN iajs-2429 58 24 a a DET iajs-2429 58 25 submodule submodule NOUN iajs-2429 58 26 of of ADP iajs-2429 58 27 𝑀 𝑀 PROPN iajs-2429 58 28 implies imply VERB iajs-2429 58 29 either either CCONJ iajs-2429 58 30 𝑓 𝑓 PRON iajs-2429 58 31 𝑀 𝑀 PROPN iajs-2429 58 32 ⊆ ⊆ NUM iajs-2429 58 33 𝐾 𝐾 PROPN iajs-2429 58 34 or or CCONJ iajs-2429 58 35 𝑔 𝑔 PROPN iajs-2429 58 36 𝑀 𝑀 PROPN iajs-2429 58 37 ⊆ ⊆ NUM iajs-2429 58 38 𝐾 𝐾 PROPN iajs-2429 59 1 [ [ X iajs-2429 59 2 3 3 NUM iajs-2429 59 3 ] ] PUNCT iajs-2429 59 4 . . PUNCT iajs-2429 60 1 equivalently equivalently ADV iajs-2429 60 2 , , PUNCT iajs-2429 60 3 𝑀 𝑀 PROPN iajs-2429 60 4 is be AUX iajs-2429 60 5 an an DET iajs-2429 60 6 s s NOUN iajs-2429 60 7 - - PUNCT iajs-2429 60 8 weakly weakly ADJ iajs-2429 60 9 second second ADJ iajs-2429 60 10 module module NOUN iajs-2429 60 11 if if SCONJ iajs-2429 60 12 and and CCONJ iajs-2429 60 13 only only ADV iajs-2429 60 14 if if SCONJ iajs-2429 60 15 for for ADP iajs-2429 60 16 each each DET iajs-2429 60 17 𝜁 𝜁 NOUN iajs-2429 60 18 , , PUNCT iajs-2429 60 19 𝜗 𝜗 PROPN iajs-2429 60 20 ∈ ∈ PROPN iajs-2429 60 21 𝑆 𝑆 PROPN iajs-2429 60 22 implies imply VERB iajs-2429 60 23 𝜁𝜗 𝜁𝜗 PROPN iajs-2429 60 24 𝑀 𝑀 PROPN iajs-2429 60 25 𝜁 𝜁 PROPN iajs-2429 60 26 𝑀 𝑀 PROPN iajs-2429 60 27 or or CCONJ iajs-2429 60 28 𝜁𝜗 𝜁𝜗 PROPN iajs-2429 60 29 𝑀 𝑀 PROPN iajs-2429 60 30 ⊇ ⊇ NOUN iajs-2429 60 31 𝜗 𝜗 X iajs-2429 60 32 𝑀 𝑀 PROPN iajs-2429 61 1 [ [ X iajs-2429 61 2 3 3 NUM iajs-2429 61 3 ] ] PUNCT iajs-2429 61 4 . . PUNCT iajs-2429 62 1 𝑀 𝑀 PROPN iajs-2429 62 2 is be AUX iajs-2429 62 3 called call VERB iajs-2429 62 4 multiplication multiplication NOUN iajs-2429 62 5 when when SCONJ iajs-2429 62 6 each each DET iajs-2429 62 7 submodule submodule NOUN iajs-2429 62 8 𝑁 𝑁 PROPN iajs-2429 62 9 of of ADP iajs-2429 62 10 𝑀 𝑀 PROPN iajs-2429 62 11 , , PUNCT iajs-2429 62 12 we we PRON iajs-2429 62 13 have have VERB iajs-2429 62 14 𝑁 𝑁 PROPN iajs-2429 62 15 𝑀𝐼 𝑀𝐼 PROPN iajs-2429 62 16 for for ADP iajs-2429 62 17 some some DET iajs-2429 62 18 ideal ideal ADJ iajs-2429 62 19 𝐼 𝐼 PROPN iajs-2429 62 20 of of ADP iajs-2429 62 21 𝑅 𝑅 PROPN iajs-2429 63 1 [ [ PROPN iajs-2429 63 2 13 13 NUM iajs-2429 63 3 ] ] PUNCT iajs-2429 63 4 . . PUNCT iajs-2429 64 1 we we PRON iajs-2429 64 2 able able ADJ iajs-2429 64 3 to to PART iajs-2429 64 4 take take VERB iajs-2429 64 5 𝐼 𝐼 PRON iajs-2429 64 6 𝑁 𝑁 PROPN iajs-2429 64 7 : : PUNCT iajs-2429 64 8 𝑀 𝑀 PROPN iajs-2429 64 9 𝑟 𝑟 NOUN iajs-2429 64 10 ∈ ∈ PROPN iajs-2429 64 11 𝑅 𝑅 PROPN iajs-2429 64 12 and and CCONJ iajs-2429 64 13 𝑀𝑟 𝑀𝑟 PROPN iajs-2429 64 14 ⊆ ⊆ PROPN iajs-2429 64 15 𝑁 𝑁 PROPN iajs-2429 64 16 is be AUX iajs-2429 64 17 an an DET iajs-2429 64 18 ideal ideal NOUN iajs-2429 64 19 of of ADP iajs-2429 64 20 𝑅 𝑅 PROPN iajs-2429 64 21 [ [ PROPN iajs-2429 64 22 13 13 NUM iajs-2429 64 23 ] ] PUNCT iajs-2429 64 24 . . PUNCT iajs-2429 65 1 𝑀 𝑀 PROPN iajs-2429 65 2 is be AUX iajs-2429 65 3 called call VERB iajs-2429 65 4 faithful faithful ADJ iajs-2429 65 5 if if SCONJ iajs-2429 65 6 0 0 NUM iajs-2429 65 7 : : PUNCT iajs-2429 65 8 𝑀 𝑀 PROPN iajs-2429 65 9 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 65 10 𝑀 𝑀 PROPN iajs-2429 65 11 𝑟 𝑟 PRON iajs-2429 65 12 ∈ ∈ PROPN iajs-2429 65 13 𝑅 𝑅 PROPN iajs-2429 65 14 and and CCONJ iajs-2429 65 15 𝑀𝑟 𝑀𝑟 PROPN iajs-2429 65 16 0 0 NUM iajs-2429 65 17 0 0 NUM iajs-2429 65 18 . . PUNCT iajs-2429 66 1 𝑀 𝑀 PROPN iajs-2429 66 2 is be AUX iajs-2429 66 3 a a DET iajs-2429 66 4 scalar scalar ADJ iajs-2429 66 5 module module NOUN iajs-2429 66 6 when when SCONJ iajs-2429 66 7 for for ADP iajs-2429 66 8 each each DET iajs-2429 66 9 𝑓 𝑓 DET iajs-2429 66 10 ∈ ∈ PROPN iajs-2429 66 11 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2429 66 12 𝑀 𝑀 PROPN iajs-2429 66 13 there there PRON iajs-2429 66 14 is be VERB iajs-2429 66 15 𝑎 𝑎 PRON iajs-2429 66 16 ∈ ∈ PROPN iajs-2429 66 17 𝑅 𝑅 PROPN iajs-2429 66 18 with with ADP iajs-2429 66 19 𝑓 𝑓 DET iajs-2429 66 20 𝑚 𝑚 NOUN iajs-2429 66 21 𝑚𝑎 𝑚𝑎 ADV iajs-2429 66 22 for for ADP iajs-2429 66 23 all all DET iajs-2429 66 24 𝑚 𝑚 ADP iajs-2429 66 25 ∈ ∈ PROPN iajs-2429 66 26 𝑀 𝑀 PROPN iajs-2429 67 1 [ [ X iajs-2429 67 2 14 14 NUM iajs-2429 67 3 ] ] PUNCT iajs-2429 67 4 . . PUNCT iajs-2429 68 1 the the DET iajs-2429 68 2 aim aim NOUN iajs-2429 68 3 of of ADP iajs-2429 68 4 this this DET iajs-2429 68 5 research research NOUN iajs-2429 68 6 is be AUX iajs-2429 68 7 to to PART iajs-2429 68 8 continue continue VERB iajs-2429 68 9 studying study VERB iajs-2429 68 10 the the DET iajs-2429 68 11 concept concept NOUN iajs-2429 68 12 of of ADP iajs-2429 68 13 semisecond semisecond ADJ iajs-2429 68 14 submodules submodule NOUN iajs-2429 68 15 . . PUNCT iajs-2429 69 1 a a DET iajs-2429 69 2 nonzero nonzero PROPN iajs-2429 69 3 submodule submodule NOUN iajs-2429 69 4 𝑁 𝑁 PROPN iajs-2429 69 5 of of ADP iajs-2429 69 6 𝑀 𝑀 PROPN iajs-2429 69 7 is be AUX iajs-2429 69 8 called call VERB iajs-2429 69 9 semisecond semisecond ADJ iajs-2429 69 10 if if SCONJ iajs-2429 69 11 for for ADP iajs-2429 69 12 each each DET iajs-2429 69 13 𝑎 𝑎 PROPN iajs-2429 69 14 ∈ ∈ PROPN iajs-2429 69 15 𝑅 𝑅 PROPN iajs-2429 69 16 , , PUNCT iajs-2429 69 17 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 69 18 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 70 1 [ [ X iajs-2429 70 2 2 2 NUM iajs-2429 70 3 ] ] PUNCT iajs-2429 70 4 . . PUNCT iajs-2429 71 1 a a DET iajs-2429 71 2 nonzero nonzero NOUN iajs-2429 71 3 module module NOUN iajs-2429 71 4 𝑀 𝑀 PROPN iajs-2429 71 5 is be AUX iajs-2429 71 6 said say VERB iajs-2429 71 7 to to PART iajs-2429 71 8 be be AUX iajs-2429 71 9 semisecond semisecond ADJ iajs-2429 71 10 if if SCONJ iajs-2429 71 11 𝑀 𝑀 PROPN iajs-2429 71 12 is be AUX iajs-2429 71 13 semisecond semisecond ADJ iajs-2429 71 14 submodule submodule NOUN iajs-2429 71 15 of of ADP iajs-2429 71 16 itself itself PRON iajs-2429 71 17 . . PUNCT iajs-2429 72 1 in in ADP iajs-2429 72 2 fact fact NOUN iajs-2429 72 3 this this DET iajs-2429 72 4 idea idea NOUN iajs-2429 72 5 is be AUX iajs-2429 72 6 the the DET iajs-2429 72 7 dual dual ADJ iajs-2429 72 8 notion notion NOUN iajs-2429 72 9 of of ADP iajs-2429 72 10 the the DET iajs-2429 72 11 concept concept NOUN iajs-2429 72 12 semiprime semiprime NOUN iajs-2429 72 13 submodules submodule NOUN iajs-2429 72 14 . . PUNCT iajs-2429 73 1 a a DET iajs-2429 73 2 proper proper ADJ iajs-2429 73 3 submodule submodule NOUN iajs-2429 73 4 of of ADP iajs-2429 73 5 𝑀 𝑀 PROPN iajs-2429 73 6 is be AUX iajs-2429 73 7 called call VERB iajs-2429 73 8 semiprime semiprime NOUN iajs-2429 73 9 if if SCONJ iajs-2429 73 10 for for ADP iajs-2429 73 11 each each DET iajs-2429 73 12 𝑎 𝑎 PROPN iajs-2429 73 13 ∈ ∈ PROPN iajs-2429 73 14 𝑅 𝑅 PROPN iajs-2429 73 15 , , PUNCT iajs-2429 73 16 𝑚 𝑚 PROPN iajs-2429 73 17 ∈ ∈ PROPN iajs-2429 73 18 𝑀 𝑀 PROPN iajs-2429 74 1 such such ADJ iajs-2429 74 2 that that SCONJ iajs-2429 74 3 𝑚𝑎 𝑚𝑎 ADP iajs-2429 74 4 ∈ ∈ NOUN iajs-2429 74 5 𝑁 𝑁 PROPN iajs-2429 74 6 implies imply VERB iajs-2429 74 7 𝑚𝑎 𝑚𝑎 NOUN iajs-2429 74 8 ∈ ∈ NOUN iajs-2429 74 9 𝑁 𝑁 PROPN iajs-2429 74 10 [ [ NOUN iajs-2429 74 11 9 9 NUM iajs-2429 74 12 ] ] PUNCT iajs-2429 74 13 . . PUNCT iajs-2429 75 1 a a DET iajs-2429 75 2 proper proper ADJ iajs-2429 75 3 ideal ideal ADJ iajs-2429 75 4 𝐼 𝐼 PROPN iajs-2429 75 5 of of ADP iajs-2429 75 6 𝑅 𝑅 PROPN iajs-2429 75 7 is be AUX iajs-2429 75 8 semiprime semiprime NOUN iajs-2429 75 9 if if SCONJ iajs-2429 75 10 for for ADP iajs-2429 75 11 each each DET iajs-2429 75 12 𝑎 𝑎 PRON iajs-2429 75 13 ∈ ∈ PROPN iajs-2429 75 14 𝑅 𝑅 PROPN iajs-2429 75 15 such such ADJ iajs-2429 75 16 that that SCONJ iajs-2429 75 17 𝑎 𝑎 ADJ iajs-2429 75 18 ∈ ∈ NOUN iajs-2429 75 19 𝐼 𝐼 PROPN iajs-2429 75 20 implies imply VERB iajs-2429 75 21 𝑎 𝑎 PRON iajs-2429 75 22 ∈ ∈ NOUN iajs-2429 75 23 𝐼 𝐼 ADP iajs-2429 75 24 [ [ X iajs-2429 75 25 7 7 NUM iajs-2429 75 26 ] ] X iajs-2429 75 27 . . PUNCT iajs-2429 76 1 equivalently equivalently ADV iajs-2429 76 2 , , PUNCT iajs-2429 76 3 a a DET iajs-2429 76 4 proper proper ADJ iajs-2429 76 5 ideal ideal ADJ iajs-2429 76 6 𝐼 𝐼 PROPN iajs-2429 76 7 of of ADP iajs-2429 76 8 𝑅 𝑅 PROPN iajs-2429 76 9 is be AUX iajs-2429 76 10 semiprime semiprime NOUN iajs-2429 76 11 if if SCONJ iajs-2429 76 12 for for ADP iajs-2429 76 13 each each DET iajs-2429 76 14 ideal ideal ADJ iajs-2429 76 15 𝐴 𝐴 PROPN iajs-2429 76 16 of of ADP iajs-2429 76 17 𝑅 𝑅 PROPN iajs-2429 76 18 such such ADJ iajs-2429 76 19 that that SCONJ iajs-2429 76 20 𝐴 𝐴 PROPN iajs-2429 76 21 ⊆ ⊆ NUM iajs-2429 76 22 𝐼 𝐼 PROPN iajs-2429 76 23 implies imply VERB iajs-2429 76 24 𝐴 𝐴 PROPN iajs-2429 76 25 ⊆ ⊆ NUM iajs-2429 76 26 𝐼 𝐼 PROPN iajs-2429 76 27 [ [ X iajs-2429 76 28 7 7 NUM iajs-2429 76 29 ] ] PUNCT iajs-2429 76 30 . . PUNCT iajs-2429 77 1 it it PRON iajs-2429 77 2 is be AUX iajs-2429 77 3 well well ADV iajs-2429 77 4 - - PUNCT iajs-2429 77 5 known know VERB iajs-2429 77 6 that that SCONJ iajs-2429 77 7 𝑅 𝑅 PROPN iajs-2429 77 8 is be AUX iajs-2429 77 9 fully fully ADV iajs-2429 77 10 semiprime semiprime NOUN iajs-2429 77 11 ( ( PUNCT iajs-2429 77 12 that that PRON iajs-2429 77 13 is be AUX iajs-2429 77 14 𝑅 𝑅 PROPN iajs-2429 77 15 in in ADP iajs-2429 77 16 which which PRON iajs-2429 77 17 every every DET iajs-2429 77 18 ideal ideal NOUN iajs-2429 77 19 is be AUX iajs-2429 77 20     SPACE iajs-2429 77 21 83 83 NUM iajs-2429 77 22     SPACE iajs-2429 77 23 ibn ibn PROPN iajs-2429 77 24 al al PROPN iajs-2429 77 25 - - PUNCT iajs-2429 77 26 haitham haitham PROPN iajs-2429 77 27 jour jour X iajs-2429 77 28 . . PROPN iajs-2429 78 1 for for ADP iajs-2429 78 2 pure pure ADJ iajs-2429 78 3 & & CCONJ iajs-2429 78 4 appl appl PROPN iajs-2429 78 5 . . PUNCT iajs-2429 79 1 sci sci PROPN iajs-2429 79 2 . . PROPN iajs-2429 80 1 33 33 NUM iajs-2429 80 2 ( ( PUNCT iajs-2429 80 3 2 2 NUM iajs-2429 80 4 ) ) PUNCT iajs-2429 80 5 2020 2020 NUM iajs-2429 80 6 semiprime semiprime NOUN iajs-2429 80 7 ) ) PUNCT iajs-2429 81 1 if if SCONJ iajs-2429 81 2 and and CCONJ iajs-2429 81 3 only only ADV iajs-2429 81 4 if if SCONJ iajs-2429 81 5 𝑅 𝑅 PROPN iajs-2429 81 6 is be AUX iajs-2429 81 7 von von PROPN iajs-2429 81 8 neumann neumann PROPN iajs-2429 81 9 regular regular PROPN iajs-2429 81 10 ( ( PUNCT iajs-2429 81 11 that that PRON iajs-2429 81 12 is be AUX iajs-2429 81 13 for for ADP iajs-2429 81 14 every every DET iajs-2429 81 15 𝑎 𝑎 PROPN iajs-2429 81 16 ∈ ∈ PROPN iajs-2429 81 17 𝑅 𝑅 PROPN iajs-2429 81 18 , , PUNCT iajs-2429 81 19 there there PRON iajs-2429 81 20 is be VERB iajs-2429 81 21 𝑏 𝑏 DET iajs-2429 81 22 ∈ ∈ PROPN iajs-2429 81 23 𝑅 𝑅 PROPN iajs-2429 81 24 such such ADJ iajs-2429 81 25 that that SCONJ iajs-2429 81 26 𝑎 𝑎 PROPN iajs-2429 81 27 𝑎𝑏𝑎 𝑎𝑏𝑎 NOUN iajs-2429 81 28 ) ) PUNCT iajs-2429 82 1 [ [ X iajs-2429 82 2 15 15 NUM iajs-2429 82 3 ] ] PUNCT iajs-2429 82 4 . . PUNCT iajs-2429 83 1 it it PRON iajs-2429 83 2 is be AUX iajs-2429 83 3 well well ADV iajs-2429 83 4 - - PUNCT iajs-2429 83 5 known know VERB iajs-2429 83 6 if if SCONJ iajs-2429 83 7 𝑅 𝑅 PROPN iajs-2429 83 8 is be AUX iajs-2429 83 9 commutative commutative ADJ iajs-2429 83 10 then then ADV iajs-2429 83 11 𝑅 𝑅 PROPN iajs-2429 83 12 is be AUX iajs-2429 83 13 von von PROPN iajs-2429 83 14 neumann neumann PROPN iajs-2429 83 15 regular regular PROPN iajs-2429 84 1 if if SCONJ iajs-2429 84 2 and and CCONJ iajs-2429 84 3 only only ADV iajs-2429 84 4 if if SCONJ iajs-2429 84 5 𝑎𝑅 𝑎𝑅 PROPN iajs-2429 84 6 𝑎 𝑎 PROPN iajs-2429 84 7 𝑅 𝑅 PROPN iajs-2429 84 8 if if SCONJ iajs-2429 84 9 and and CCONJ iajs-2429 84 10 only only ADV iajs-2429 84 11 if if SCONJ iajs-2429 84 12 every every DET iajs-2429 84 13 ideal ideal NOUN iajs-2429 84 14 of of ADP iajs-2429 84 15 𝑅 𝑅 PROPN iajs-2429 84 16 is be AUX iajs-2429 84 17 pure pure ADJ iajs-2429 84 18 ( ( PUNCT iajs-2429 84 19 that that PRON iajs-2429 84 20 is be AUX iajs-2429 84 21 𝐼 𝐼 ADP iajs-2429 84 22 ∩ ∩ ADJ iajs-2429 84 23 𝐽 𝐽 PROPN iajs-2429 84 24 𝐼𝐽 𝐼𝐽 NOUN iajs-2429 84 25 for for ADP iajs-2429 84 26 each each DET iajs-2429 84 27 ideal ideal ADJ iajs-2429 84 28 𝐼 𝐼 PROPN iajs-2429 84 29 and and CCONJ iajs-2429 84 30 𝐽 𝐽 PROPN iajs-2429 84 31 of of ADP iajs-2429 84 32 𝑅 𝑅 PROPN iajs-2429 84 33 ) ) PUNCT iajs-2429 84 34 if if SCONJ iajs-2429 84 35 and and CCONJ iajs-2429 84 36 only only ADV iajs-2429 84 37 if if SCONJ iajs-2429 84 38 𝑅 𝑅 PROPN iajs-2429 84 39 is be AUX iajs-2429 84 40 fully fully ADV iajs-2429 84 41 idempotent idempotent ADJ iajs-2429 84 42 . . PUNCT iajs-2429 85 1 and and CCONJ iajs-2429 85 2 𝑀 𝑀 PROPN iajs-2429 85 3 is be AUX iajs-2429 85 4 called call VERB iajs-2429 85 5 regular regular ADV iajs-2429 85 6 if if SCONJ iajs-2429 85 7 for for ADP iajs-2429 85 8 every every DET iajs-2429 85 9 𝑚 𝑚 PROPN iajs-2429 85 10 ∈ ∈ PROPN iajs-2429 85 11 𝑀 𝑀 PROPN iajs-2429 85 12 and and CCONJ iajs-2429 85 13 for for ADP iajs-2429 85 14 every every DET iajs-2429 85 15 𝑎 𝑎 PRON iajs-2429 85 16 ∈ ∈ PROPN iajs-2429 85 17 𝑅 𝑅 NOUN iajs-2429 85 18 we we PRON iajs-2429 85 19 have have VERB iajs-2429 85 20 𝑚𝑎 𝑚𝑎 NOUN iajs-2429 85 21 𝑚𝑎𝑟𝑎 𝑚𝑎𝑟𝑎 NOUN iajs-2429 85 22 for for ADP iajs-2429 85 23 some some DET iajs-2429 85 24 𝑟 𝑟 NOUN iajs-2429 85 25 ∈ ∈ NOUN iajs-2429 85 26 𝑅. 𝑅. ADV iajs-2429 85 27 if if SCONJ iajs-2429 85 28 𝑀 𝑀 PROPN iajs-2429 85 29 is be AUX iajs-2429 85 30 regular regular ADJ iajs-2429 85 31 then then ADV iajs-2429 85 32 every every DET iajs-2429 85 33 submodule submodule NOUN iajs-2429 85 34 of of ADP iajs-2429 85 35 𝑀 𝑀 PROPN iajs-2429 85 36 is be AUX iajs-2429 85 37 pure pure ADJ iajs-2429 85 38 ( ( PUNCT iajs-2429 85 39 that that PRON iajs-2429 85 40 is be AUX iajs-2429 85 41 every every DET iajs-2429 85 42 submodule submodule NOUN iajs-2429 85 43 𝑁 𝑁 PROPN iajs-2429 85 44 of of ADP iajs-2429 85 45 𝑀 𝑀 PROPN iajs-2429 85 46 satisfying satisfy VERB iajs-2429 85 47 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 85 48 𝑀𝐼 𝑀𝐼 PROPN iajs-2429 85 49 ∩ ∩ NOUN iajs-2429 85 50 𝑁 𝑁 PROPN iajs-2429 85 51 for for ADP iajs-2429 85 52 each each DET iajs-2429 85 53 ideal ideal ADJ iajs-2429 85 54 𝐼 𝐼 PROPN iajs-2429 85 55 of of ADP iajs-2429 85 56 𝑅 𝑅 PROPN iajs-2429 85 57 ) ) PUNCT iajs-2429 86 1 [ [ X iajs-2429 86 2 15 15 NUM iajs-2429 86 3 ] ] PUNCT iajs-2429 86 4 . . PUNCT iajs-2429 87 1 if if SCONJ iajs-2429 87 2 𝑅 𝑅 PROPN iajs-2429 87 3 is be AUX iajs-2429 87 4 commutative commutative ADJ iajs-2429 87 5 then then ADV iajs-2429 87 6 𝑀 𝑀 PROPN iajs-2429 87 7 is be AUX iajs-2429 87 8 regular regular ADJ iajs-2429 87 9 if if SCONJ iajs-2429 87 10 and and CCONJ iajs-2429 87 11 only only ADV iajs-2429 87 12 if if SCONJ iajs-2429 87 13 for for ADP iajs-2429 87 14 every every DET iajs-2429 87 15 𝑚 𝑚 PROPN iajs-2429 87 16 ∈ ∈ PROPN iajs-2429 87 17 𝑀 𝑀 PROPN iajs-2429 87 18 and and CCONJ iajs-2429 87 19 for for ADP iajs-2429 87 20 every every DET iajs-2429 87 21 𝑎 𝑎 PRON iajs-2429 87 22 ∈ ∈ PROPN iajs-2429 87 23 𝑅 𝑅 NOUN iajs-2429 87 24 we we PRON iajs-2429 87 25 have have VERB iajs-2429 87 26 𝑚𝑎 𝑚𝑎 NOUN iajs-2429 87 27 𝑚𝑎 𝑚𝑎 ADP iajs-2429 87 28 𝑟 𝑟 NOUN iajs-2429 87 29 for for ADP iajs-2429 87 30 some some DET iajs-2429 87 31 𝑟 𝑟 PRON iajs-2429 87 32 ∈ ∈ NOUN iajs-2429 87 33 𝑅. 𝑅. NOUN iajs-2429 87 34 also also ADV iajs-2429 87 35 𝑅 𝑅 PROPN iajs-2429 87 36 is be AUX iajs-2429 87 37 boolean boolean ADJ iajs-2429 87 38 ring ring NOUN iajs-2429 87 39 if if SCONJ iajs-2429 87 40 𝑎 𝑎 PRON iajs-2429 87 41 𝑎 𝑎 NOUN iajs-2429 87 42 for for ADP iajs-2429 87 43 every every DET iajs-2429 87 44 𝑎 𝑎 PRON iajs-2429 87 45 ∈ ∈ PROPN iajs-2429 87 46 𝑅 𝑅 PROPN iajs-2429 87 47 [ [ NOUN iajs-2429 87 48 7 7 NUM iajs-2429 87 49 ] ] PUNCT iajs-2429 87 50 . . PUNCT iajs-2429 87 51     SPACE iajs-2429 88 1 thus thus ADV iajs-2429 88 2 a a DET iajs-2429 88 3 boolean boolean ADJ iajs-2429 88 4 ring ring NOUN iajs-2429 88 5 is be AUX iajs-2429 88 6 von von PROPN iajs-2429 88 7 neumann neumann PROPN iajs-2429 88 8 . . PUNCT iajs-2429 89 1 we we PRON iajs-2429 89 2 call call VERB iajs-2429 89 3 a a DET iajs-2429 89 4 module module NOUN iajs-2429 89 5 𝑀 𝑀 NOUN iajs-2429 89 6 is be AUX iajs-2429 89 7 rickart rickart NOUN iajs-2429 89 8 when when SCONJ iajs-2429 89 9 for for ADP iajs-2429 89 10 every every DET iajs-2429 89 11 𝑓 𝑓 PRON iajs-2429 89 12 ∈ ∈ PROPN iajs-2429 89 13 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2429 89 14 𝑀 𝑀 PROPN iajs-2429 89 15 , , PUNCT iajs-2429 89 16 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 PROPN iajs-2429 89 17 is be AUX iajs-2429 89 18 a a DET iajs-2429 89 19 direct direct ADJ iajs-2429 89 20 summand summand NOUN iajs-2429 89 21 of of ADP iajs-2429 89 22 𝑀 𝑀 PROPN iajs-2429 90 1 [ [ X iajs-2429 90 2 16 16 NUM iajs-2429 90 3 ] ] PUNCT iajs-2429 90 4 . . PUNCT iajs-2429 91 1 𝑀 𝑀 PROPN iajs-2429 91 2 is be AUX iajs-2429 91 3 a a DET iajs-2429 91 4 dual dual ADJ iajs-2429 91 5 rickart rickart NOUN iajs-2429 91 6 module module NOUN iajs-2429 91 7 when when SCONJ iajs-2429 91 8 for for ADP iajs-2429 91 9 every every DET iajs-2429 91 10 𝑓 𝑓 PRON iajs-2429 91 11 ∈ ∈ PROPN iajs-2429 91 12 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2429 91 13 𝑀 𝑀 PROPN iajs-2429 91 14 , , PUNCT iajs-2429 91 15 i i PRON iajs-2429 91 16 m m VERB iajs-2429 91 17 𝑓 𝑓 PRON iajs-2429 91 18 is be AUX iajs-2429 91 19 a a DET iajs-2429 91 20 direct direct ADJ iajs-2429 91 21 summand summand NOUN iajs-2429 91 22 of of ADP iajs-2429 91 23 𝑀 𝑀 PROPN iajs-2429 92 1 [ [ X iajs-2429 92 2 16 16 NUM iajs-2429 92 3 ] ] PUNCT iajs-2429 92 4 . . PUNCT iajs-2429 93 1 it it PRON iajs-2429 93 2 is be AUX iajs-2429 93 3 wellknown wellknown ADJ iajs-2429 93 4 that that SCONJ iajs-2429 93 5 for for ADP iajs-2429 93 6 each each DET iajs-2429 93 7 𝑎 𝑎 PRON iajs-2429 93 8 ∈ ∈ PROPN iajs-2429 93 9 𝑅 𝑅 NOUN iajs-2429 93 10 we we PRON iajs-2429 93 11 can can AUX iajs-2429 93 12 define define VERB iajs-2429 93 13 𝑓 𝑓 DET iajs-2429 93 14 : : PUNCT iajs-2429 93 15 𝑅 𝑅 PROPN iajs-2429 93 16 → → SYM iajs-2429 93 17 𝑅 𝑅 PROPN iajs-2429 93 18 by by ADP iajs-2429 93 19 𝑓 𝑓 PRON iajs-2429 93 20 𝑟 𝑟 NOUN iajs-2429 93 21 𝑎𝑟 𝑎𝑟 NOUN iajs-2429 93 22 for for ADP iajs-2429 93 23 each each DET iajs-2429 93 24 𝑟 𝑟 PRON iajs-2429 93 25 ∈ ∈ PROPN iajs-2429 93 26 𝑅 𝑅 PROPN iajs-2429 93 27 then then ADV iajs-2429 93 28 𝐼𝑚𝑓 𝐼𝑚𝑓 PROPN iajs-2429 93 29 𝑎𝑅. 𝑎𝑅. PROPN iajs-2429 93 30 this this PRON iajs-2429 93 31 means mean VERB iajs-2429 93 32 𝑅 𝑅 PROPN iajs-2429 93 33 is be AUX iajs-2429 93 34 von von PROPN iajs-2429 93 35 neumann neumann PROPN iajs-2429 93 36 regular regular PROPN iajs-2429 94 1 if if SCONJ iajs-2429 94 2 and and CCONJ iajs-2429 94 3 only only ADV iajs-2429 94 4 if if SCONJ iajs-2429 94 5 𝑅 𝑅 PROPN iajs-2429 94 6 is be AUX iajs-2429 94 7 dual dual ADJ iajs-2429 94 8 rickart rickart NOUN iajs-2429 94 9 as as ADP iajs-2429 94 10 𝑅module 𝑅module PROPN iajs-2429 94 11 . . PUNCT iajs-2429 95 1 a a DET iajs-2429 95 2 nonzero nonzero PROPN iajs-2429 95 3 submodule submodule NOUN iajs-2429 95 4 𝑁 𝑁 PROPN iajs-2429 95 5 of of ADP iajs-2429 95 6 𝑀 𝑀 PROPN iajs-2429 95 7 is be AUX iajs-2429 95 8 weak weak ADJ iajs-2429 95 9 semisecond semisecond NOUN iajs-2429 95 10 whenever whenever SCONJ iajs-2429 95 11 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 95 12 ⊆ ⊆ PROPN iajs-2429 95 13 𝐾 𝐾 NOUN iajs-2429 95 14 where where SCONJ iajs-2429 95 15 𝑎 𝑎 DET iajs-2429 95 16 ∈ ∈ PROPN iajs-2429 95 17 𝑅 𝑅 PROPN iajs-2429 95 18 and and CCONJ iajs-2429 95 19 𝐾 𝐾 PROPN iajs-2429 95 20 a a DET iajs-2429 95 21 submodule submodule NOUN iajs-2429 95 22 of of ADP iajs-2429 95 23 𝑀 𝑀 PROPN iajs-2429 95 24 implies imply VERB iajs-2429 95 25 either either CCONJ iajs-2429 95 26 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 95 27 ⊆ ⊆ PROPN iajs-2429 95 28 𝐾 𝐾 PROPN iajs-2429 95 29 or or CCONJ iajs-2429 95 30 𝑎 𝑎 PROPN iajs-2429 95 31 ∈ ∈ NOUN iajs-2429 95 32 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 96 1 𝑁 𝑁 PROPN iajs-2429 96 2 [ [ X iajs-2429 96 3 17 17 NUM iajs-2429 96 4 ] ] PUNCT iajs-2429 96 5 . . PUNCT iajs-2429 97 1 a a DET iajs-2429 97 2 nonzero nonzero PROPN iajs-2429 97 3 submodule submodule NOUN iajs-2429 97 4 𝑁 𝑁 PROPN iajs-2429 97 5 of of ADP iajs-2429 97 6 𝑀 𝑀 PROPN iajs-2429 97 7 is be AUX iajs-2429 97 8 called call VERB iajs-2429 97 9 a a DET iajs-2429 97 10 strongly strongly ADV iajs-2429 97 11 2 2 NUM iajs-2429 97 12 - - PUNCT iajs-2429 97 13 absorbing absorb VERB iajs-2429 97 14 second second ADJ iajs-2429 97 15 submodule submodule NOUN iajs-2429 97 16 if if SCONJ iajs-2429 97 17 for for ADP iajs-2429 97 18 each each DET iajs-2429 97 19 𝑎 𝑎 NOUN iajs-2429 97 20 , , PUNCT iajs-2429 97 21 𝑏 𝑏 PROPN iajs-2429 97 22 ∈ ∈ PROPN iajs-2429 97 23 𝑅 𝑅 PROPN iajs-2429 97 24 , , PUNCT iajs-2429 97 25 we we PRON iajs-2429 97 26 have have VERB iajs-2429 97 27 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 97 28 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 97 29 or or CCONJ iajs-2429 97 30 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 97 31 𝑁𝑏 𝑁𝑏 PROPN iajs-2429 97 32 or or CCONJ iajs-2429 97 33 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 97 34 0 0 PUNCT iajs-2429 98 1 [ [ X iajs-2429 98 2 18 18 NUM iajs-2429 98 3 ] ] PUNCT iajs-2429 98 4 . . PUNCT iajs-2429 99 1 a a DET iajs-2429 99 2 module module NOUN iajs-2429 99 3 𝑀 𝑀 PROPN iajs-2429 99 4 is be AUX iajs-2429 99 5 called call VERB iajs-2429 99 6 cacellation cacellation NOUN iajs-2429 99 7 if if SCONJ iajs-2429 99 8 𝑀𝐼 𝑀𝐼 PROPN iajs-2429 99 9 𝑀𝐽 𝑀𝐽 PROPN iajs-2429 99 10 implies imply VERB iajs-2429 99 11 𝐼 𝐼 PROPN iajs-2429 99 12 𝐽 𝐽 PROPN iajs-2429 99 13 for for ADP iajs-2429 99 14 each each DET iajs-2429 99 15 ideal ideal ADJ iajs-2429 99 16 𝐼 𝐼 PROPN iajs-2429 99 17 and and CCONJ iajs-2429 99 18 𝐽 𝐽 PROPN iajs-2429 99 19 of of ADP iajs-2429 99 20 𝑅 𝑅 PROPN iajs-2429 100 1 [ [ PROPN iajs-2429 100 2 19 19 NUM iajs-2429 100 3 ] ] PUNCT iajs-2429 100 4 . . PUNCT iajs-2429 101 1 other other ADJ iajs-2429 101 2 works work VERB iajs-2429 101 3 within within ADP iajs-2429 101 4 [ [ X iajs-2429 101 5 20 20 NUM iajs-2429 101 6 - - SYM iajs-2429 101 7 23 23 NUM iajs-2429 101 8 ] ] PUNCT iajs-2429 101 9 . . PUNCT iajs-2429 102 1 is be AUX iajs-2429 102 2 related relate VERB iajs-2429 102 3 topics topic NOUN iajs-2429 102 4 . . PUNCT iajs-2429 103 1 the the DET iajs-2429 103 2 paper paper NOUN iajs-2429 103 3 contains contain VERB iajs-2429 103 4 five five NUM iajs-2429 103 5 branches branch NOUN iajs-2429 103 6 and and CCONJ iajs-2429 103 7 better well ADV iajs-2429 103 8 say say VERB iajs-2429 103 9 “ " PUNCT iajs-2429 103 10 sections section NOUN iajs-2429 103 11 ” " PUNCT iajs-2429 103 12 ) ) PUNCT iajs-2429 103 13 . . PUNCT iajs-2429 104 1 in in ADP iajs-2429 104 2 second second ADJ iajs-2429 104 3 part part NOUN iajs-2429 104 4 , , PUNCT iajs-2429 104 5 we we PRON iajs-2429 104 6 give give VERB iajs-2429 104 7 other other ADJ iajs-2429 104 8 descriptions description NOUN iajs-2429 104 9 of of ADP iajs-2429 104 10 the the DET iajs-2429 104 11 semisecond semisecond ADJ iajs-2429 104 12 submodules submodule NOUN iajs-2429 104 13 idea idea NOUN iajs-2429 104 14 ( ( PUNCT iajs-2429 104 15 theorem theorem VERB iajs-2429 104 16 2.2 2.2 NUM iajs-2429 104 17 , , PUNCT iajs-2429 104 18 theorem theorem VERB iajs-2429 104 19 2.4 2.4 NUM iajs-2429 104 20 , , PUNCT iajs-2429 104 21 and and CCONJ iajs-2429 104 22 proposition proposition NOUN iajs-2429 104 23 2.8 2.8 NUM iajs-2429 104 24 ) ) PUNCT iajs-2429 104 25 . . PUNCT iajs-2429 105 1 more more ADJ iajs-2429 105 2 examples example NOUN iajs-2429 105 3 and and CCONJ iajs-2429 105 4 information information NOUN iajs-2429 105 5 about about ADP iajs-2429 105 6 this this DET iajs-2429 105 7 idea idea NOUN iajs-2429 105 8 are be AUX iajs-2429 105 9 provided provide VERB iajs-2429 105 10 ( ( PUNCT iajs-2429 105 11 remarks remark NOUN iajs-2429 105 12 and and CCONJ iajs-2429 105 13 examples example NOUN iajs-2429 105 14 2.3 2.3 NUM iajs-2429 105 15 ) ) PUNCT iajs-2429 105 16 . . PUNCT iajs-2429 106 1 we we PRON iajs-2429 106 2 study study VERB iajs-2429 106 3 the the DET iajs-2429 106 4 homomorphic homomorphic ADJ iajs-2429 106 5 image image NOUN iajs-2429 106 6 and and CCONJ iajs-2429 106 7 the the DET iajs-2429 106 8 direct direct ADJ iajs-2429 106 9 sum sum NOUN iajs-2429 106 10 of of ADP iajs-2429 106 11 this this DET iajs-2429 106 12 class class NOUN iajs-2429 106 13 of of ADP iajs-2429 106 14 modules module NOUN iajs-2429 106 15 ( ( PUNCT iajs-2429 106 16 proposition proposition NOUN iajs-2429 106 17 2.5 2.5 NUM iajs-2429 106 18 and and CCONJ iajs-2429 106 19 propsition propsition PROPN iajs-2429 106 20 2.6 2.6 NUM iajs-2429 106 21 ) ) PUNCT iajs-2429 106 22 . . PUNCT iajs-2429 107 1 section section NOUN iajs-2429 107 2 three three NUM iajs-2429 107 3 includes include VERB iajs-2429 107 4 ( ( PUNCT iajs-2429 107 5 theorem theorem ADJ iajs-2429 107 6 3.1 3.1 NUM iajs-2429 107 7 ) ) PUNCT iajs-2429 107 8 is be AUX iajs-2429 107 9 the the DET iajs-2429 107 10 most most ADV iajs-2429 107 11 important important ADJ iajs-2429 107 12 tool tool NOUN iajs-2429 107 13 to to PART iajs-2429 107 14 describe describe VERB iajs-2429 107 15 semisecond semisecond ADJ iajs-2429 107 16 submodules submodule NOUN iajs-2429 107 17 . . PUNCT iajs-2429 108 1 more more ADJ iajs-2429 108 2 characterizations characterization NOUN iajs-2429 108 3 are be AUX iajs-2429 108 4 supplied supply VERB iajs-2429 108 5 ( ( PUNCT iajs-2429 108 6 corollary corollary ADJ iajs-2429 108 7 3.9 3.9 NUM iajs-2429 108 8 and and CCONJ iajs-2429 108 9 theorem theorem VERB iajs-2429 108 10 3.12 3.12 NUM iajs-2429 108 11 ) ) PUNCT iajs-2429 108 12 . . PUNCT iajs-2429 109 1 section section NOUN iajs-2429 109 2 four four NUM iajs-2429 109 3 is be AUX iajs-2429 109 4 devoted devote VERB iajs-2429 109 5 to to ADP iajs-2429 109 6 finding find VERB iajs-2429 109 7 any any DET iajs-2429 109 8 relationships relationship NOUN iajs-2429 109 9 between between ADP iajs-2429 109 10 semisecond semisecond ADJ iajs-2429 109 11 submodules submodule NOUN iajs-2429 109 12 and and CCONJ iajs-2429 109 13 related related ADJ iajs-2429 109 14 modules module NOUN iajs-2429 109 15 . . PUNCT iajs-2429 110 1 among among ADP iajs-2429 110 2 other other ADJ iajs-2429 110 3 observations observation NOUN iajs-2429 110 4 , , PUNCT iajs-2429 110 5 we we PRON iajs-2429 110 6 see see VERB iajs-2429 110 7 that that SCONJ iajs-2429 110 8 every every DET iajs-2429 110 9 nonzero nonzero ADJ iajs-2429 110 10 regular regular ADJ iajs-2429 110 11 module module NOUN iajs-2429 110 12 over over ADP iajs-2429 110 13 a a DET iajs-2429 110 14 commutative commutative ADJ iajs-2429 110 15 ring ring NOUN iajs-2429 110 16 is be AUX iajs-2429 110 17 semisecond semisecond ADJ iajs-2429 110 18 ( ( PUNCT iajs-2429 110 19 theorem theorem VERB iajs-2429 110 20 4.1 4.1 NUM iajs-2429 110 21 ) ) PUNCT iajs-2429 110 22 . . PUNCT iajs-2429 111 1 the the DET iajs-2429 111 2 semisecond semisecond NOUN iajs-2429 111 3 and and CCONJ iajs-2429 111 4 von von PROPN iajs-2429 111 5 neumann neumann PROPN iajs-2429 111 6 regular regular ADJ iajs-2429 111 7 concepts concept NOUN iajs-2429 111 8 are be AUX iajs-2429 111 9 coincident coincident ADJ iajs-2429 111 10 in in ADP iajs-2429 111 11 the the DET iajs-2429 111 12 commutative commutative ADJ iajs-2429 111 13 rings ring NOUN iajs-2429 111 14 ( ( PUNCT iajs-2429 111 15 theorem theorem VERB iajs-2429 111 16 4.7 4.7 NUM iajs-2429 111 17 ) ) PUNCT iajs-2429 111 18 . . PUNCT iajs-2429 112 1 in in ADP iajs-2429 112 2 section section NOUN iajs-2429 112 3 five five NUM iajs-2429 112 4 , , PUNCT iajs-2429 112 5 we we PRON iajs-2429 112 6 present present VERB iajs-2429 112 7 the the DET iajs-2429 112 8 concept concept NOUN iajs-2429 112 9 s s NOUN iajs-2429 112 10 - - PUNCT iajs-2429 112 11 semisecond semisecond NOUN iajs-2429 112 12 submodules submodule NOUN iajs-2429 112 13 and and CCONJ iajs-2429 112 14 the the DET iajs-2429 112 15 basic basic ADJ iajs-2429 112 16     SPACE iajs-2429 112 17 properties property NOUN iajs-2429 112 18 of of ADP iajs-2429 112 19 this this DET iajs-2429 112 20 modules module NOUN iajs-2429 112 21     SPACE iajs-2429 112 22 is be AUX iajs-2429 112 23 investigated investigate VERB iajs-2429 112 24 . . PUNCT iajs-2429 112 25       SPACE iajs-2429 113 1 in in ADP iajs-2429 113 2 what what PRON iajs-2429 113 3 follows follow VERB iajs-2429 113 4 , , PUNCT iajs-2429 113 5 ℤ ℤ PROPN iajs-2429 113 6 , , PUNCT iajs-2429 113 7 ℚ ℚ PROPN iajs-2429 113 8 , , PUNCT iajs-2429 113 9 ℤ ℤ PROPN iajs-2429 113 10 , , PUNCT iajs-2429 113 11 ℤ ℤ PROPN iajs-2429 113 12 ℤ ℤ PROPN iajs-2429 113 13 ℤ ℤ PROPN iajs-2429 113 14 and and CCONJ iajs-2429 113 15 𝑀𝑎𝑡 𝑀𝑎𝑡 PROPN iajs-2429 113 16 𝑅 𝑅 PROPN iajs-2429 113 17 we we PRON iajs-2429 113 18 denote denote VERB iajs-2429 113 19 respectively respectively ADV iajs-2429 113 20 , , PUNCT iajs-2429 113 21 integers integer NOUN iajs-2429 113 22 , , PUNCT iajs-2429 113 23 rational rational ADJ iajs-2429 113 24 numbers number NOUN iajs-2429 113 25 , , PUNCT iajs-2429 113 26     SPACE iajs-2429 113 27 the the DET iajs-2429 113 28 𝑝-prüfer 𝑝-prüfer PROPN iajs-2429 113 29 group group NOUN iajs-2429 113 30 , , PUNCT iajs-2429 113 31 the the DET iajs-2429 113 32 residue residue NOUN iajs-2429 113 33 ring ring NOUN iajs-2429 113 34 modulo modulo NOUN iajs-2429 113 35 𝑛 𝑛 ADP iajs-2429 113 36 and and CCONJ iajs-2429 113 37 an an DET iajs-2429 113 38 𝑛 𝑛 PRON iajs-2429 113 39 𝑛 𝑛 DET iajs-2429 113 40 matrix matrix NOUN iajs-2429 113 41 ring ring NOUN iajs-2429 113 42 over over ADP iajs-2429 113 43 𝑅 𝑅 PROPN iajs-2429 113 44 . . PUNCT iajs-2429 114 1 2 2 NUM iajs-2429 114 2 . . NOUN iajs-2429 114 3 semisecond semisecond ADJ iajs-2429 114 4 submodules submodule NOUN iajs-2429 114 5 we we PRON iajs-2429 114 6 give give VERB iajs-2429 114 7 a a DET iajs-2429 114 8 characterization characterization NOUN iajs-2429 114 9 of of ADP iajs-2429 114 10 semisecond semisecond ADJ iajs-2429 114 11 submodules submodule NOUN iajs-2429 114 12 , , PUNCT iajs-2429 114 13 first first ADV iajs-2429 114 14 we we PRON iajs-2429 114 15 recall recall VERB iajs-2429 114 16 the the DET iajs-2429 114 17 main main ADJ iajs-2429 114 18 definition definition NOUN iajs-2429 114 19 . . PUNCT iajs-2429 115 1 definition definition NOUN iajs-2429 115 2 ( ( PUNCT iajs-2429 115 3 2.1 2.1 NUM iajs-2429 115 4 ) ) PUNCT iajs-2429 116 1 [ [ X iajs-2429 116 2 2 2 NUM iajs-2429 116 3 ] ] PUNCT iajs-2429 116 4 . . PUNCT iajs-2429 117 1 a a DET iajs-2429 117 2 nonzero nonzero PROPN iajs-2429 117 3 submodule submodule NOUN iajs-2429 117 4 𝑁 𝑁 PROPN iajs-2429 117 5 of of ADP iajs-2429 117 6 𝑅-module 𝑅-module PROPN iajs-2429 117 7 𝑀 𝑀 PROPN iajs-2429 117 8 is be AUX iajs-2429 117 9 called call VERB iajs-2429 117 10 semisecond semisecond ADJ iajs-2429 117 11 if if SCONJ iajs-2429 117 12 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 117 13 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 117 14 for for ADP iajs-2429 117 15 each each DET iajs-2429 117 16 𝑎 𝑎 PRON iajs-2429 117 17 ∈ ∈ NOUN iajs-2429 117 18 𝑅. 𝑅. NOUN iajs-2429 117 19 theorem theorem NOUN iajs-2429 117 20 ( ( PUNCT iajs-2429 117 21 2.2 2.2 NUM iajs-2429 117 22 ): ): PUNCT iajs-2429 117 23 the the DET iajs-2429 117 24 following follow VERB iajs-2429 117 25 assertions assertion NOUN iajs-2429 117 26 are be AUX iajs-2429 117 27 equivalent equivalent ADJ iajs-2429 117 28 ( ( PUNCT iajs-2429 117 29 1 1 X iajs-2429 117 30 ) ) PUNCT iajs-2429 117 31 𝑁 𝑁 NOUN iajs-2429 117 32 is be AUX iajs-2429 117 33 a a DET iajs-2429 117 34 semisecond semisecond ADJ iajs-2429 117 35 submodule submodule NOUN iajs-2429 117 36 of of ADP iajs-2429 117 37 an an DET iajs-2429 117 38 𝑅-module 𝑅-module PROPN iajs-2429 117 39 𝑀 𝑀 PROPN iajs-2429 117 40 ( ( PUNCT iajs-2429 117 41 2 2 X iajs-2429 117 42 ) ) PUNCT iajs-2429 117 43 𝑁 𝑁 PROPN iajs-2429 117 44 0 0 NUM iajs-2429 118 1 and and CCONJ iajs-2429 118 2 whenever whenever SCONJ iajs-2429 118 3 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 118 4 ⊆ ⊆ NUM iajs-2429 118 5 𝐾 𝐾 PROPN iajs-2429 118 6 , , PUNCT iajs-2429 118 7 where where SCONJ iajs-2429 118 8 𝑎 𝑎 PROPN iajs-2429 118 9 ∈ ∈ PROPN iajs-2429 118 10 𝑅 𝑅 PROPN iajs-2429 118 11 and and CCONJ iajs-2429 118 12 𝐾 𝐾 PROPN iajs-2429 118 13 a a DET iajs-2429 118 14 submodule submodule NOUN iajs-2429 118 15 of of ADP iajs-2429 118 16 𝑀 𝑀 PROPN iajs-2429 118 17 implies imply VERB iajs-2429 118 18 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 118 19 ⊆ ⊆ PROPN iajs-2429 118 20 𝐾 𝐾 PROPN iajs-2429 118 21     SPACE iajs-2429 118 22 84 84 NUM iajs-2429 118 23     SPACE iajs-2429 118 24 ibn ibn PROPN iajs-2429 118 25 al al PROPN iajs-2429 118 26 - - PUNCT iajs-2429 118 27 haitham haitham PROPN iajs-2429 118 28 jour jour X iajs-2429 118 29 . . PROPN iajs-2429 119 1 for for ADP iajs-2429 119 2 pure pure ADJ iajs-2429 119 3 & & CCONJ iajs-2429 119 4 appl appl PROPN iajs-2429 119 5 . . PUNCT iajs-2429 120 1 sci sci PROPN iajs-2429 120 2 . . PROPN iajs-2429 121 1 33 33 NUM iajs-2429 121 2 ( ( PUNCT iajs-2429 121 3 2 2 NUM iajs-2429 121 4 ) ) PUNCT iajs-2429 121 5 2020 2020 NUM iajs-2429 121 6 proof proof NOUN iajs-2429 121 7 . . PUNCT iajs-2429 122 1 ( ( PUNCT iajs-2429 122 2 1 1 X iajs-2429 122 3 ) ) PUNCT iajs-2429 122 4   NOUN iajs-2429 122 5 ( ( PUNCT iajs-2429 122 6 2 2 X iajs-2429 122 7 ) ) PUNCT iajs-2429 122 8 let let VERB iajs-2429 122 9 𝑎 𝑎 PRON iajs-2429 122 10 ∈ ∈ PROPN iajs-2429 122 11 𝑅 𝑅 PROPN iajs-2429 122 12 and and CCONJ iajs-2429 122 13 𝐾 𝐾 PROPN iajs-2429 122 14 a a DET iajs-2429 122 15 submodule submodule NOUN iajs-2429 122 16 of of ADP iajs-2429 122 17 𝑀 𝑀 PROPN iajs-2429 122 18 with with ADP iajs-2429 122 19 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 122 20 ⊆ ⊆ NUM iajs-2429 122 21 𝐾. 𝐾. NOUN iajs-2429 122 22 because because SCONJ iajs-2429 122 23 𝑁 𝑁 PROPN iajs-2429 122 24 is be AUX iajs-2429 122 25 semisecond semisecond ADJ iajs-2429 122 26 then then ADV iajs-2429 122 27 𝑁 𝑁 PROPN iajs-2429 122 28 0 0 PUNCT iajs-2429 122 29 and and CCONJ iajs-2429 122 30 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 122 31 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 122 32 for for ADP iajs-2429 122 33 each each DET iajs-2429 122 34 𝑎 𝑎 PRON iajs-2429 122 35 ∈ ∈ PROPN iajs-2429 122 36 𝑅 𝑅 PROPN iajs-2429 122 37 implies imply VERB iajs-2429 122 38 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 122 39 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 122 40 ⊆ ⊆ NUM iajs-2429 122 41 𝐾 𝐾 PROPN iajs-2429 122 42 as as SCONJ iajs-2429 122 43 desired desire VERB iajs-2429 122 44 . . PUNCT iajs-2429 123 1 ( ( PUNCT iajs-2429 123 2 3 3 X iajs-2429 123 3 ) ) PUNCT iajs-2429 123 4   NOUN iajs-2429 123 5 ( ( PUNCT iajs-2429 123 6 1 1 X iajs-2429 123 7 ) ) PUNCT iajs-2429 123 8 assume assume VERB iajs-2429 123 9 𝑁 𝑁 PROPN iajs-2429 123 10 0 0 NUM iajs-2429 123 11 and and CCONJ iajs-2429 123 12 𝑎 𝑎 PROPN iajs-2429 123 13 ∈ ∈ PROPN iajs-2429 123 14 𝑅 𝑅 PROPN iajs-2429 123 15 then then ADV iajs-2429 123 16 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 123 17 ⊆ ⊆ NUM iajs-2429 123 18 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 123 19 . . PUNCT iajs-2429 124 1 by by ADP iajs-2429 124 2 hypothesis hypothesis NOUN iajs-2429 124 3 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 124 4 ⊆ ⊆ NUM iajs-2429 124 5 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 124 6 and and CCONJ iajs-2429 124 7 hence hence ADV iajs-2429 124 8 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 124 9 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 124 10 as as SCONJ iajs-2429 124 11 required require VERB iajs-2429 124 12 . . PUNCT iajs-2429 125 1 remarks remark NOUN iajs-2429 125 2 and and CCONJ iajs-2429 125 3 examples example NOUN iajs-2429 125 4 ( ( PUNCT iajs-2429 125 5 2.3 2.3 NUM iajs-2429 125 6 ) ) PUNCT iajs-2429 125 7 ( ( PUNCT iajs-2429 125 8 1 1 X iajs-2429 125 9 ) ) PUNCT iajs-2429 125 10 obviously obviously ADV iajs-2429 125 11 semisecond semisecond ADJ iajs-2429 125 12 submodules submodule NOUN iajs-2429 125 13 are be AUX iajs-2429 125 14 weak weak ADJ iajs-2429 125 15 semisecond semisecond NOUN iajs-2429 125 16 but but CCONJ iajs-2429 125 17 the the DET iajs-2429 125 18 converse converse NOUN iajs-2429 125 19 fails fail VERB iajs-2429 125 20 for for ADP iajs-2429 125 21 more more ADJ iajs-2429 125 22 information information NOUN iajs-2429 125 23 see see VERB iajs-2429 125 24 [ [ X iajs-2429 125 25 17 17 NUM iajs-2429 125 26 ] ] PUNCT iajs-2429 125 27 . . PUNCT iajs-2429 126 1 ( ( PUNCT iajs-2429 126 2 2 2 X iajs-2429 126 3 ) ) PUNCT iajs-2429 126 4 it it PRON iajs-2429 126 5 is be AUX iajs-2429 126 6 clear clear ADJ iajs-2429 126 7 that that SCONJ iajs-2429 126 8 weakly weakly ADJ iajs-2429 126 9 second second ADJ iajs-2429 126 10 submodules submodule NOUN iajs-2429 126 11 are be AUX iajs-2429 126 12 semisecond semisecond ADJ iajs-2429 126 13 . . PUNCT iajs-2429 127 1 the the DET iajs-2429 127 2 converse converse NOUN iajs-2429 127 3 is be AUX iajs-2429 127 4 not not PART iajs-2429 127 5 hold hold NOUN iajs-2429 127 6 in in ADP iajs-2429 127 7 general general ADJ iajs-2429 127 8 , , PUNCT iajs-2429 127 9 ℤ ℤ PROPN iajs-2429 127 10 as as SCONJ iajs-2429 127 11 ℤ-module ℤ-module PROPN iajs-2429 127 12 is be AUX iajs-2429 127 13 semisecond semisecond ADJ iajs-2429 127 14 since since SCONJ iajs-2429 127 15 ℤ ℤ PROPN iajs-2429 127 16 . . PUNCT iajs-2429 128 1 𝑎 𝑎 X iajs-2429 128 2 ℤ ℤ NOUN iajs-2429 128 3 . . PUNCT iajs-2429 129 1 𝑎 𝑎 X iajs-2429 129 2 for for ADP iajs-2429 129 3 each each DET iajs-2429 129 4 𝑎 𝑎 PRON iajs-2429 129 5 ∈ ∈ ADJ iajs-2429 129 6 ℤ ℤ NOUN iajs-2429 129 7 but but CCONJ iajs-2429 129 8 ℤ ℤ PROPN iajs-2429 129 9 is be AUX iajs-2429 129 10 not not PART iajs-2429 129 11 weakly weakly ADV iajs-2429 129 12 second second ADJ iajs-2429 129 13 because because SCONJ iajs-2429 129 14 ℤ ℤ PROPN iajs-2429 129 15 . . PUNCT iajs-2429 129 16 3 3 NUM iajs-2429 129 17 ℤ ℤ NOUN iajs-2429 129 18 . . PUNCT iajs-2429 130 1 2.3 2.3 NUM iajs-2429 130 2 0 0 NUM iajs-2429 130 3 ℤ ℤ NOUN iajs-2429 130 4 . . PUNCT iajs-2429 131 1 2 2 X iajs-2429 131 2 . . X iajs-2429 131 3 ( ( PUNCT iajs-2429 131 4 3 3 NUM iajs-2429 131 5 ) ) PUNCT iajs-2429 131 6 as as ADP iajs-2429 131 7 another another DET iajs-2429 131 8 example example NOUN iajs-2429 131 9 of of ADP iajs-2429 131 10 ( ( PUNCT iajs-2429 131 11 2 2 NUM iajs-2429 131 12 ) ) PUNCT iajs-2429 131 13 , , PUNCT iajs-2429 131 14 let let VERB iajs-2429 131 15 𝑁 𝑁 PROPN iajs-2429 131 16 ℤ ℤ PROPN iajs-2429 131 17 ⊕ ⊕ PROPN iajs-2429 131 18 ℤ ℤ PROPN iajs-2429 131 19 be be AUX iajs-2429 131 20 a a DET iajs-2429 131 21 submodule submodule NOUN iajs-2429 131 22 of of ADP iajs-2429 131 23 𝑀 𝑀 PROPN iajs-2429 131 24 ℤ ℤ PROPN iajs-2429 131 25 ⊕ ⊕ PROPN iajs-2429 131 26 ℤ ℤ PROPN iajs-2429 131 27 as as ADP iajs-2429 131 28 ℤ-module ℤ-module PROPN iajs-2429 131 29 where where SCONJ iajs-2429 131 30 𝑝 𝑝 NOUN iajs-2429 131 31 and and CCONJ iajs-2429 131 32 𝑞 𝑞 ADP iajs-2429 131 33 prime prime ADJ iajs-2429 131 34 numbers number NOUN iajs-2429 131 35 . . PUNCT iajs-2429 132 1 then then ADV iajs-2429 132 2 𝑁 𝑁 PROPN iajs-2429 132 3 is be AUX iajs-2429 132 4 semisecond semisecond ADJ iajs-2429 132 5 since since SCONJ iajs-2429 132 6 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 132 7 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 132 8 for for ADP iajs-2429 132 9 each each DET iajs-2429 132 10 𝑎 𝑎 PRON iajs-2429 132 11 ∈ ∈ ADJ iajs-2429 132 12 ℤ ℤ NOUN iajs-2429 132 13 but but CCONJ iajs-2429 132 14 𝑁 𝑁 PROPN iajs-2429 132 15 is be AUX iajs-2429 132 16 not not PART iajs-2429 132 17 a a DET iajs-2429 132 18 weakly weakly ADJ iajs-2429 132 19 second second ADJ iajs-2429 132 20 submodule submodule NOUN iajs-2429 132 21 of of ADP iajs-2429 132 22 𝑀 𝑀 PROPN iajs-2429 133 1 because because SCONJ iajs-2429 133 2 𝑁. 𝑁. PROPN iajs-2429 133 3 𝑝. 𝑝. NOUN iajs-2429 133 4 𝑞 𝑞 X iajs-2429 133 5 0 0 PUNCT iajs-2429 133 6 while while SCONJ iajs-2429 133 7 𝑁. 𝑁. PROPN iajs-2429 133 8 𝑝 𝑝 PROPN iajs-2429 133 9 0 0 NUM iajs-2429 133 10 ⊕ ⊕ PROPN iajs-2429 133 11 ℤ ℤ PROPN iajs-2429 133 12 and and CCONJ iajs-2429 133 13 𝑁. 𝑁. PROPN iajs-2429 133 14 𝑞 𝑞 X iajs-2429 133 15 ℤ ℤ PROPN iajs-2429 133 16 ⊕ ⊕ PROPN iajs-2429 133 17 0 0 NUM iajs-2429 134 1 . . PUNCT iajs-2429 135 1 ( ( PUNCT iajs-2429 135 2 4 4 X iajs-2429 135 3 ) ) PUNCT iajs-2429 135 4 clearly clearly ADV iajs-2429 135 5 every every DET iajs-2429 135 6 module module NOUN iajs-2429 135 7 over over ADP iajs-2429 135 8 boolean boolean ADJ iajs-2429 135 9 ring ring NOUN iajs-2429 135 10 is be AUX iajs-2429 135 11 semisecond semisecond ADJ iajs-2429 135 12 . . PUNCT iajs-2429 136 1 ( ( PUNCT iajs-2429 136 2 5 5 X iajs-2429 136 3 ) ) PUNCT iajs-2429 136 4 secondary secondary ADJ iajs-2429 136 5 and and CCONJ iajs-2429 136 6 weakly weakly ADJ iajs-2429 136 7 secondary secondary ADJ iajs-2429 136 8 submodules submodule NOUN iajs-2429 136 9 not not PART iajs-2429 136 10 necessarily necessarily ADV iajs-2429 136 11 semisecond semisecond ADJ iajs-2429 136 12 . . PUNCT iajs-2429 137 1 consider consider VERB iajs-2429 137 2 ℤ ℤ PROPN iajs-2429 137 3 as as ADP iajs-2429 137 4 ℤ-module ℤ-module PROPN iajs-2429 137 5 is be AUX iajs-2429 137 6 secondary secondary ADJ iajs-2429 137 7 ( ( PUNCT iajs-2429 137 8 and and CCONJ iajs-2429 137 9 hence hence ADV iajs-2429 137 10 weakly weakly ADV iajs-2429 137 11 secondary secondary ADJ iajs-2429 137 12 ) ) PUNCT iajs-2429 137 13 see see VERB iajs-2429 138 1 [ [ X iajs-2429 138 2 4 4 NUM iajs-2429 138 3 ] ] PUNCT iajs-2429 138 4 . . PUNCT iajs-2429 139 1 but but CCONJ iajs-2429 139 2 𝑀 𝑀 PROPN iajs-2429 139 3 is be AUX iajs-2429 139 4 not not PART iajs-2429 139 5 semisecond semisecond ADJ iajs-2429 139 6 because because SCONJ iajs-2429 139 7 ℤ ℤ PROPN iajs-2429 139 8 . . PUNCT iajs-2429 139 9 2 2 NUM iajs-2429 139 10 ℤ ℤ NOUN iajs-2429 139 11 . . PUNCT iajs-2429 139 12 2 2 NUM iajs-2429 139 13 . . PUNCT iajs-2429 140 1 ( ( PUNCT iajs-2429 140 2 6 6 X iajs-2429 140 3 ) ) PUNCT iajs-2429 140 4 semisecond semisecond NOUN iajs-2429 140 5 submodules submodule NOUN iajs-2429 140 6 also also ADV iajs-2429 140 7 need need AUX iajs-2429 140 8 not not PART iajs-2429 140 9 be be AUX iajs-2429 140 10 secondary secondary ADJ iajs-2429 140 11 or or CCONJ iajs-2429 140 12 weakly weakly ADJ iajs-2429 140 13 secondary secondary ADJ iajs-2429 140 14 submodules submodule NOUN iajs-2429 140 15 . . PUNCT iajs-2429 141 1 for for ADP iajs-2429 141 2 example example NOUN iajs-2429 141 3 : : PUNCT iajs-2429 141 4 ℤ ℤ PROPN iajs-2429 141 5 as as SCONJ iajs-2429 141 6 ℤ-module ℤ-module PROPN iajs-2429 141 7 is be AUX iajs-2429 141 8 semisecond semisecond ADJ iajs-2429 141 9 by by ADP iajs-2429 141 10 ( ( PUNCT iajs-2429 141 11 2 2 NUM iajs-2429 141 12 ) ) PUNCT iajs-2429 141 13 but but CCONJ iajs-2429 141 14 ℤ ℤ PROPN iajs-2429 141 15 is be AUX iajs-2429 141 16 not not PART iajs-2429 141 17 weakly weakly ADV iajs-2429 141 18 secondary secondary ADJ iajs-2429 141 19 and and CCONJ iajs-2429 141 20 hence hence ADV iajs-2429 141 21 it it PRON iajs-2429 141 22 is be AUX iajs-2429 141 23 not not PART iajs-2429 141 24 secondary secondary ADJ iajs-2429 141 25 see see VERB iajs-2429 141 26 [ [ X iajs-2429 141 27 4 4 NUM iajs-2429 141 28 ] ] PUNCT iajs-2429 141 29 . . PUNCT iajs-2429 142 1 ( ( PUNCT iajs-2429 142 2 7 7 X iajs-2429 142 3 ) ) PUNCT iajs-2429 142 4 it it PRON iajs-2429 142 5 is be AUX iajs-2429 142 6 obvious obvious ADJ iajs-2429 142 7 that that SCONJ iajs-2429 142 8 coquasi coquasi NOUN iajs-2429 142 9 - - PUNCT iajs-2429 142 10 dedekind dedekind NOUN iajs-2429 142 11 ( ( PUNCT iajs-2429 142 12 or or CCONJ iajs-2429 142 13 simple simple ADJ iajs-2429 142 14 or or CCONJ iajs-2429 142 15 divisible divisible ADJ iajs-2429 142 16 ) ) PUNCT iajs-2429 142 17 submodule submodule NOUN iajs-2429 142 18   NOUN iajs-2429 142 19 second second PROPN iajs-2429 142 20 submodule submodule PROPN iajs-2429 142 21   NOUN iajs-2429 142 22 strongly strongly ADV iajs-2429 142 23 2 2 NUM iajs-2429 142 24 - - PUNCT iajs-2429 142 25 absorbing absorbing ADJ iajs-2429 142 26 second second ADJ iajs-2429 142 27 submodules submodule NOUN iajs-2429 142 28   NOUN iajs-2429 142 29 weakly weakly ADJ iajs-2429 142 30 second second ADJ iajs-2429 142 31 submodules submodule NOUN iajs-2429 142 32   NOUN iajs-2429 142 33 semisecond semisecond ADJ iajs-2429 142 34 submodules submodules PROPN iajs-2429 142 35   NOUN iajs-2429 142 36 weak weak ADJ iajs-2429 142 37 semisecond semisecond ADJ iajs-2429 142 38 submodules submodule NOUN iajs-2429 142 39 . . PUNCT iajs-2429 143 1 the the DET iajs-2429 143 2 converse converse NOUN iajs-2429 143 3 is be AUX iajs-2429 143 4 not not PART iajs-2429 143 5 true true ADJ iajs-2429 143 6 in in ADP iajs-2429 143 7 general general ADJ iajs-2429 143 8 , , PUNCT iajs-2429 143 9 𝑀 𝑀 PROPN iajs-2429 143 10 ℤ ℤ PROPN iajs-2429 143 11 ⊕ ⊕ PROPN iajs-2429 143 12 ℤ ℤ PROPN iajs-2429 143 13 as as SCONJ iajs-2429 143 14 ℤ-module ℤ-module PROPN iajs-2429 143 15 is be AUX iajs-2429 143 16 semisecond semisecond ADJ iajs-2429 143 17 but but CCONJ iajs-2429 143 18 it it PRON iajs-2429 143 19 is be AUX iajs-2429 143 20 not not PART iajs-2429 143 21 strongly strongly ADV iajs-2429 143 22 2 2 NUM iajs-2429 143 23 absorbing absorb VERB iajs-2429 143 24 second,(and second,(and NOUN iajs-2429 143 25 hence hence ADV iajs-2429 143 26 not not PART iajs-2429 143 27 weakly weakly ADV iajs-2429 143 28 second second ADJ iajs-2429 143 29 ) ) PUNCT iajs-2429 143 30 since since SCONJ iajs-2429 143 31 𝑀. 𝑀. PROPN iajs-2429 143 32 3 3 NUM iajs-2429 143 33 𝑀2.3 𝑀2.3 NOUN iajs-2429 143 34 0 0 NUM iajs-2429 143 35 ⊕ ⊕ PROPN iajs-2429 143 36 ℤ ℤ NOUN iajs-2429 143 37 𝑀. 𝑀. PROPN iajs-2429 143 38 2 2 NUM iajs-2429 143 39 and and CCONJ iajs-2429 143 40 𝑀. 𝑀. PROPN iajs-2429 143 41 2.3 2.3 NUM iajs-2429 143 42 0 0 NUM iajs-2429 143 43 . . PUNCT iajs-2429 144 1 ( ( PUNCT iajs-2429 144 2 8) 8) NUM iajs-2429 144 3 if if SCONJ iajs-2429 144 4 𝑁 𝑁 PROPN iajs-2429 144 5 is be AUX iajs-2429 144 6 a a DET iajs-2429 144 7 maximal maximal ADJ iajs-2429 144 8 ( ( PUNCT iajs-2429 144 9 and and CCONJ iajs-2429 144 10 hence hence ADV iajs-2429 144 11 prime prime ADJ iajs-2429 144 12 ) ) PUNCT iajs-2429 144 13 submodule submodule NOUN iajs-2429 144 14 then then ADV iajs-2429 144 15 𝑁 𝑁 PROPN iajs-2429 144 16 may may AUX iajs-2429 144 17 not not PART iajs-2429 144 18 be be AUX iajs-2429 144 19 semisecond semisecond ADJ iajs-2429 144 20 . . PUNCT iajs-2429 145 1 for for ADP iajs-2429 145 2 example example NOUN iajs-2429 145 3 , , PUNCT iajs-2429 145 4 𝑁 𝑁 PROPN iajs-2429 145 5 𝑝ℤ 𝑝ℤ PROPN iajs-2429 145 6 is be AUX iajs-2429 145 7 a a DET iajs-2429 145 8 maximal maximal ADJ iajs-2429 145 9 submodule submodule NOUN iajs-2429 145 10 of of ADP iajs-2429 145 11 ℤ ℤ PROPN iajs-2429 145 12 as as ADP iajs-2429 145 13 ℤ-module ℤ-module PROPN iajs-2429 145 14 but but CCONJ iajs-2429 145 15 𝑁 𝑁 PROPN iajs-2429 145 16 is be AUX iajs-2429 145 17 not not PART iajs-2429 145 18 semisecond semisecond ADJ iajs-2429 145 19 since since SCONJ iajs-2429 145 20 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 145 21 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 145 22 for for ADP iajs-2429 145 23 every every DET iajs-2429 145 24 𝑎 𝑎 PRON iajs-2429 145 25 ∈ ∈ ADJ iajs-2429 145 26 ℤ ℤ NOUN iajs-2429 145 27 and and CCONJ iajs-2429 145 28 any any DET iajs-2429 145 29 prime prime ADJ iajs-2429 145 30 number number NOUN iajs-2429 145 31 𝑝. 𝑝. NOUN iajs-2429 145 32 ( ( PUNCT iajs-2429 145 33 9 9 X iajs-2429 145 34 ) ) PUNCT iajs-2429 145 35 let let VERB iajs-2429 145 36 𝑁and 𝑁and PROPN iajs-2429 145 37 𝐻 𝐻 PROPN iajs-2429 145 38 be be AUX iajs-2429 145 39 submodules submodule NOUN iajs-2429 145 40 of of ADP iajs-2429 145 41 an an DET iajs-2429 145 42 𝑅-module 𝑅-module PROPN iajs-2429 145 43 𝑀 𝑀 PROPN iajs-2429 145 44 with with ADP iajs-2429 145 45 𝑁 𝑁 PROPN iajs-2429 145 46 ⊆ ⊆ NUM iajs-2429 145 47 𝐻 𝐻 PROPN iajs-2429 145 48 ⊆ ⊆ NUM iajs-2429 145 49 𝑀. 𝑀. NOUN iajs-2429 145 50 if if SCONJ iajs-2429 145 51 𝑁 𝑁 PROPN iajs-2429 145 52 is be AUX iajs-2429 145 53 a a DET iajs-2429 145 54 smisecond smisecond ADJ iajs-2429 145 55 submodule submodule NOUN iajs-2429 145 56 of of ADP iajs-2429 145 57 𝑀 𝑀 PROPN iajs-2429 145 58 then then ADV iajs-2429 145 59 𝐻 𝐻 PROPN iajs-2429 145 60 needs needs AUX iajs-2429 145 61 not not PART iajs-2429 145 62 be be AUX iajs-2429 145 63 a a DET iajs-2429 145 64 semisecond semisecond ADJ iajs-2429 145 65 submodule submodule NOUN iajs-2429 145 66 of of ADP iajs-2429 145 67 𝑀. 𝑀. PROPN iajs-2429 145 68 let let VERB iajs-2429 145 69 𝑁 𝑁 PROPN iajs-2429 145 70 ℤ ℤ PROPN iajs-2429 145 71 . . PUNCT iajs-2429 145 72 2 2 NUM iajs-2429 145 73 and and CCONJ iajs-2429 145 74 𝐻 𝐻 PROPN iajs-2429 145 75 ℤ ℤ PROPN iajs-2429 145 76 𝑀 𝑀 PROPN iajs-2429 145 77 submodules submodule NOUN iajs-2429 145 78 of of ADP iajs-2429 145 79 𝑀 𝑀 PROPN iajs-2429 145 80 ℤ ℤ PROPN iajs-2429 145 81 as as ADP iajs-2429 145 82 ℤ-module ℤ-module PROPN iajs-2429 145 83 where where SCONJ iajs-2429 145 84 𝑁 𝑁 PROPN iajs-2429 145 85 is be AUX iajs-2429 145 86 a a DET iajs-2429 145 87 simple simple ADJ iajs-2429 145 88 submodule submodule NOUN iajs-2429 145 89 so so SCONJ iajs-2429 145 90 it it PRON iajs-2429 145 91 is be AUX iajs-2429 145 92 semisecond semisecond ADJ iajs-2429 145 93 while while SCONJ iajs-2429 145 94 𝐻 𝐻 PROPN iajs-2429 145 95 is be AUX iajs-2429 145 96 not not PART iajs-2429 145 97 semisecond semisecond ADJ iajs-2429 145 98 by by ADP iajs-2429 145 99 ( ( PUNCT iajs-2429 145 100 5 5 NUM iajs-2429 145 101 ) ) PUNCT iajs-2429 145 102 . . PUNCT iajs-2429 146 1 ( ( PUNCT iajs-2429 146 2 10 10 NUM iajs-2429 146 3 ) ) PUNCT iajs-2429 146 4 let let VERB iajs-2429 146 5 𝑁 𝑁 PROPN iajs-2429 146 6 and and CCONJ iajs-2429 146 7 𝐻 𝐻 PROPN iajs-2429 146 8 be be VERB iajs-2429 146 9 submodules submodule NOUN iajs-2429 146 10 of of ADP iajs-2429 146 11 an an DET iajs-2429 146 12 𝑅-module 𝑅-module PROPN iajs-2429 146 13 𝑀 𝑀 PROPN iajs-2429 146 14 with with ADP iajs-2429 146 15 𝑁 𝑁 PROPN iajs-2429 146 16 ⊆ ⊆ NUM iajs-2429 146 17 𝐻 𝐻 PROPN iajs-2429 146 18 ⊆ ⊆ NUM iajs-2429 146 19 𝑀. 𝑀. NOUN iajs-2429 146 20 if if SCONJ iajs-2429 146 21 𝐻 𝐻 PROPN iajs-2429 146 22 is be AUX iajs-2429 146 23 a a DET iajs-2429 146 24 sermisecond sermisecond ADJ iajs-2429 146 25 submodule submodule NOUN iajs-2429 146 26 of of ADP iajs-2429 146 27 , , PUNCT iajs-2429 146 28 then then ADV iajs-2429 146 29 𝑁 𝑁 PROPN iajs-2429 146 30 needs needs AUX iajs-2429 146 31 not not PART iajs-2429 146 32 be be AUX iajs-2429 146 33 a a DET iajs-2429 146 34 semisecond semisecond ADJ iajs-2429 146 35 submodule submodule NOUN iajs-2429 146 36 of of ADP iajs-2429 146 37 𝑀. 𝑀. PROPN iajs-2429 146 38 let let VERB iajs-2429 146 39 𝑁 𝑁 PROPN iajs-2429 146 40 ℤ ℤ PROPN iajs-2429 146 41 be be AUX iajs-2429 146 42 a a DET iajs-2429 146 43 submodule submodule NOUN iajs-2429 146 44 of of ADP iajs-2429 146 45 𝑀 𝑀 PROPN iajs-2429 146 46 ℤ ℤ PROPN iajs-2429 146 47 as as ADP iajs-2429 146 48 ℤ-module ℤ-module PROPN iajs-2429 146 49 . . PROPN iajs-2429 147 1 since since SCONJ iajs-2429 147 2 𝑀 𝑀 PROPN iajs-2429 147 3 is be AUX iajs-2429 147 4 a a DET iajs-2429 147 5 divisible divisible ADJ iajs-2429 147 6 module module NOUN iajs-2429 147 7 then then ADV iajs-2429 147 8 𝑀 𝑀 PROPN iajs-2429 147 9 is be AUX iajs-2429 147 10 semisecond semisecond ADJ iajs-2429 147 11 but but CCONJ iajs-2429 147 12 𝑁 𝑁 PROPN iajs-2429 147 13 is be AUX iajs-2429 147 14 not not PART iajs-2429 147 15 semisecond semisecond ADJ iajs-2429 147 16 because because SCONJ iajs-2429 147 17 𝑁. 𝑁. PROPN iajs-2429 147 18 𝑝 𝑝 NOUN iajs-2429 147 19 0 0 NUM iajs-2429 147 20 𝑁𝑝 𝑁𝑝 PROPN iajs-2429 147 21 ℤ ℤ PROPN iajs-2429 147 22 . . PUNCT iajs-2429 148 1 ( ( PUNCT iajs-2429 148 2 11 11 NUM iajs-2429 148 3 ) ) PUNCT iajs-2429 148 4 as as ADP iajs-2429 148 5 another another DET iajs-2429 148 6 example example NOUN iajs-2429 148 7 of of ADP iajs-2429 148 8 ( ( PUNCT iajs-2429 148 9 10 10 NUM iajs-2429 148 10 ) ) PUNCT iajs-2429 148 11 , , PUNCT iajs-2429 148 12 ℚ ℚ PROPN iajs-2429 148 13 as as SCONJ iajs-2429 148 14 ℤ-module ℤ-module PROPN iajs-2429 148 15 is be AUX iajs-2429 148 16 divisible divisible ADJ iajs-2429 148 17 so so SCONJ iajs-2429 148 18 it it PRON iajs-2429 148 19 is be AUX iajs-2429 148 20 semisecond semisecond ADJ iajs-2429 148 21 but but CCONJ iajs-2429 148 22 the the DET iajs-2429 148 23 submodule submodule NOUN iajs-2429 148 24 ℤ ℤ PROPN iajs-2429 148 25 is be AUX iajs-2429 148 26 not not PART iajs-2429 148 27 semisecond semisecond ADJ iajs-2429 148 28 . . PUNCT iajs-2429 148 29     SPACE iajs-2429 148 30 85 85 NUM iajs-2429 148 31     SPACE iajs-2429 148 32 ibn ibn PROPN iajs-2429 148 33 al al PROPN iajs-2429 148 34 - - PUNCT iajs-2429 148 35 haitham haitham PROPN iajs-2429 148 36 jour jour X iajs-2429 148 37 . . PROPN iajs-2429 149 1 for for ADP iajs-2429 149 2 pure pure ADJ iajs-2429 149 3 & & CCONJ iajs-2429 149 4 appl appl PROPN iajs-2429 149 5 . . PUNCT iajs-2429 150 1 sci sci PROPN iajs-2429 150 2 . . PROPN iajs-2429 151 1 33 33 NUM iajs-2429 151 2 ( ( PUNCT iajs-2429 151 3 2 2 NUM iajs-2429 151 4 ) ) PUNCT iajs-2429 151 5 2020 2020 NUM iajs-2429 151 6 theorem theorem NOUN iajs-2429 151 7 ( ( PUNCT iajs-2429 151 8 2.4 2.4 NUM iajs-2429 151 9 ): ): PUNCT iajs-2429 151 10 the the DET iajs-2429 151 11 following follow VERB iajs-2429 151 12 assertions assertion NOUN iajs-2429 151 13 are be AUX iajs-2429 151 14     SPACE iajs-2429 151 15 equiavalent equiavalent NOUN iajs-2429 151 16 ( ( PUNCT iajs-2429 151 17 1 1 X iajs-2429 151 18 ) ) PUNCT iajs-2429 151 19 𝑁 𝑁 NOUN iajs-2429 151 20 is be AUX iajs-2429 151 21 a a DET iajs-2429 151 22 semisecond semisecond ADJ iajs-2429 151 23 submodule submodule NOUN iajs-2429 151 24 of of ADP iajs-2429 151 25 an an DET iajs-2429 151 26 𝑅-module 𝑅-module PROPN iajs-2429 151 27 𝑀. 𝑀. PROPN iajs-2429 151 28 ( ( PUNCT iajs-2429 151 29 2 2 NUM iajs-2429 151 30 ) ) PUNCT iajs-2429 151 31 𝑁 𝑁 PROPN iajs-2429 151 32 0 0 NUM iajs-2429 151 33 and and CCONJ iajs-2429 151 34 for for ADP iajs-2429 151 35 each each DET iajs-2429 151 36 𝑎 𝑎 NOUN iajs-2429 151 37 , , PUNCT iajs-2429 151 38 𝑏 𝑏 PROPN iajs-2429 151 39 ∈ ∈ PROPN iajs-2429 151 40 𝑅 𝑅 PROPN iajs-2429 151 41 and and CCONJ iajs-2429 151 42 𝐾 𝐾 PROPN iajs-2429 151 43 a a DET iajs-2429 151 44 finite finite ADJ iajs-2429 151 45 intersection intersection NOUN iajs-2429 151 46 of of ADP iajs-2429 151 47 completely completely ADV iajs-2429 151 48 irreducible irreducible ADJ iajs-2429 151 49 submodules submodule NOUN iajs-2429 151 50 of of ADP iajs-2429 151 51 𝑀 𝑀 PROPN iajs-2429 151 52 with with ADP iajs-2429 151 53 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 151 54 ⊆ ⊆ PROPN iajs-2429 151 55 𝐾 𝐾 PROPN iajs-2429 151 56 implies imply VERB iajs-2429 151 57 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 151 58 ⊆ ⊆ NUM iajs-2429 151 59 𝐾. 𝐾. PROPN iajs-2429 151 60 proof proof NOUN iajs-2429 151 61 . . PUNCT iajs-2429 152 1 ( ( PUNCT iajs-2429 152 2 1 1 X iajs-2429 152 3 ) ) PUNCT iajs-2429 152 4   NOUN iajs-2429 152 5 ( ( PUNCT iajs-2429 152 6 2 2 X iajs-2429 152 7 ) ) PUNCT iajs-2429 152 8 it it PRON iajs-2429 152 9 is be AUX iajs-2429 152 10 clear clear ADJ iajs-2429 152 11 . . PUNCT iajs-2429 153 1 ( ( PUNCT iajs-2429 153 2 3 3 X iajs-2429 153 3 ) ) PUNCT iajs-2429 153 4   NOUN iajs-2429 153 5 ( ( PUNCT iajs-2429 153 6 1 1 X iajs-2429 153 7 ) ) PUNCT iajs-2429 153 8 let let VERB iajs-2429 153 9 0 0 NUM iajs-2429 154 1 𝑁 𝑁 NOUN iajs-2429 154 2 and and CCONJ iajs-2429 154 3 𝐾 𝐾 PROPN iajs-2429 154 4 are be AUX iajs-2429 154 5 submodules submodule NOUN iajs-2429 154 6 of of ADP iajs-2429 154 7 𝑀 𝑀 PROPN iajs-2429 154 8 with with ADP iajs-2429 154 9 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 154 10 ⊆ ⊆ PROPN iajs-2429 154 11 𝐾 𝐾 PROPN iajs-2429 154 12 where where SCONJ iajs-2429 154 13 𝑎 𝑎 DET iajs-2429 154 14 ∈ ∈ NOUN iajs-2429 154 15 𝑅. 𝑅. NOUN iajs-2429 154 16 suppose suppose VERB iajs-2429 154 17 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 154 18 ⊈ ⊈ PROPN iajs-2429 154 19 𝐾 𝐾 PROPN iajs-2429 154 20 implies imply VERB iajs-2429 154 21 𝐾 𝐾 PROPN iajs-2429 154 22 ∩ ∩ ADJ iajs-2429 154 23 ∈∧ ∈∧ NOUN iajs-2429 154 24 𝐻 𝐻 NOUN iajs-2429 154 25 for for ADP iajs-2429 154 26 some some DET iajs-2429 154 27 collection collection NOUN iajs-2429 154 28 𝐻 𝐻 NOUN iajs-2429 154 29 ∈∧ ∈∧ NOUN iajs-2429 154 30 of of ADP iajs-2429 154 31 completely completely ADV iajs-2429 154 32 irreducible irreducible ADJ iajs-2429 154 33 submodules submodule NOUN iajs-2429 154 34 of of ADP iajs-2429 154 35 𝑀. 𝑀. PROPN iajs-2429 154 36 we we PRON iajs-2429 154 37 have have VERB iajs-2429 154 38 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 154 39 ⊈∩ ⊈∩ PROPN iajs-2429 154 40 ∈∧ ∈∧ NOUN iajs-2429 154 41 𝐻 𝐻 PROPN iajs-2429 154 42 . . PUNCT iajs-2429 155 1 so so ADV iajs-2429 155 2 there there PRON iajs-2429 155 3 exists exist VERB iajs-2429 155 4 𝑖 𝑖 ADP iajs-2429 155 5 ∈∧ ∈∧ NOUN iajs-2429 155 6 such such ADJ iajs-2429 155 7 that that SCONJ iajs-2429 155 8 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 155 9 ⊈ ⊈ PROPN iajs-2429 155 10 𝐻 𝐻 PROPN iajs-2429 155 11 . . PUNCT iajs-2429 156 1 on on ADP iajs-2429 156 2 the the DET iajs-2429 156 3 other other ADJ iajs-2429 156 4 hand hand NOUN iajs-2429 156 5 , , PUNCT iajs-2429 156 6 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 156 7 ⊆ ⊆ NUM iajs-2429 156 8 𝐾 𝐾 PROPN iajs-2429 156 9 ∩ ∩ ADJ iajs-2429 156 10 ∈∧ ∈∧ NOUN iajs-2429 156 11 𝐻 𝐻 PROPN iajs-2429 156 12 and and CCONJ iajs-2429 156 13 hence hence ADV iajs-2429 156 14 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 156 15 ⊆ ⊆ NUM iajs-2429 156 16 𝐾 𝐾 PROPN iajs-2429 156 17 ⊆ ⊆ NUM iajs-2429 156 18 ⋂ ⋂ PROPN iajs-2429 156 19 𝐻 𝐻 PROPN iajs-2429 156 20 for for ADP iajs-2429 156 21 some some DET iajs-2429 156 22 positive positive ADJ iajs-2429 156 23 integer integer NOUN iajs-2429 156 24 𝑛 𝑛 ADP iajs-2429 156 25 because because SCONJ iajs-2429 156 26 𝐾 𝐾 PROPN iajs-2429 156 27 ⊆ ⊆ NUM iajs-2429 156 28 𝐻 𝐻 PROPN iajs-2429 156 29 for for ADP iajs-2429 156 30 each each DET iajs-2429 156 31 𝑖 𝑖 DET iajs-2429 156 32 ∈∧. ∈∧. NOUN iajs-2429 156 33 by by ADP iajs-2429 156 34 hypothesis hypothesis NOUN iajs-2429 156 35 , , PUNCT iajs-2429 156 36 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 156 37 ⊆ ⊆ NUM iajs-2429 156 38 ⋂ ⋂ PROPN iajs-2429 156 39 𝐻 𝐻 NOUN iajs-2429 156 40 .then .then PUNCT iajs-2429 157 1 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 157 2 ⊆ ⊆ NUM iajs-2429 157 3 𝐻 𝐻 NOUN iajs-2429 157 4     SPACE iajs-2429 157 5 which which PRON iajs-2429 157 6 is be AUX iajs-2429 157 7 a a DET iajs-2429 157 8 contradiction contradiction NOUN iajs-2429 157 9 as as SCONJ iajs-2429 157 10 required require VERB iajs-2429 157 11 . . PUNCT iajs-2429 158 1 proposition proposition NOUN iajs-2429 158 2 ( ( PUNCT iajs-2429 158 3 2.5 2.5 NUM iajs-2429 158 4 ): ): PUNCT iajs-2429 158 5 every every DET iajs-2429 158 6 nonzero nonzero ADJ iajs-2429 158 7 homomorphic homomorphic ADJ iajs-2429 158 8 image image NOUN iajs-2429 158 9 of of ADP iajs-2429 158 10 semisecond semisecond ADJ iajs-2429 158 11 submodule submodule NOUN iajs-2429 158 12 is be AUX iajs-2429 158 13 smisecond smisecond ADJ iajs-2429 158 14 . . PUNCT iajs-2429 159 1 proof proof NOUN iajs-2429 159 2 . . PUNCT iajs-2429 160 1 let let VERB iajs-2429 160 2 𝐴 𝐴 PROPN iajs-2429 160 3 and and CCONJ iajs-2429 160 4 𝐵 𝐵 NOUN iajs-2429 160 5 be be AUX iajs-2429 160 6 𝑅-modules 𝑅-modules PROPN iajs-2429 160 7 and and CCONJ iajs-2429 160 8 0 0 NUM iajs-2429 160 9 𝑓 𝑓 X iajs-2429 160 10 : : PUNCT iajs-2429 160 11 𝐴 𝐴 PROPN iajs-2429 160 12 → → SYM iajs-2429 160 13 𝐵 𝐵 PROPN iajs-2429 160 14 an an DET iajs-2429 160 15 𝑅-homomorphism 𝑅-homomorphism PROPN iajs-2429 160 16 . . PUNCT iajs-2429 161 1 let let VERB iajs-2429 161 2 𝑁 𝑁 PROPN iajs-2429 161 3 be be AUX iajs-2429 161 4 a a DET iajs-2429 161 5 semisecond semisecond ADJ iajs-2429 161 6 submodule submodule NOUN iajs-2429 161 7 of of ADP iajs-2429 161 8 𝐴. 𝐴. PROPN iajs-2429 161 9 firstly firstly ADV iajs-2429 161 10 , , PUNCT iajs-2429 161 11 since since SCONJ iajs-2429 161 12 𝑓 𝑓 PRON iajs-2429 161 13 0 0 NUM iajs-2429 161 14 implies imply VERB iajs-2429 161 15 𝑓 𝑓 DET iajs-2429 161 16 𝑁 𝑁 PROPN iajs-2429 161 17 0 0 NUM iajs-2429 161 18 . . PUNCT iajs-2429 162 1 for for ADP iajs-2429 162 2 each each DET iajs-2429 162 3 𝑎 𝑎 PROPN iajs-2429 162 4 ∈ ∈ PROPN iajs-2429 162 5 𝑅 𝑅 PROPN iajs-2429 162 6 then then ADV iajs-2429 162 7 𝑓 𝑓 DET iajs-2429 162 8 𝑁 𝑁 PROPN iajs-2429 162 9 𝑎 𝑎 PROPN iajs-2429 162 10 𝑓 𝑓 PRON iajs-2429 162 11 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 162 12 𝑓 𝑓 PROPN iajs-2429 162 13 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 162 14 𝑓 𝑓 PROPN iajs-2429 162 15 𝑁 𝑁 PROPN iajs-2429 162 16 𝑎 𝑎 ADJ iajs-2429 162 17 . . PUNCT iajs-2429 163 1 proposition proposition NOUN iajs-2429 163 2 ( ( PUNCT iajs-2429 163 3 2.6 2.6 NUM iajs-2429 163 4 ): ): PUNCT iajs-2429 163 5 let let VERB iajs-2429 163 6 𝑁 𝑁 PROPN iajs-2429 163 7 and and CCONJ iajs-2429 163 8 𝑁 𝑁 PROPN iajs-2429 163 9 be be VERB iajs-2429 163 10 non non ADJ iajs-2429 163 11 - - ADJ iajs-2429 163 12 zero zero NUM iajs-2429 163 13 submodules submodule NOUN iajs-2429 163 14 of of ADP iajs-2429 163 15 𝑀 𝑀 PROPN iajs-2429 163 16 and and CCONJ iajs-2429 163 17 𝑀 𝑀 PROPN iajs-2429 163 18 𝑅-modules 𝑅-modules PROPN iajs-2429 163 19 respectively respectively ADV iajs-2429 163 20 . . PUNCT iajs-2429 164 1 then then ADV iajs-2429 164 2 𝑁 𝑁 PROPN iajs-2429 164 3 𝑁 𝑁 PROPN iajs-2429 164 4 ⊕ ⊕ PROPN iajs-2429 164 5 𝑁 𝑁 PROPN iajs-2429 164 6 is be AUX iajs-2429 164 7 a a DET iajs-2429 164 8 semisecond semisecond ADJ iajs-2429 164 9 submodule submodule NOUN iajs-2429 164 10 of of ADP iajs-2429 164 11 𝑀 𝑀 PROPN iajs-2429 164 12 𝑀 𝑀 PROPN iajs-2429 164 13 ⊕ ⊕ PROPN iajs-2429 164 14 𝑀 𝑀 PROPN iajs-2429 165 1 if if SCONJ iajs-2429 165 2 and and CCONJ iajs-2429 165 3 only only ADV iajs-2429 165 4 if if SCONJ iajs-2429 165 5 𝑁 𝑁 PROPN iajs-2429 165 6 and and CCONJ iajs-2429 165 7 𝑁 𝑁 PROPN iajs-2429 165 8 are be AUX iajs-2429 165 9 semisecond semisecond ADJ iajs-2429 165 10 submodules submodule NOUN iajs-2429 165 11 of of ADP iajs-2429 165 12 𝑀 𝑀 PROPN iajs-2429 165 13 and and CCONJ iajs-2429 165 14 𝑀 𝑀 PROPN iajs-2429 165 15 respectively respectively ADV iajs-2429 165 16 . . PUNCT iajs-2429 166 1 proof proof NOUN iajs-2429 166 2 . . PUNCT iajs-2429 167 1 ( ( PUNCT iajs-2429 167 2   NOUN iajs-2429 167 3 ) ) PUNCT iajs-2429 167 4 let let VERB iajs-2429 167 5 𝑎 𝑎 PRON iajs-2429 167 6 ∈ ∈ PROPN iajs-2429 167 7 𝑅 𝑅 PROPN iajs-2429 167 8 then then ADV iajs-2429 167 9 𝑁 𝑁 PROPN iajs-2429 167 10 ⊕ ⊕ PROPN iajs-2429 167 11 𝑁 𝑁 NOUN iajs-2429 167 12 𝑎 𝑎 ADJ iajs-2429 167 13 𝑁 𝑁 PROPN iajs-2429 167 14 ⊕ ⊕ NOUN iajs-2429 167 15 𝑁 𝑁 NOUN iajs-2429 167 16 𝑎 𝑎 NOUN iajs-2429 167 17 and and CCONJ iajs-2429 167 18 hence hence ADV iajs-2429 167 19 𝑁 𝑁 PROPN iajs-2429 167 20 𝑎 𝑎 PROPN iajs-2429 167 21 ⊕ ⊕ NOUN iajs-2429 167 22 𝑁 𝑁 NOUN iajs-2429 167 23 𝑎 𝑎 ADJ iajs-2429 167 24 𝑁 𝑁 PROPN iajs-2429 167 25 𝑎 𝑎 PROPN iajs-2429 167 26 ⊕ ⊕ NOUN iajs-2429 167 27 𝑁 𝑁 NOUN iajs-2429 167 28 𝑎 𝑎 NOUN iajs-2429 167 29 implies imply VERB iajs-2429 167 30 𝑁 𝑁 PROPN iajs-2429 167 31 𝑎 𝑎 ADJ iajs-2429 167 32 𝑁 𝑁 PROPN iajs-2429 167 33 𝑎 𝑎 NOUN iajs-2429 167 34 and and CCONJ iajs-2429 167 35 𝑁 𝑁 PROPN iajs-2429 167 36 𝑎 𝑎 VERB iajs-2429 167 37 𝑁 𝑁 PROPN iajs-2429 167 38 𝑎 𝑎 NOUN iajs-2429 167 39 as as ADP iajs-2429 167 40 required require VERB iajs-2429 167 41 . . PUNCT iajs-2429 168 1 ( ( PUNCT iajs-2429 168 2   NOUN iajs-2429 168 3 ) ) PUNCT iajs-2429 168 4 it it PRON iajs-2429 168 5 is be AUX iajs-2429 168 6 clear clear ADJ iajs-2429 168 7 . . PUNCT iajs-2429 169 1 corollary corollary ADJ iajs-2429 169 2 ( ( PUNCT iajs-2429 169 3 2.7 2.7 NUM iajs-2429 169 4 ): ): PUNCT iajs-2429 169 5 every every DET iajs-2429 169 6 non non ADJ iajs-2429 169 7 - - ADJ iajs-2429 169 8 zero zero ADJ iajs-2429 169 9 direct direct ADJ iajs-2429 169 10 summand summand NOUN iajs-2429 169 11 of of ADP iajs-2429 169 12 a a DET iajs-2429 169 13 semisecond semisecond ADJ iajs-2429 169 14 module module NOUN iajs-2429 169 15 is be AUX iajs-2429 169 16 semisecond semisecond ADJ iajs-2429 169 17 . . PUNCT iajs-2429 170 1 proposition proposition NOUN iajs-2429 170 2 ( ( PUNCT iajs-2429 170 3 2.8 2.8 NUM iajs-2429 170 4 ): ): PUNCT iajs-2429 170 5 the the DET iajs-2429 170 6 following following ADJ iajs-2429 170 7 statements statement NOUN iajs-2429 170 8 are be AUX iajs-2429 170 9 equivalent equivalent ADJ iajs-2429 170 10 ( ( PUNCT iajs-2429 170 11 1 1 X iajs-2429 170 12 ) ) PUNCT iajs-2429 170 13 𝑁 𝑁 NOUN iajs-2429 170 14 is be AUX iajs-2429 170 15 a a DET iajs-2429 170 16 semisecond semisecond ADJ iajs-2429 170 17 submodule submodule NOUN iajs-2429 170 18 of of ADP iajs-2429 170 19 𝑅-module 𝑅-module PROPN iajs-2429 170 20 𝑀. 𝑀. PROPN iajs-2429 170 21 ( ( PUNCT iajs-2429 170 22 2 2 NUM iajs-2429 170 23 ) ) PUNCT iajs-2429 170 24 is be AUX iajs-2429 170 25 a a DET iajs-2429 170 26 semisecond semisecond ADJ iajs-2429 170 27 submodule submodule NOUN iajs-2429 170 28 of of ADP iajs-2429 170 29 𝑅-module 𝑅-module PROPN iajs-2429 170 30 for for ADP iajs-2429 170 31 each each DET iajs-2429 170 32 submodule submodule NOUN iajs-2429 170 33 𝐻 𝐻 PROPN iajs-2429 170 34 of of ADP iajs-2429 170 35 𝑀 𝑀 PROPN iajs-2429 170 36 contained contain VERB iajs-2429 170 37 in in ADP iajs-2429 170 38 𝑁. 𝑁. PROPN iajs-2429 170 39 proof proof NOUN iajs-2429 170 40 . . PUNCT iajs-2429 171 1 ( ( PUNCT iajs-2429 171 2 1 1 X iajs-2429 171 3 ) ) PUNCT iajs-2429 171 4   NOUN iajs-2429 171 5 ( ( PUNCT iajs-2429 171 6 2 2 X iajs-2429 171 7 ) ) PUNCT iajs-2429 171 8 let let VERB iajs-2429 171 9 𝑁 𝑁 PROPN iajs-2429 171 10 be be AUX iajs-2429 171 11 a a DET iajs-2429 171 12 semisecond semisecond ADJ iajs-2429 171 13 submodule submodule NOUN iajs-2429 171 14 𝑀 𝑀 PROPN iajs-2429 171 15 and and CCONJ iajs-2429 171 16 𝜋 𝜋 NOUN iajs-2429 171 17 : : PUNCT iajs-2429 171 18 𝑀 𝑀 PROPN iajs-2429 171 19 → → PUNCT iajs-2429 171 20 be be AUX iajs-2429 171 21 the the DET iajs-2429 171 22 natural natural ADJ iajs-2429 171 23 homomorphism homomorphism NOUN iajs-2429 171 24 for for SCONJ iajs-2429 171 25 each each DET iajs-2429 171 26 submodule submodule NOUN iajs-2429 171 27 𝐻 𝐻 PROPN iajs-2429 171 28 of of ADP iajs-2429 171 29 𝑀 𝑀 PROPN iajs-2429 171 30 contained contain VERB iajs-2429 171 31 in in ADP iajs-2429 171 32 𝑁 𝑁 PROPN iajs-2429 171 33 so so ADV iajs-2429 171 34 by by ADP iajs-2429 171 35 proposition proposition NOUN iajs-2429 171 36 2.5 2.5 NUM iajs-2429 171 37 , , PUNCT iajs-2429 171 38 𝜋 𝜋 PRON iajs-2429 171 39 𝑁 𝑁 PROPN iajs-2429 171 40 is be AUX iajs-2429 171 41 a a DET iajs-2429 171 42 semisecond semisecond ADJ iajs-2429 171 43 submodule submodule NOUN iajs-2429 171 44 . . PUNCT iajs-2429 172 1 ( ( PUNCT iajs-2429 172 2 2 2 X iajs-2429 172 3 ) ) PUNCT iajs-2429 172 4   NOUN iajs-2429 172 5 ( ( PUNCT iajs-2429 172 6 1 1 X iajs-2429 172 7 ) ) PUNCT iajs-2429 172 8 it it PRON iajs-2429 172 9 is be AUX iajs-2429 172 10 clear clear ADJ iajs-2429 172 11 by by ADP iajs-2429 172 12 taking take VERB iajs-2429 172 13 𝐻 𝐻 PROPN iajs-2429 172 14 0 0 NUM iajs-2429 172 15 . . NOUN iajs-2429 173 1 3 3 NUM iajs-2429 173 2 . . X iajs-2429 174 1 more more ADJ iajs-2429 174 2     SPACE iajs-2429 174 3 characterizations characterization NOUN iajs-2429 174 4 and and CCONJ iajs-2429 174 5 facts fact NOUN iajs-2429 174 6 about about ADP iajs-2429 174 7 semisecond semisecond NOUN iajs-2429 174 8 submodules submodule NOUN iajs-2429 174 9 theorem theorem NOUN iajs-2429 174 10 ( ( PUNCT iajs-2429 174 11 3.1 3.1 NUM iajs-2429 174 12 ): ): PUNCT iajs-2429 174 13 the the DET iajs-2429 174 14 following following ADJ iajs-2429 174 15 statements statement NOUN iajs-2429 174 16 are be AUX iajs-2429 174 17     SPACE iajs-2429 174 18 equivalent equivalent ADJ iajs-2429 174 19 ( ( PUNCT iajs-2429 174 20 1 1 X iajs-2429 174 21 ) ) PUNCT iajs-2429 174 22 𝑁 𝑁 NOUN iajs-2429 174 23 is be AUX iajs-2429 174 24 a a DET iajs-2429 174 25 semisecond semisecond ADJ iajs-2429 174 26 submodule submodule NOUN iajs-2429 174 27 of of ADP iajs-2429 174 28 an an DET iajs-2429 174 29 𝑅-module 𝑅-module PROPN iajs-2429 174 30 𝑀. 𝑀. PROPN iajs-2429 174 31 ( ( PUNCT iajs-2429 174 32 2 2 NUM iajs-2429 174 33 ) ) PUNCT iajs-2429 174 34 𝑁 𝑁 PROPN iajs-2429 174 35 0 0 NUM iajs-2429 174 36 and and CCONJ iajs-2429 174 37 𝐾 𝐾 PROPN iajs-2429 174 38 : : PUNCT iajs-2429 174 39 𝑁 𝑁 PROPN iajs-2429 174 40 is be AUX iajs-2429 174 41 a a DET iajs-2429 174 42 semiprime semiprime NOUN iajs-2429 174 43 ideal ideal NOUN iajs-2429 174 44 of of ADP iajs-2429 174 45 𝑅 𝑅 PROPN iajs-2429 174 46 for for ADP iajs-2429 174 47 each each DET iajs-2429 174 48 submodule submodule NOUN iajs-2429 174 49 𝐾 𝐾 PROPN iajs-2429 174 50 ⊉ ⊉ PROPN iajs-2429 174 51 𝑁 𝑁 PROPN iajs-2429 174 52 in in ADP iajs-2429 174 53 𝑀. 𝑀. NOUN iajs-2429 174 54 proof proof NOUN iajs-2429 174 55 . . PUNCT iajs-2429 175 1 ( ( PUNCT iajs-2429 175 2 1 1 X iajs-2429 175 3 ) ) PUNCT iajs-2429 175 4   NOUN iajs-2429 175 5 ( ( PUNCT iajs-2429 175 6 2 2 X iajs-2429 175 7 ) ) PUNCT iajs-2429 175 8 assume assume VERB iajs-2429 175 9 𝑁 𝑁 PROPN iajs-2429 175 10 is be AUX iajs-2429 175 11 a a DET iajs-2429 175 12 semisecond semisecond ADJ iajs-2429 175 13 submodule submodule NOUN iajs-2429 175 14 of of ADP iajs-2429 175 15 an an DET iajs-2429 175 16 𝑅-module 𝑅-module PROPN iajs-2429 175 17 𝑀 𝑀 PROPN iajs-2429 175 18 and and CCONJ iajs-2429 175 19 𝐾 𝐾 PROPN iajs-2429 175 20 a a DET iajs-2429 175 21 submodule submodule NOUN iajs-2429 175 22 of of ADP iajs-2429 175 23 𝑀 𝑀 PROPN iajs-2429 175 24 such such ADJ iajs-2429 175 25 that that SCONJ iajs-2429 175 26 𝑁 𝑁 PROPN iajs-2429 175 27 ⊈ ⊈ PROPN iajs-2429 175 28 𝐾 𝐾 PROPN iajs-2429 175 29 implies imply VERB iajs-2429 175 30 𝐾 𝐾 PROPN iajs-2429 175 31 : : PUNCT iajs-2429 175 32 𝑁 𝑁 NOUN iajs-2429 175 33 𝑅. 𝑅. ADV iajs-2429 175 34 let let VERB iajs-2429 175 35 𝑎 𝑎 PRON iajs-2429 175 36 ∈ ∈ PROPN iajs-2429 175 37 𝑅 𝑅 NOUN iajs-2429 175 38 with with ADP iajs-2429 175 39 𝑎 𝑎 PROPN iajs-2429 175 40 ∈ ∈ PROPN iajs-2429 175 41 𝐾 𝐾 NOUN iajs-2429 175 42 : : PUNCT iajs-2429 175 43 𝑁 𝑁 PROPN iajs-2429 175 44 implies imply VERB iajs-2429 175 45 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 175 46 ⊆ ⊆ PROPN iajs-2429 175 47 𝐾 𝐾 PROPN iajs-2429 175 48 thus thus ADV iajs-2429 175 49 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 175 50 ⊆ ⊆ PROPN iajs-2429 175 51 𝐾 𝐾 PROPN iajs-2429 175 52 and and CCONJ iajs-2429 175 53 hence hence ADV iajs-2429 175 54 𝑎 𝑎 PROPN iajs-2429 175 55 ∈ ∈ ADJ iajs-2429 175 56 𝐾 𝐾 NOUN iajs-2429 175 57 : : PUNCT iajs-2429 175 58 𝑁 𝑁 PROPN iajs-2429 175 59 as as SCONJ iajs-2429 175 60 required require VERB iajs-2429 175 61 . . PUNCT iajs-2429 175 62     SPACE iajs-2429 176 1 86 86 NUM iajs-2429 176 2     SPACE iajs-2429 176 3 ibn ibn PROPN iajs-2429 176 4 al al PROPN iajs-2429 176 5 - - PUNCT iajs-2429 176 6 haitham haitham PROPN iajs-2429 176 7 jour jour X iajs-2429 176 8 . . PROPN iajs-2429 176 9 for for ADP iajs-2429 176 10 pure pure ADJ iajs-2429 176 11 & & CCONJ iajs-2429 176 12 appl appl PROPN iajs-2429 176 13 . . PUNCT iajs-2429 177 1 sci sci PROPN iajs-2429 177 2 . . PROPN iajs-2429 178 1 33 33 NUM iajs-2429 178 2 ( ( PUNCT iajs-2429 178 3 2 2 NUM iajs-2429 178 4 ) ) PUNCT iajs-2429 178 5 2020 2020 NUM iajs-2429 178 6 ( ( PUNCT iajs-2429 178 7 2 2 NUM iajs-2429 178 8 ) ) PUNCT iajs-2429 178 9   NOUN iajs-2429 178 10 ( ( PUNCT iajs-2429 178 11 1 1 X iajs-2429 178 12 ) ) PUNCT iajs-2429 178 13 let let VERB iajs-2429 178 14 𝑁 𝑁 PROPN iajs-2429 178 15 and and CCONJ iajs-2429 178 16 𝐾 𝐾 PROPN iajs-2429 178 17 be be AUX iajs-2429 178 18 submodules submodule NOUN iajs-2429 178 19 of of ADP iajs-2429 178 20 an an DET iajs-2429 178 21 𝑅-module 𝑅-module PROPN iajs-2429 178 22 𝑀 𝑀 PROPN iajs-2429 178 23 such such ADJ iajs-2429 178 24 that that SCONJ iajs-2429 178 25 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 178 26 ⊆ ⊆ PROPN iajs-2429 178 27 𝐾 𝐾 NOUN iajs-2429 178 28 where where SCONJ iajs-2429 178 29 𝑎 𝑎 DET iajs-2429 178 30 ∈ ∈ NOUN iajs-2429 178 31 𝑅. 𝑅. VERB iajs-2429 178 32 in in ADP iajs-2429 178 33 case case NOUN iajs-2429 178 34 𝑁 𝑁 PROPN iajs-2429 178 35 ⊆ ⊆ NUM iajs-2429 178 36 𝐾 𝐾 PROPN iajs-2429 178 37 then then ADV iajs-2429 178 38 already already ADV iajs-2429 178 39 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 178 40 ⊆ ⊆ NUM iajs-2429 178 41 𝐾. 𝐾. NOUN iajs-2429 178 42 if if SCONJ iajs-2429 178 43 𝑁 𝑁 PROPN iajs-2429 178 44 ⊈ ⊈ PROPN iajs-2429 178 45 𝐾 𝐾 PROPN iajs-2429 178 46 then then ADV iajs-2429 178 47 𝐾 𝐾 PROPN iajs-2429 178 48 : : PUNCT iajs-2429 178 49 𝑁 𝑁 PROPN iajs-2429 178 50 is be AUX iajs-2429 178 51 a a DET iajs-2429 178 52 semiprime semiprime NOUN iajs-2429 178 53 ideal ideal NOUN iajs-2429 178 54 of of ADP iajs-2429 178 55 𝑅 𝑅 PROPN iajs-2429 178 56 by by ADP iajs-2429 178 57 hypothesis hypothesis NOUN iajs-2429 178 58 and and CCONJ iajs-2429 178 59 𝑎 𝑎 NOUN iajs-2429 178 60 ∈ ∈ PROPN iajs-2429 178 61 𝐾 𝐾 NOUN iajs-2429 178 62 : : PUNCT iajs-2429 178 63 𝑁 𝑁 PROPN iajs-2429 178 64 implies imply VERB iajs-2429 178 65 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 178 66 ⊆ ⊆ NUM iajs-2429 178 67 𝐾 𝐾 PROPN iajs-2429 178 68 as as SCONJ iajs-2429 178 69 desired desire VERB iajs-2429 178 70 . . PUNCT iajs-2429 179 1 corollary corollary ADJ iajs-2429 179 2 ( ( PUNCT iajs-2429 179 3 3.2 3.2 NUM iajs-2429 179 4 ): ): PUNCT iajs-2429 179 5 every every DET iajs-2429 179 6 submodule submodule NOUN iajs-2429 179 7 of of ADP iajs-2429 179 8 a a DET iajs-2429 179 9 module module NOUN iajs-2429 179 10 over over ADP iajs-2429 179 11 a a DET iajs-2429 179 12 fully fully ADJ iajs-2429 179 13 semiprime semiprime NOUN iajs-2429 179 14 ( ( PUNCT iajs-2429 179 15 that that PRON iajs-2429 179 16 is be AUX iajs-2429 179 17 von von PROPN iajs-2429 179 18 neumann neumann PROPN iajs-2429 179 19 regular regular PROPN iajs-2429 179 20 ) ) PUNCT iajs-2429 179 21 ring ring NOUN iajs-2429 179 22 is be AUX iajs-2429 179 23 semisecond semisecond ADJ iajs-2429 179 24 . . PUNCT iajs-2429 180 1 proof proof NOUN iajs-2429 180 2 . . PUNCT iajs-2429 181 1 directly directly ADV iajs-2429 181 2 via via ADP iajs-2429 181 3 theorem theorem ADJ iajs-2429 181 4 3.1 3.1 NUM iajs-2429 181 5 . . PUNCT iajs-2429 181 6 corollary corollary NOUN iajs-2429 181 7 ( ( PUNCT iajs-2429 181 8 3.3 3.3 NUM iajs-2429 181 9 ): ): PUNCT iajs-2429 181 10 if if SCONJ iajs-2429 181 11 𝑁 𝑁 PROPN iajs-2429 181 12 is be AUX iajs-2429 181 13 a a DET iajs-2429 181 14 semisecond semisecond ADJ iajs-2429 181 15 submodule submodule NOUN iajs-2429 181 16 of of ADP iajs-2429 181 17 an an DET iajs-2429 181 18 𝑅-module 𝑅-module PROPN iajs-2429 181 19 𝑀 𝑀 PROPN iajs-2429 181 20 then then ADV iajs-2429 181 21 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 181 22 𝑁 𝑁 PROPN iajs-2429 181 23 is be AUX iajs-2429 181 24 a a DET iajs-2429 181 25 semiprime semiprime NOUN iajs-2429 181 26 ideal ideal NOUN iajs-2429 181 27 of of ADP iajs-2429 181 28 𝑅. 𝑅. ADJ iajs-2429 181 29 proof proof NOUN iajs-2429 181 30 . . PUNCT iajs-2429 182 1 directly directly ADV iajs-2429 182 2 via via ADP iajs-2429 182 3 theorem theorem PROPN iajs-2429 182 4 3.1 3.1 NUM iajs-2429 182 5 . . PUNCT iajs-2429 182 6 examples example NOUN iajs-2429 182 7 ( ( PUNCT iajs-2429 182 8 3.4 3.4 NUM iajs-2429 182 9 ): ): PUNCT iajs-2429 182 10 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 182 11 𝑁 𝑁 PROPN iajs-2429 182 12 0 0 NUM iajs-2429 182 13 is be AUX iajs-2429 182 14 a a DET iajs-2429 182 15 semiprime semiprime NOUN iajs-2429 182 16 ideal ideal NOUN iajs-2429 182 17 of of ADP iajs-2429 182 18 ℤ ℤ PROPN iajs-2429 182 19 for for ADP iajs-2429 182 20 every every DET iajs-2429 182 21 nonzero nonzero PROPN iajs-2429 182 22 submodule submodule NOUN iajs-2429 182 23 𝑁 𝑁 PROPN iajs-2429 182 24 of of ADP iajs-2429 182 25 the the DET iajs-2429 182 26 ℤ-module ℤ-module PROPN iajs-2429 182 27 ℤ ℤ PROPN iajs-2429 182 28 while while SCONJ iajs-2429 182 29 𝑁 𝑁 PROPN iajs-2429 182 30 is be AUX iajs-2429 182 31 not not PART iajs-2429 182 32 semisecond semisecond ADJ iajs-2429 182 33 . . PUNCT iajs-2429 183 1 corollary corollary ADJ iajs-2429 183 2 ( ( PUNCT iajs-2429 183 3 3.5 3.5 NUM iajs-2429 183 4 ): ): PUNCT iajs-2429 183 5 if if SCONJ iajs-2429 183 6 𝑁 𝑁 PROPN iajs-2429 183 7 is be AUX iajs-2429 183 8 a a DET iajs-2429 183 9 semisecond semisecond ADJ iajs-2429 183 10 submodule submodule NOUN iajs-2429 183 11 of of ADP iajs-2429 183 12 an an DET iajs-2429 183 13 𝑅-module 𝑅-module PROPN iajs-2429 183 14 𝑀 𝑀 PROPN iajs-2429 183 15 then then ADV iajs-2429 183 16 for for ADP iajs-2429 183 17 every every DET iajs-2429 183 18 submodule submodule NOUN iajs-2429 183 19 𝐾 𝐾 PROPN iajs-2429 183 20 ⊉ ⊉ PROPN iajs-2429 183 21 𝑁 𝑁 PROPN iajs-2429 183 22 in in ADP iajs-2429 183 23 𝑀 𝑀 PROPN iajs-2429 183 24 we we PRON iajs-2429 183 25 have have VERB iajs-2429 183 26 𝐾 𝐾 NOUN iajs-2429 183 27 : : PUNCT iajs-2429 183 28 𝑁 𝑁 PROPN iajs-2429 183 29 𝐾 𝐾 PROPN iajs-2429 183 30 : : PUNCT iajs-2429 183 31 𝑁𝑏 𝑁𝑏 PROPN iajs-2429 183 32 for for ADP iajs-2429 183 33 each each DET iajs-2429 183 34 𝑏 𝑏 PRON iajs-2429 183 35 ∈ ∈ NOUN iajs-2429 183 36 𝑅. 𝑅. NOUN iajs-2429 183 37 proof proof NOUN iajs-2429 183 38 . . PUNCT iajs-2429 184 1 let let VERB iajs-2429 184 2 𝑎 𝑎 NOUN iajs-2429 184 3 ∈ ∈ PROPN iajs-2429 184 4 𝐾 𝐾 NOUN iajs-2429 184 5 : : PUNCT iajs-2429 184 6 𝑁 𝑁 PROPN iajs-2429 184 7 then then ADV iajs-2429 184 8 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 184 9 ⊆ ⊆ PROPN iajs-2429 184 10 𝐾 𝐾 PROPN iajs-2429 184 11 implies imply VERB iajs-2429 184 12 for for ADP iajs-2429 184 13 each each DET iajs-2429 184 14 𝑏 𝑏 PROPN iajs-2429 184 15 ∈ ∈ PROPN iajs-2429 184 16 𝑅 𝑅 PROPN iajs-2429 184 17 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 184 18 ⊆ ⊆ NUM iajs-2429 184 19 𝐾 𝐾 NOUN iajs-2429 184 20 so so ADV iajs-2429 184 21 𝑎 𝑎 PRON iajs-2429 184 22 ∈ ∈ PROPN iajs-2429 184 23 𝐾 𝐾 NOUN iajs-2429 184 24 : : PUNCT iajs-2429 184 25 𝑁𝑏 𝑁𝑏 PROPN iajs-2429 184 26 . . PUNCT iajs-2429 185 1 conversly conversly ADV iajs-2429 185 2 , , PUNCT iajs-2429 185 3 let let VERB iajs-2429 185 4 𝑎 𝑎 PROPN iajs-2429 185 5 ∈ ∈ PROPN iajs-2429 185 6 𝐾 𝐾 NOUN iajs-2429 185 7 : : PUNCT iajs-2429 185 8 𝑁𝑏 𝑁𝑏 PROPN iajs-2429 185 9 then then ADV iajs-2429 185 10 𝑁𝑎𝑏 𝑁𝑎𝑏 PROPN iajs-2429 185 11 ⊆ ⊆ PROPN iajs-2429 185 12 𝐾 𝐾 PROPN iajs-2429 185 13 implies imply VERB iajs-2429 185 14 𝑎𝑏 𝑎𝑏 PROPN iajs-2429 185 15 ∈ ∈ PROPN iajs-2429 185 16 𝐾 𝐾 PROPN iajs-2429 185 17 : : PUNCT iajs-2429 185 18 𝑁 𝑁 PROPN iajs-2429 185 19 and and CCONJ iajs-2429 185 20 we we PRON iajs-2429 185 21 can can AUX iajs-2429 185 22 take take VERB iajs-2429 185 23 𝑏 𝑏 PRON iajs-2429 185 24 𝑎 𝑎 NOUN iajs-2429 185 25 then then ADV iajs-2429 185 26 𝑎 𝑎 PROPN iajs-2429 185 27 ∈ ∈ ADJ iajs-2429 185 28 𝐾 𝐾 NOUN iajs-2429 185 29 : : PUNCT iajs-2429 185 30 𝑁 𝑁 PROPN iajs-2429 185 31 . . PUNCT iajs-2429 186 1 via via ADP iajs-2429 186 2 theorem theorem ADJ iajs-2429 186 3 3.1 3.1 NUM iajs-2429 186 4 , , PUNCT iajs-2429 186 5 𝐾 𝐾 NOUN iajs-2429 186 6 : : PUNCT iajs-2429 186 7 𝑁 𝑁 PROPN iajs-2429 186 8 is be AUX iajs-2429 186 9 a a DET iajs-2429 186 10 semiprime semiprime NOUN iajs-2429 186 11 ideal ideal NOUN iajs-2429 186 12 of of ADP iajs-2429 186 13 𝑅 𝑅 PROPN iajs-2429 186 14 implies imply VERB iajs-2429 186 15 𝑎 𝑎 PROPN iajs-2429 186 16 ∈ ∈ PROPN iajs-2429 186 17 𝐾 𝐾 NOUN iajs-2429 186 18 : : PUNCT iajs-2429 186 19 𝑁 𝑁 PROPN iajs-2429 186 20 as as SCONJ iajs-2429 186 21 required require VERB iajs-2429 186 22 . . PUNCT iajs-2429 187 1 corollary corollary ADJ iajs-2429 187 2 ( ( PUNCT iajs-2429 187 3 3.6 3.6 NUM iajs-2429 187 4 ): ): PUNCT iajs-2429 187 5 if if SCONJ iajs-2429 187 6 𝑁 𝑁 PROPN iajs-2429 187 7 is be AUX iajs-2429 187 8 a a DET iajs-2429 187 9 semisecond semisecond ADJ iajs-2429 187 10 submodule submodule NOUN iajs-2429 187 11 of of ADP iajs-2429 187 12 an an DET iajs-2429 187 13 𝑅-module 𝑅-module PROPN iajs-2429 187 14 𝑀 𝑀 PROPN iajs-2429 187 15 then then ADV iajs-2429 187 16 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 187 17 𝑁 𝑁 PROPN iajs-2429 187 18 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 187 19 𝑁𝑏 𝑁𝑏 PROPN iajs-2429 187 20 for for ADP iajs-2429 187 21 each each DET iajs-2429 187 22 𝑏 𝑏 PRON iajs-2429 187 23 ∈ ∈ NOUN iajs-2429 187 24 𝑅. 𝑅. NOUN iajs-2429 187 25 proof proof NOUN iajs-2429 187 26 . . PUNCT iajs-2429 188 1 directly directly ADV iajs-2429 188 2 by by ADP iajs-2429 188 3 corollary corollary ADJ iajs-2429 188 4 3.5 3.5 NUM iajs-2429 188 5 . . PUNCT iajs-2429 189 1 theorem theorem NOUN iajs-2429 189 2 ( ( PUNCT iajs-2429 189 3 3.7 3.7 NUM iajs-2429 189 4 ): ): PUNCT iajs-2429 189 5 the the DET iajs-2429 189 6 following following ADJ iajs-2429 189 7 statements statement NOUN iajs-2429 189 8 are be AUX iajs-2429 189 9 equivalent equivalent ADJ iajs-2429 189 10 ( ( PUNCT iajs-2429 189 11 1 1 X iajs-2429 189 12 ) ) PUNCT iajs-2429 189 13 𝑁 𝑁 NOUN iajs-2429 189 14 is be AUX iajs-2429 189 15 a a DET iajs-2429 189 16 semisecond semisecond ADJ iajs-2429 189 17 submodule submodule NOUN iajs-2429 189 18 of of ADP iajs-2429 189 19 an an DET iajs-2429 189 20 𝑅-module 𝑅-module PROPN iajs-2429 189 21 𝑀. 𝑀. PROPN iajs-2429 189 22 ( ( PUNCT iajs-2429 189 23 2 2 NUM iajs-2429 189 24 ) ) PUNCT iajs-2429 189 25 𝑁 𝑁 PROPN iajs-2429 189 26 0 0 NUM iajs-2429 189 27 and and CCONJ iajs-2429 189 28 for for ADP iajs-2429 189 29 each each DET iajs-2429 189 30 ideals ideal NOUN iajs-2429 189 31 𝐼 𝐼 ADP iajs-2429 189 32 of of ADP iajs-2429 189 33 𝑅 𝑅 PROPN iajs-2429 189 34 such such ADJ iajs-2429 189 35 that that SCONJ iajs-2429 189 36 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 189 37 ⊆ ⊆ PROPN iajs-2429 189 38 𝐾 𝐾 PROPN iajs-2429 189 39 implies imply VERB iajs-2429 189 40 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 189 41 ⊆ ⊆ NUM iajs-2429 189 42 𝐾. 𝐾. PROPN iajs-2429 189 43 proof proof NOUN iajs-2429 189 44 . . PUNCT iajs-2429 190 1 ( ( PUNCT iajs-2429 190 2 1 1 X iajs-2429 190 3 ) ) PUNCT iajs-2429 190 4   NOUN iajs-2429 190 5 ( ( PUNCT iajs-2429 190 6 2 2 NUM iajs-2429 190 7 ) ) PUNCT iajs-2429 190 8 first first ADV iajs-2429 190 9 since since SCONJ iajs-2429 190 10 𝑁 𝑁 PROPN iajs-2429 190 11 is be AUX iajs-2429 190 12 a a DET iajs-2429 190 13 semisecond semisecond ADJ iajs-2429 190 14 submodule submodule NOUN iajs-2429 190 15 of of ADP iajs-2429 190 16 an an DET iajs-2429 190 17 𝑅-module 𝑅-module PROPN iajs-2429 190 18 𝑀 𝑀 PROPN iajs-2429 190 19 then then ADV iajs-2429 190 20 𝑁 𝑁 PROPN iajs-2429 190 21 0 0 NUM iajs-2429 190 22 . . PUNCT iajs-2429 191 1 let let VERB iajs-2429 191 2 𝐼 𝐼 PRON iajs-2429 191 3 be be AUX iajs-2429 191 4 an an DET iajs-2429 191 5 ideal ideal NOUN iajs-2429 191 6 of of ADP iajs-2429 191 7 𝑅 𝑅 PROPN iajs-2429 191 8 and and CCONJ iajs-2429 191 9 𝐾 𝐾 PROPN iajs-2429 191 10 a a DET iajs-2429 191 11 submodule submodule NOUN iajs-2429 191 12 of of ADP iajs-2429 191 13 𝑀. 𝑀. PROPN iajs-2429 192 1 if if SCONJ iajs-2429 192 2 𝑁 𝑁 PROPN iajs-2429 192 3 ⊈ ⊈ PROPN iajs-2429 192 4 𝐾 𝐾 NOUN iajs-2429 192 5 we we PRON iajs-2429 192 6 have have VERB iajs-2429 192 7 either either CCONJ iajs-2429 192 8 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 192 9 ⊈ ⊈ PROPN iajs-2429 192 10 𝐾 𝐾 PROPN iajs-2429 192 11 and and CCONJ iajs-2429 192 12 so so ADV iajs-2429 192 13 nothing nothing PRON iajs-2429 192 14 to to PART iajs-2429 192 15 prove prove VERB iajs-2429 192 16 or or CCONJ iajs-2429 192 17 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 192 18 ⊆ ⊆ PROPN iajs-2429 192 19 𝐾 𝐾 PROPN iajs-2429 192 20 it it PRON iajs-2429 192 21 follows follow VERB iajs-2429 192 22 𝐼 𝐼 ADP iajs-2429 192 23 ⊆ ⊆ NUM iajs-2429 192 24 𝐾 𝐾 NOUN iajs-2429 192 25 : : PUNCT iajs-2429 192 26 𝑁 𝑁 PROPN iajs-2429 192 27 and and CCONJ iajs-2429 192 28 by by ADP iajs-2429 192 29 theorem theorem ADJ iajs-2429 192 30 3.1 3.1 NUM iajs-2429 192 31 , , PUNCT iajs-2429 192 32 𝐾 𝐾 NOUN iajs-2429 192 33 : : PUNCT iajs-2429 192 34 𝑁 𝑁 PROPN iajs-2429 192 35 is be AUX iajs-2429 192 36 a a DET iajs-2429 192 37 semiprime semiprime NOUN iajs-2429 192 38 ideal ideal NOUN iajs-2429 192 39 of of ADP iajs-2429 192 40 𝑅 𝑅 PROPN iajs-2429 192 41 so so ADV iajs-2429 192 42 𝐼 𝐼 PROPN iajs-2429 192 43 ⊆ ⊆ NUM iajs-2429 192 44 𝐾 𝐾 PROPN iajs-2429 192 45 : : PUNCT iajs-2429 192 46 𝑁 𝑁 PROPN iajs-2429 192 47 and and CCONJ iajs-2429 192 48 hence hence ADV iajs-2429 192 49 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 192 50 ⊆ ⊆ NUM iajs-2429 192 51 𝐾. 𝐾. PROPN iajs-2429 192 52 in in ADP iajs-2429 192 53 case case NOUN iajs-2429 192 54 𝑁 𝑁 PROPN iajs-2429 192 55 ⊆ ⊆ NUM iajs-2429 192 56 𝐾 𝐾 PROPN iajs-2429 192 57 then then ADV iajs-2429 192 58 the the DET iajs-2429 192 59 result result NOUN iajs-2429 192 60 already already ADV iajs-2429 192 61 is be AUX iajs-2429 192 62 obtained obtain VERB iajs-2429 192 63 . . PUNCT iajs-2429 193 1 ( ( PUNCT iajs-2429 193 2 2 2 NUM iajs-2429 193 3 ) ) PUNCT iajs-2429 193 4   NOUN iajs-2429 193 5 ( ( PUNCT iajs-2429 193 6 1 1 X iajs-2429 193 7 ) ) PUNCT iajs-2429 193 8 let let VERB iajs-2429 193 9 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 193 10 ⊆ ⊆ NUM iajs-2429 193 11 𝐾 𝐾 PROPN iajs-2429 193 12 , , PUNCT iajs-2429 193 13 where where SCONJ iajs-2429 193 14 𝑎 𝑎 PROPN iajs-2429 193 15 ∈ ∈ PROPN iajs-2429 193 16 𝑅 𝑅 PROPN iajs-2429 193 17 and and CCONJ iajs-2429 193 18 𝐾 𝐾 PROPN iajs-2429 193 19 a a DET iajs-2429 193 20 submodule submodule NOUN iajs-2429 193 21 of of ADP iajs-2429 193 22 𝑀 𝑀 PROPN iajs-2429 193 23 , , PUNCT iajs-2429 193 24 then then ADV iajs-2429 193 25 𝑁 𝑁 PROPN iajs-2429 193 26 𝑎 𝑎 PRON iajs-2429 193 27 ⊆ ⊆ NUM iajs-2429 193 28 𝐾. 𝐾. NOUN iajs-2429 193 29 by by ADP iajs-2429 193 30 hypothesis hypothesis NOUN iajs-2429 193 31 𝑁 𝑁 PROPN iajs-2429 193 32 𝑎 𝑎 NOUN iajs-2429 193 33 ⊆ ⊆ NUM iajs-2429 193 34 𝐾 𝐾 NOUN iajs-2429 193 35 where where SCONJ iajs-2429 193 36 𝑎 𝑎 NOUN iajs-2429 193 37 is be AUX iajs-2429 193 38 the the DET iajs-2429 193 39 principal principal ADJ iajs-2429 193 40 ideal ideal NOUN iajs-2429 193 41 generated generate VERB iajs-2429 193 42 by by ADP iajs-2429 193 43 𝑎 𝑎 PROPN iajs-2429 193 44 and and CCONJ iajs-2429 193 45 hence hence ADV iajs-2429 193 46 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 193 47 ⊆ ⊆ PROPN iajs-2429 193 48 𝐾 𝐾 PROPN iajs-2429 193 49 as as SCONJ iajs-2429 193 50 dsired dsire VERB iajs-2429 193 51 . . PUNCT iajs-2429 194 1 corollary corollary ADJ iajs-2429 194 2 ( ( PUNCT iajs-2429 194 3 3.8 3.8 NUM iajs-2429 194 4 ): ): PUNCT iajs-2429 194 5 the the DET iajs-2429 194 6 following following ADJ iajs-2429 194 7 statements statement NOUN iajs-2429 194 8 are be AUX iajs-2429 194 9     SPACE iajs-2429 194 10 equivalent equivalent ADJ iajs-2429 194 11 ( ( PUNCT iajs-2429 194 12 1 1 X iajs-2429 194 13 ) ) PUNCT iajs-2429 194 14 𝑁 𝑁 NOUN iajs-2429 194 15 is be AUX iajs-2429 194 16 a a DET iajs-2429 194 17 semisecond semisecond ADJ iajs-2429 194 18 submodule submodule NOUN iajs-2429 194 19 of of ADP iajs-2429 194 20 an an DET iajs-2429 194 21 𝑅-module 𝑅-module PROPN iajs-2429 194 22 𝑀. 𝑀. PROPN iajs-2429 194 23 ( ( PUNCT iajs-2429 194 24 2 2 NUM iajs-2429 194 25 ) ) PUNCT iajs-2429 194 26 𝑁 𝑁 PROPN iajs-2429 194 27 0 0 NUM iajs-2429 194 28 and and CCONJ iajs-2429 194 29 for for ADP iajs-2429 194 30 each each DET iajs-2429 194 31 ideal ideal ADJ iajs-2429 194 32 𝐼 𝐼 PROPN iajs-2429 194 33 of of ADP iajs-2429 194 34 𝑅 𝑅 PROPN iajs-2429 194 35 and and CCONJ iajs-2429 194 36 𝐾 𝐾 PROPN iajs-2429 194 37 a a DET iajs-2429 194 38 submodule submodule NOUN iajs-2429 194 39 of of ADP iajs-2429 194 40 𝑀 𝑀 PROPN iajs-2429 194 41 such such ADJ iajs-2429 194 42 that that SCONJ iajs-2429 194 43 𝑁 𝑁 PROPN iajs-2429 194 44 ⊈ ⊈ PROPN iajs-2429 194 45 𝐾 𝐾 PROPN iajs-2429 194 46 and and CCONJ iajs-2429 194 47 𝐼 𝐼 PROPN iajs-2429 194 48 ⊆ ⊆ NUM iajs-2429 194 49 𝐾 𝐾 NOUN iajs-2429 194 50 : : PUNCT iajs-2429 194 51 𝑁 𝑁 PROPN iajs-2429 194 52 implies imply VERB iajs-2429 194 53 𝐼 𝐼 ADP iajs-2429 194 54 ⊆ ⊆ NUM iajs-2429 194 55 𝐾 𝐾 NOUN iajs-2429 194 56 : : PUNCT iajs-2429 194 57 𝑁 𝑁 PROPN iajs-2429 194 58 . . PUNCT iajs-2429 195 1 proof proof NOUN iajs-2429 195 2 . . PUNCT iajs-2429 196 1 directly directly ADV iajs-2429 196 2 via via ADP iajs-2429 196 3 corollary corollary ADJ iajs-2429 196 4 3.7 3.7 NUM iajs-2429 196 5 . . PUNCT iajs-2429 197 1 corollary corollary ADJ iajs-2429 197 2 ( ( PUNCT iajs-2429 197 3 3.9 3.9 NUM iajs-2429 197 4 ): ): PUNCT iajs-2429 197 5 the the DET iajs-2429 197 6 following following ADJ iajs-2429 197 7 statements statement NOUN iajs-2429 197 8 are be AUX iajs-2429 197 9     SPACE iajs-2429 197 10 equivalent equivalent ADJ iajs-2429 197 11 ( ( PUNCT iajs-2429 197 12 1 1 X iajs-2429 197 13 ) ) PUNCT iajs-2429 197 14 𝑁 𝑁 NOUN iajs-2429 197 15 is be AUX iajs-2429 197 16 a a DET iajs-2429 197 17 semisecond semisecond ADJ iajs-2429 197 18 submodule submodule NOUN iajs-2429 197 19 of of ADP iajs-2429 197 20 an an DET iajs-2429 197 21 𝑅-module 𝑅-module PROPN iajs-2429 197 22 𝑀. 𝑀. PROPN iajs-2429 197 23 ( ( PUNCT iajs-2429 197 24 2 2 NUM iajs-2429 197 25 ) ) PUNCT iajs-2429 197 26 𝑁 𝑁 PROPN iajs-2429 197 27 0 0 NUM iajs-2429 197 28 and and CCONJ iajs-2429 197 29 for for SCONJ iajs-2429 197 30 each each DET iajs-2429 197 31 ideal ideal ADJ iajs-2429 197 32 𝐼 𝐼 PROPN iajs-2429 197 33 of of ADP iajs-2429 197 34 𝑅 𝑅 PROPN iajs-2429 197 35 implies imply VERB iajs-2429 197 36 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 197 37 𝑁𝐼. 𝑁𝐼. ADP iajs-2429 197 38 proof proof NOUN iajs-2429 197 39 . . PUNCT iajs-2429 198 1 ( ( PUNCT iajs-2429 198 2 1 1 X iajs-2429 198 3 ) ) PUNCT iajs-2429 198 4   NOUN iajs-2429 198 5 ( ( PUNCT iajs-2429 198 6 2 2 NUM iajs-2429 198 7 ) ) PUNCT iajs-2429 198 8 first first ADV iajs-2429 198 9 since since SCONJ iajs-2429 198 10 𝑁 𝑁 PROPN iajs-2429 198 11 is be AUX iajs-2429 198 12 a a DET iajs-2429 198 13 semisecond semisecond ADJ iajs-2429 198 14 submodule submodule NOUN iajs-2429 198 15 of of ADP iajs-2429 198 16 an an DET iajs-2429 198 17 𝑅-module 𝑅-module PROPN iajs-2429 198 18 𝑀 𝑀 PROPN iajs-2429 198 19 then then ADV iajs-2429 198 20 𝑁 𝑁 PROPN iajs-2429 198 21 0 0 NUM iajs-2429 198 22 . . PUNCT iajs-2429 199 1 let let VERB iajs-2429 199 2 𝐼 𝐼 PRON iajs-2429 199 3 be be AUX iajs-2429 199 4 an an DET iajs-2429 199 5 ideal ideal NOUN iajs-2429 199 6 of of ADP iajs-2429 199 7 𝑅 𝑅 PROPN iajs-2429 199 8 then then ADV iajs-2429 199 9 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 199 10 ⊆ ⊆ NUM iajs-2429 199 11 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 199 12 so so ADV iajs-2429 199 13 by by ADP iajs-2429 199 14 theorem theorem NOUN iajs-2429 199 15 3.7 3.7 NUM iajs-2429 199 16 , , PUNCT iajs-2429 199 17 we we PRON iajs-2429 199 18 have have VERB iajs-2429 199 19 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 199 20 ⊆ ⊆ NUM iajs-2429 199 21 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 199 22 and and CCONJ iajs-2429 199 23 thus thus ADV iajs-2429 199 24 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 199 25 𝑁𝐼. 𝑁𝐼. CCONJ iajs-2429 199 26     SPACE iajs-2429 199 27 87 87 NUM iajs-2429 199 28     SPACE iajs-2429 199 29 ibn ibn PROPN iajs-2429 199 30 al al PROPN iajs-2429 199 31 - - PUNCT iajs-2429 199 32 haitham haitham PROPN iajs-2429 199 33 jour jour X iajs-2429 199 34 . . PROPN iajs-2429 200 1 for for ADP iajs-2429 200 2 pure pure ADJ iajs-2429 200 3 & & CCONJ iajs-2429 200 4 appl appl PROPN iajs-2429 200 5 . . PUNCT iajs-2429 201 1 sci sci PROPN iajs-2429 201 2 . . PROPN iajs-2429 202 1 33 33 NUM iajs-2429 202 2 ( ( PUNCT iajs-2429 202 3 2 2 NUM iajs-2429 202 4 ) ) PUNCT iajs-2429 202 5 2020 2020 NUM iajs-2429 202 6 ( ( PUNCT iajs-2429 202 7 2 2 NUM iajs-2429 202 8 ) ) PUNCT iajs-2429 202 9   NOUN iajs-2429 202 10 ( ( PUNCT iajs-2429 202 11 1 1 X iajs-2429 202 12 ) ) PUNCT iajs-2429 202 13 it it PRON iajs-2429 202 14 is be AUX iajs-2429 202 15 clear clear ADJ iajs-2429 202 16 . . PUNCT iajs-2429 203 1 theorem theorem NOUN iajs-2429 203 2 ( ( PUNCT iajs-2429 203 3 3.10 3.10 NUM iajs-2429 203 4 ): ): PUNCT iajs-2429 203 5 let let VERB iajs-2429 203 6 𝑁 𝑁 PROPN iajs-2429 203 7 be be AUX iajs-2429 203 8 a a DET iajs-2429 203 9 submodule submodule NOUN iajs-2429 203 10 of of ADP iajs-2429 203 11 an an DET iajs-2429 203 12 𝑅-module 𝑅-module PROPN iajs-2429 203 13 𝑀. 𝑀. PROPN iajs-2429 203 14 if if SCONJ iajs-2429 203 15 for for SCONJ iajs-2429 203 16 each each DET iajs-2429 203 17 𝑎 𝑎 PROPN iajs-2429 203 18 ∈ ∈ PROPN iajs-2429 203 19 𝑅 𝑅 PROPN iajs-2429 203 20 , , PUNCT iajs-2429 203 21 𝑎 𝑎 PROPN iajs-2429 203 22 𝑅 𝑅 PROPN iajs-2429 203 23 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 203 24 𝑁 𝑁 PROPN iajs-2429 203 25 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 203 26 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 203 27 𝑁 𝑁 NOUN iajs-2429 203 28 then then ADV iajs-2429 203 29 𝑁 𝑁 PROPN iajs-2429 203 30 is be AUX iajs-2429 203 31 semisecond semisecond ADJ iajs-2429 203 32 . . PUNCT iajs-2429 204 1 proof proof NOUN iajs-2429 204 2 . . PUNCT iajs-2429 205 1 assume assume VERB iajs-2429 205 2 for for ADP iajs-2429 205 3 each each DET iajs-2429 205 4 𝑎 𝑎 PROPN iajs-2429 205 5 ∈ ∈ PROPN iajs-2429 205 6 𝑅 𝑅 PROPN iajs-2429 205 7 , , PUNCT iajs-2429 205 8 𝑎 𝑎 PROPN iajs-2429 205 9 𝑅 𝑅 PROPN iajs-2429 205 10 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 205 11 𝑁 𝑁 PROPN iajs-2429 205 12 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 205 13 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 205 14 𝑁 𝑁 NOUN iajs-2429 205 15 then then ADV iajs-2429 205 16 𝑎 𝑎 NOUN iajs-2429 205 17 𝑏 𝑏 NOUN iajs-2429 205 18 𝑎 𝑎 PROPN iajs-2429 205 19 𝑐 𝑐 NOUN iajs-2429 205 20 for for ADP iajs-2429 205 21 some some DET iajs-2429 205 22 𝑏 𝑏 NOUN iajs-2429 205 23 , , PUNCT iajs-2429 205 24 𝑐 𝑐 PROPN iajs-2429 205 25 ∈ ∈ PROPN iajs-2429 205 26 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 205 27 𝑁 𝑁 PROPN iajs-2429 205 28 implies imply VERB iajs-2429 205 29 𝑎 𝑎 NUM iajs-2429 205 30 𝑎 𝑎 NOUN iajs-2429 205 31 ∈ ∈ NOUN iajs-2429 205 32 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 205 33 𝑁 𝑁 PROPN iajs-2429 205 34 and and CCONJ iajs-2429 205 35 hence hence ADV iajs-2429 205 36 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 205 37 𝑁𝑎. 𝑁𝑎. PROPN iajs-2429 205 38 theorem theorem NOUN iajs-2429 205 39 ( ( PUNCT iajs-2429 205 40 3.11 3.11 NUM iajs-2429 205 41 ): ): PUNCT iajs-2429 205 42 if if SCONJ iajs-2429 205 43 𝑁 𝑁 PROPN iajs-2429 205 44 is be AUX iajs-2429 205 45 a a DET iajs-2429 205 46 semisecond semisecond ADJ iajs-2429 205 47 finitely finitely ADV iajs-2429 205 48 generated generate VERB iajs-2429 205 49 submodule submodule NOUN iajs-2429 205 50 of of ADP iajs-2429 205 51 an an DET iajs-2429 205 52 𝑅-module 𝑅-module PROPN iajs-2429 205 53 𝑀 𝑀 PROPN iajs-2429 205 54 then then ADV iajs-2429 205 55 for for SCONJ iajs-2429 205 56 each each DET iajs-2429 205 57 𝑎 𝑎 PROPN iajs-2429 205 58 ∈ ∈ PROPN iajs-2429 205 59 𝑅 𝑅 PROPN iajs-2429 205 60 , , PUNCT iajs-2429 205 61 𝑎 𝑎 PROPN iajs-2429 205 62 𝑅 𝑅 PROPN iajs-2429 205 63 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 205 64 𝑁 𝑁 PROPN iajs-2429 205 65 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 205 66 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 205 67 𝑁 𝑁 PROPN iajs-2429 205 68 . . PUNCT iajs-2429 206 1 proof proof NOUN iajs-2429 206 2 . . PUNCT iajs-2429 207 1 let let VERB iajs-2429 207 2 𝑎 𝑎 PRON iajs-2429 207 3 ∈ ∈ PROPN iajs-2429 207 4 𝑅 𝑅 PROPN iajs-2429 207 5 then then ADV iajs-2429 207 6 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 207 7 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 207 8 that that PRON iajs-2429 207 9 is be AUX iajs-2429 207 10 𝑁 𝑁 PROPN iajs-2429 207 11 𝑎𝑅 𝑎𝑅 PROPN iajs-2429 207 12 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 207 13 𝑁 𝑁 PROPN iajs-2429 207 14 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 207 15 . . PUNCT iajs-2429 208 1 by by ADP iajs-2429 208 2 hypothesis hypothesis NOUN iajs-2429 208 3 𝑁 𝑁 PROPN iajs-2429 208 4 is be AUX iajs-2429 208 5 finitely finitely ADV iajs-2429 208 6 generated generate VERB iajs-2429 208 7 . . PUNCT iajs-2429 209 1 it it PRON iajs-2429 209 2 is be AUX iajs-2429 209 3 not not PART iajs-2429 209 4 hard hard ADJ iajs-2429 209 5 to to PART iajs-2429 209 6 see see VERB iajs-2429 209 7 that that SCONJ iajs-2429 209 8 𝑁 𝑁 PROPN iajs-2429 209 9 𝑎𝑅 𝑎𝑅 PROPN iajs-2429 209 10 is be AUX iajs-2429 209 11 also also ADV iajs-2429 209 12 finitely finitely ADV iajs-2429 209 13 generated generate VERB iajs-2429 209 14 . . PUNCT iajs-2429 210 1 via via ADP iajs-2429 210 2 [ [ X iajs-2429 210 3 23 23 NUM iajs-2429 210 4 , , PUNCT iajs-2429 210 5 corollary corollary ADJ iajs-2429 210 6 2.5 2.5 NUM iajs-2429 210 7 ] ] PUNCT iajs-2429 210 8 , , PUNCT iajs-2429 210 9 it it PRON iajs-2429 210 10 follows follow VERB iajs-2429 210 11 that that SCONJ iajs-2429 210 12 𝑥 𝑥 PROPN iajs-2429 210 13 1 1 NUM iajs-2429 210 14 ∈ ∈ NOUN iajs-2429 210 15 𝑅𝑎 𝑅𝑎 PROPN iajs-2429 210 16 and and CCONJ iajs-2429 210 17 𝑁𝑥 𝑁𝑥 PROPN iajs-2429 210 18 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 210 19 0 0 NUM iajs-2429 210 20 . . PUNCT iajs-2429 211 1 let let VERB iajs-2429 211 2 𝑥 𝑥 DET iajs-2429 211 3 1 1 NUM iajs-2429 211 4 𝑎𝑡 𝑎𝑡 NOUN iajs-2429 211 5 for for ADP iajs-2429 211 6 some some DET iajs-2429 211 7 𝑡 𝑡 PROPN iajs-2429 211 8 ∈ ∈ PROPN iajs-2429 211 9 𝑅 𝑅 PROPN iajs-2429 211 10 then then ADV iajs-2429 211 11 𝑥 𝑥 PROPN iajs-2429 211 12 𝑎𝑡 𝑎𝑡 DET iajs-2429 211 13 1 1 NUM iajs-2429 211 14 implies imply VERB iajs-2429 211 15 𝑁 𝑁 PROPN iajs-2429 211 16 𝑎𝑡 𝑎𝑡 ADP iajs-2429 211 17 1 1 NUM iajs-2429 211 18 𝑎 𝑎 NOUN iajs-2429 211 19 0 0 NUM iajs-2429 211 20 . . PUNCT iajs-2429 212 1 this this PRON iajs-2429 212 2 means mean VERB iajs-2429 212 3 𝑎 𝑎 VERB iajs-2429 212 4 𝑡 𝑡 PROPN iajs-2429 212 5 𝑎 𝑎 PROPN iajs-2429 212 6 ∈ ∈ NOUN iajs-2429 212 7 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 212 8 𝑁 𝑁 NOUN iajs-2429 212 9 so so ADV iajs-2429 212 10 𝑎 𝑎 PRON iajs-2429 212 11 𝑡 𝑡 NOUN iajs-2429 212 12 𝑎 𝑎 VERB iajs-2429 212 13 𝑏 𝑏 NOUN iajs-2429 212 14 for for ADP iajs-2429 212 15 some some DET iajs-2429 212 16 𝑏 𝑏 PRON iajs-2429 212 17 ∈ ∈ NOUN iajs-2429 212 18 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 212 19 𝑁 𝑁 PROPN iajs-2429 212 20 implies imply VERB iajs-2429 212 21 𝑎 𝑎 NUM iajs-2429 212 22 𝑎 𝑎 NOUN iajs-2429 212 23 𝑡 𝑡 NOUN iajs-2429 212 24 𝑏 𝑏 NOUN iajs-2429 212 25 and and CCONJ iajs-2429 212 26 hence hence ADV iajs-2429 212 27 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 212 28 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 212 29 𝑁 𝑁 PROPN iajs-2429 212 30 ⊆ ⊆ NUM iajs-2429 212 31 𝑎 𝑎 PRON iajs-2429 212 32 𝑅 𝑅 NOUN iajs-2429 212 33 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 212 34 𝑁 𝑁 PROPN iajs-2429 212 35 . . PUNCT iajs-2429 213 1 then then ADV iajs-2429 213 2 𝑎 𝑎 PROPN iajs-2429 213 3 𝑅 𝑅 PROPN iajs-2429 213 4 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 213 5 𝑁 𝑁 PROPN iajs-2429 213 6 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 213 7 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 213 8 𝑁 𝑁 PROPN iajs-2429 213 9 . . PUNCT iajs-2429 214 1 theorem theorem PROPN iajs-2429 214 2 ( ( PUNCT iajs-2429 214 3 3.12 3.12 NUM iajs-2429 214 4 ): ): PUNCT iajs-2429 214 5 let let VERB iajs-2429 214 6 𝑁 𝑁 PROPN iajs-2429 214 7 be be AUX iajs-2429 214 8 a a DET iajs-2429 214 9 finitely finitely ADV iajs-2429 214 10 generated generate VERB iajs-2429 214 11 submodule submodule NOUN iajs-2429 214 12 of of ADP iajs-2429 214 13 a a DET iajs-2429 214 14 module module NOUN iajs-2429 214 15 𝑀 𝑀 PROPN iajs-2429 214 16 over over ADP iajs-2429 214 17 a a DET iajs-2429 214 18 commutative commutative ADJ iajs-2429 214 19 ring ring NOUN iajs-2429 214 20 𝑅. 𝑅. ADV iajs-2429 214 21 the the DET iajs-2429 214 22 following follow VERB iajs-2429 214 23 statements statement NOUN iajs-2429 214 24 are be AUX iajs-2429 214 25 equivalent equivalent ADJ iajs-2429 214 26 ( ( PUNCT iajs-2429 214 27 1 1 X iajs-2429 214 28 ) ) PUNCT iajs-2429 214 29 𝑁 𝑁 NOUN iajs-2429 214 30 is be AUX iajs-2429 214 31 semisecond semisecond ADJ iajs-2429 214 32 . . PUNCT iajs-2429 215 1 ( ( PUNCT iajs-2429 215 2 2 2 X iajs-2429 215 3 ) ) PUNCT iajs-2429 215 4 for for ADP iajs-2429 215 5 each each DET iajs-2429 215 6 𝑎 𝑎 PROPN iajs-2429 215 7 ∈ ∈ PROPN iajs-2429 215 8 𝑅 𝑅 PROPN iajs-2429 215 9 , , PUNCT iajs-2429 215 10 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 215 11 𝑁𝑟 𝑁𝑟 PROPN iajs-2429 215 12 𝑁𝑟 𝑁𝑟 PROPN iajs-2429 215 13 for for ADP iajs-2429 215 14 some some DET iajs-2429 215 15 𝑟 𝑟 NOUN iajs-2429 215 16 ∈ ∈ NOUN iajs-2429 215 17 𝑅. 𝑅. NOUN iajs-2429 215 18 proof proof NOUN iajs-2429 215 19 . . PUNCT iajs-2429 216 1 ( ( PUNCT iajs-2429 216 2 1 1 X iajs-2429 216 3 ) ) PUNCT iajs-2429 216 4   NOUN iajs-2429 216 5 ( ( PUNCT iajs-2429 216 6 2 2 NUM iajs-2429 216 7 ) ) PUNCT iajs-2429 216 8 by by ADP iajs-2429 216 9 theorem theorem NOUN iajs-2429 216 10 3.11 3.11 NUM iajs-2429 216 11 , , PUNCT iajs-2429 216 12 for for ADP iajs-2429 216 13 each each DET iajs-2429 216 14 𝑎 𝑎 PROPN iajs-2429 216 15 ∈ ∈ PROPN iajs-2429 216 16 𝑅 𝑅 PROPN iajs-2429 216 17 , , PUNCT iajs-2429 216 18 𝑎 𝑎 PROPN iajs-2429 216 19 𝑅 𝑅 PROPN iajs-2429 216 20 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 216 21 𝑁 𝑁 PROPN iajs-2429 216 22 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 216 23 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 216 24 𝑁 𝑁 PROPN iajs-2429 216 25 . . PUNCT iajs-2429 217 1 then then ADV iajs-2429 217 2 𝑎 𝑎 X iajs-2429 217 3 𝑡 𝑡 NOUN iajs-2429 217 4 𝑏 𝑏 NOUN iajs-2429 217 5 𝑎𝑠 𝑎𝑠 PROPN iajs-2429 217 6 𝑐 𝑐 PROPN iajs-2429 217 7 for for ADP iajs-2429 217 8 some some DET iajs-2429 217 9 𝑠 𝑠 PROPN iajs-2429 217 10 , , PUNCT iajs-2429 217 11 𝑡 𝑡 PROPN iajs-2429 217 12 ∈ ∈ PROPN iajs-2429 217 13 𝑅 𝑅 PROPN iajs-2429 217 14 and and CCONJ iajs-2429 217 15 𝑏 𝑏 NOUN iajs-2429 217 16 , , PUNCT iajs-2429 217 17 𝑐 𝑐 PROPN iajs-2429 217 18 ∈ ∈ PROPN iajs-2429 217 19 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 217 20 𝑁 𝑁 PROPN iajs-2429 217 21 . . PUNCT iajs-2429 218 1 by by ADP iajs-2429 218 2 choosing choose VERB iajs-2429 218 3 𝑠 𝑠 PRON iajs-2429 218 4 1 1 NUM iajs-2429 218 5 we we PRON iajs-2429 218 6 have have VERB iajs-2429 218 7 𝑎 𝑎 NUM iajs-2429 218 8 𝑎 𝑎 NOUN iajs-2429 218 9 𝑡 𝑡 NOUN iajs-2429 218 10 𝑑 𝑑 NOUN iajs-2429 218 11 for for ADP iajs-2429 218 12 some some DET iajs-2429 218 13 𝑑 𝑑 NOUN iajs-2429 218 14 𝑏 𝑏 PROPN iajs-2429 218 15 𝑐 𝑐 PROPN iajs-2429 218 16 ∈ ∈ PROPN iajs-2429 218 17 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 218 18 𝑁 𝑁 NOUN iajs-2429 218 19 thus thus ADV iajs-2429 218 20 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 219 1 ⊆ ⊆ NUM iajs-2429 219 2 𝑎𝑡𝑅 𝑎𝑡𝑅 PROPN iajs-2429 219 3 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 219 4 𝑁 𝑁 PROPN iajs-2429 219 5 implies imply VERB iajs-2429 219 6 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 219 7 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 219 8 𝑁 𝑁 PROPN iajs-2429 219 9 ⊆ ⊆ NUM iajs-2429 219 10 𝑎𝑡𝑅 𝑎𝑡𝑅 PROPN iajs-2429 219 11 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 219 12 𝑁 𝑁 NOUN iajs-2429 219 13 hence hence ADV iajs-2429 219 14 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 219 15 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 219 16 𝑁 𝑁 PROPN iajs-2429 219 17 𝑎𝑡𝑅 𝑎𝑡𝑅 PROPN iajs-2429 219 18 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 219 19 𝑁 𝑁 PROPN iajs-2429 219 20 . . PUNCT iajs-2429 220 1 put put VERB iajs-2429 220 2 𝑟 𝑟 NOUN iajs-2429 220 3 𝑎𝑡 𝑎𝑡 ADP iajs-2429 220 4 it it PRON iajs-2429 220 5 follows follow VERB iajs-2429 220 6 𝑎 𝑎 PRON iajs-2429 220 7 𝑟 𝑟 PRON iajs-2429 220 8 ∈ ∈ NOUN iajs-2429 220 9 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 220 10 𝑁 𝑁 PROPN iajs-2429 220 11 . . PUNCT iajs-2429 221 1 therefore therefore ADV iajs-2429 221 2 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 221 3 𝑁𝑟 𝑁𝑟 PROPN iajs-2429 221 4 but but CCONJ iajs-2429 221 5 𝑎𝑡 𝑎𝑡 NUM iajs-2429 221 6 𝑎 𝑎 PROPN iajs-2429 221 7 𝑡 𝑡 NOUN iajs-2429 221 8 𝑑𝑡 𝑑𝑡 ADP iajs-2429 221 9 that that PRON iajs-2429 221 10 is be AUX iajs-2429 221 11 𝑟 𝑟 X iajs-2429 221 12 𝑟 𝑟 X iajs-2429 221 13 ∈ ∈ NOUN iajs-2429 221 14 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 221 15 𝑁 𝑁 NOUN iajs-2429 221 16 thus thus ADV iajs-2429 221 17 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 221 18 𝑁𝑟 𝑁𝑟 PROPN iajs-2429 221 19 𝑁𝑟 𝑁𝑟 PROPN iajs-2429 221 20 as as SCONJ iajs-2429 221 21 desired desire VERB iajs-2429 221 22 . . PUNCT iajs-2429 222 1 ( ( PUNCT iajs-2429 222 2 2 2 X iajs-2429 222 3 ) ) PUNCT iajs-2429 222 4   NOUN iajs-2429 222 5 ( ( PUNCT iajs-2429 222 6 1 1 NUM iajs-2429 222 7 ) ) PUNCT iajs-2429 222 8 for for ADP iajs-2429 222 9 each each DET iajs-2429 222 10 𝑎 𝑎 PROPN iajs-2429 222 11 ∈ ∈ PROPN iajs-2429 222 12 𝑅 𝑅 PROPN iajs-2429 222 13 , , PUNCT iajs-2429 222 14 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 222 15 𝑁𝑎𝑎 𝑁𝑎𝑎 PROPN iajs-2429 222 16 𝑁𝑟𝑎 𝑁𝑟𝑎 PROPN iajs-2429 222 17 𝑁𝑎𝑟 𝑁𝑎𝑟 PROPN iajs-2429 222 18 𝑁𝑟𝑟 𝑁𝑟𝑟 PROPN iajs-2429 222 19 𝑁𝑟 𝑁𝑟 PROPN iajs-2429 222 20 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 222 21 implies imply VERB iajs-2429 222 22 𝑁 𝑁 PROPN iajs-2429 222 23 is be AUX iajs-2429 222 24 semisecond semisecond ADJ iajs-2429 222 25 . . PUNCT iajs-2429 223 1 4 4 X iajs-2429 223 2 . . NOUN iajs-2429 223 3 semisecond semisecond ADJ iajs-2429 223 4 submodules submodule NOUN iajs-2429 223 5 and and CCONJ iajs-2429 223 6 related related ADJ iajs-2429 223 7 concepts concept NOUN iajs-2429 223 8 let let VERB iajs-2429 223 9 us we PRON iajs-2429 223 10 start start VERB iajs-2429 223 11 by by ADP iajs-2429 223 12 the the DET iajs-2429 223 13 following follow VERB iajs-2429 223 14 observation observation NOUN iajs-2429 223 15 ( ( PUNCT iajs-2429 223 16 observation observation NOUN iajs-2429 223 17 ) ) PUNCT iajs-2429 223 18 theorem theorem NOUN iajs-2429 223 19 ( ( PUNCT iajs-2429 223 20 4.1 4.1 NUM iajs-2429 223 21 ): ): PUNCT iajs-2429 223 22 every every DET iajs-2429 223 23 non non ADJ iajs-2429 223 24 - - ADJ iajs-2429 223 25 zero zero ADJ iajs-2429 223 26 regular regular ADJ iajs-2429 223 27 module module NOUN iajs-2429 223 28 over over ADP iajs-2429 223 29 a a DET iajs-2429 223 30 commutative commutative ADJ iajs-2429 223 31 ring ring NOUN iajs-2429 223 32 is be AUX iajs-2429 223 33 semisecond semisecond ADJ iajs-2429 223 34 . . PUNCT iajs-2429 224 1 proof proof NOUN iajs-2429 224 2 . . PUNCT iajs-2429 225 1 let let VERB iajs-2429 225 2 𝑀 𝑀 PROPN iajs-2429 225 3 be be AUX iajs-2429 225 4 a a DET iajs-2429 225 5 nonzero nonzero NOUN iajs-2429 225 6 regular regular ADJ iajs-2429 225 7 𝑅-module 𝑅-module PROPN iajs-2429 225 8 . . PUNCT iajs-2429 226 1 we we PRON iajs-2429 226 2 show show VERB iajs-2429 226 3 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 226 4 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 226 5 for for ADP iajs-2429 226 6 each each DET iajs-2429 226 7 𝑎 𝑎 PRON iajs-2429 226 8 ∈ ∈ NOUN iajs-2429 226 9 𝑅. 𝑅. NOUN iajs-2429 226 10 let let VERB iajs-2429 226 11 𝑥 𝑥 PRON iajs-2429 226 12 ∈ ∈ PROPN iajs-2429 226 13 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 226 14 implies imply VERB iajs-2429 226 15 𝑥 𝑥 PRON iajs-2429 226 16 𝑚𝑎 𝑚𝑎 ADV iajs-2429 226 17 for for ADP iajs-2429 226 18 some some DET iajs-2429 226 19 𝑚 𝑚 PROPN iajs-2429 226 20 ∈ ∈ PROPN iajs-2429 226 21 𝑀 𝑀 PROPN iajs-2429 226 22 it it PRON iajs-2429 226 23 follows follow VERB iajs-2429 226 24 𝑚𝑎 𝑚𝑎 ADP iajs-2429 226 25 𝑚𝑎𝑟𝑎 𝑚𝑎𝑟𝑎 NOUN iajs-2429 226 26 𝑚𝑎 𝑚𝑎 ADP iajs-2429 226 27 𝑟 𝑟 NOUN iajs-2429 226 28 ∈ ∈ PROPN iajs-2429 226 29 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 226 30 for for ADP iajs-2429 226 31 some some DET iajs-2429 226 32 𝑟 𝑟 PRON iajs-2429 226 33 ∈ ∈ NOUN iajs-2429 226 34 𝑅. 𝑅. NOUN iajs-2429 226 35 example example NOUN iajs-2429 226 36 ( ( PUNCT iajs-2429 226 37 4.2 4.2 NUM iajs-2429 226 38 ): ): PUNCT iajs-2429 226 39 ( ( PUNCT iajs-2429 226 40 1 1 X iajs-2429 226 41 ) ) PUNCT iajs-2429 226 42 every every DET iajs-2429 226 43 regular regular ADJ iajs-2429 226 44 ideal ideal NOUN iajs-2429 226 45 𝐼 𝐼 PROPN iajs-2429 226 46 of of ADP iajs-2429 226 47 commutative commutative ADJ iajs-2429 226 48 ring ring NOUN iajs-2429 226 49 𝑅 𝑅 PROPN iajs-2429 226 50 is be AUX iajs-2429 226 51 a a DET iajs-2429 226 52 semisecond semisecond NOUN iajs-2429 226 53 as as ADP iajs-2429 226 54 𝑅-module 𝑅-module PROPN iajs-2429 226 55 . . PUNCT iajs-2429 227 1 ( ( PUNCT iajs-2429 227 2 2 2 X iajs-2429 227 3 ) ) PUNCT iajs-2429 227 4 ℤ ℤ PROPN iajs-2429 227 5 and and CCONJ iajs-2429 227 6 ℚ ℚ PROPN iajs-2429 227 7 as as SCONJ iajs-2429 227 8 ℤ-modules ℤ-modules PROPN iajs-2429 227 9 are be AUX iajs-2429 227 10 semisecond semisecond ADJ iajs-2429 227 11 but but CCONJ iajs-2429 227 12 not not PART iajs-2429 227 13 regular regular ADJ iajs-2429 227 14 . . PUNCT iajs-2429 228 1 corollary corollary ADJ iajs-2429 228 2 ( ( PUNCT iajs-2429 228 3 4.3 4.3 NUM iajs-2429 228 4 ): ): PUNCT iajs-2429 228 5 every every DET iajs-2429 228 6 non non ADJ iajs-2429 228 7 - - ADJ iajs-2429 228 8 zero zero NUM iajs-2429 228 9 module module NOUN iajs-2429 228 10 over over ADP iajs-2429 228 11 commutative commutative ADJ iajs-2429 228 12 von von PROPN iajs-2429 228 13 neumann neumann PROPN iajs-2429 228 14 regular regular PROPN iajs-2429 228 15 ring ring NOUN iajs-2429 228 16 is be AUX iajs-2429 228 17 semisecond semisecond ADJ iajs-2429 228 18 . . PUNCT iajs-2429 229 1 proof proof NOUN iajs-2429 229 2 . . PUNCT iajs-2429 230 1 since since SCONJ iajs-2429 230 2 every every DET iajs-2429 230 3 module module NOUN iajs-2429 230 4 over over ADP iajs-2429 230 5 von von PROPN iajs-2429 230 6 neumann neumann PROPN iajs-2429 230 7 regular regular PROPN iajs-2429 230 8 ring ring NOUN iajs-2429 230 9 is be AUX iajs-2429 230 10 regular regular ADJ iajs-2429 230 11 so so SCONJ iajs-2429 230 12 the the DET iajs-2429 230 13 result result NOUN iajs-2429 230 14 follows follow VERB iajs-2429 230 15 by by ADP iajs-2429 230 16 theorem theorem ADJ iajs-2429 230 17 4.1 4.1 NUM iajs-2429 230 18 . . PUNCT iajs-2429 230 19 corollary corollary ADJ iajs-2429 230 20 ( ( PUNCT iajs-2429 230 21 4.4 4.4 NUM iajs-2429 230 22 ): ): PUNCT iajs-2429 230 23 every every DET iajs-2429 230 24 nonzero nonzero PROPN iajs-2429 230 25 submodule submodule NOUN iajs-2429 230 26 of of ADP iajs-2429 230 27 a a DET iajs-2429 230 28 regular regular ADJ iajs-2429 230 29 module module NOUN iajs-2429 230 30 over over ADP iajs-2429 230 31 commutative commutative ADJ iajs-2429 230 32 ring ring NOUN iajs-2429 230 33 is be AUX iajs-2429 230 34 semisecond semisecond ADJ iajs-2429 230 35 . . PUNCT iajs-2429 231 1 proof proof NOUN iajs-2429 231 2 . . PUNCT iajs-2429 232 1 since since SCONJ iajs-2429 232 2 every every DET iajs-2429 232 3 submodule submodule NOUN iajs-2429 232 4 of of ADP iajs-2429 232 5 a a DET iajs-2429 232 6 regular regular ADJ iajs-2429 232 7 module module NOUN iajs-2429 232 8 is be AUX iajs-2429 232 9 regular regular ADJ iajs-2429 232 10 , , PUNCT iajs-2429 232 11 so so ADV iajs-2429 232 12 by by ADP iajs-2429 232 13 theorem theorem NOUN iajs-2429 232 14 4.1 4.1 NUM iajs-2429 232 15 we we PRON iajs-2429 232 16 already already ADV iajs-2429 232 17 have have VERB iajs-2429 232 18 the the DET iajs-2429 232 19 result result NOUN iajs-2429 232 20 . . PUNCT iajs-2429 232 21     SPACE iajs-2429 233 1 88 88 NUM iajs-2429 233 2     SPACE iajs-2429 233 3 ibn ibn PROPN iajs-2429 233 4 al al PROPN iajs-2429 233 5 - - PUNCT iajs-2429 233 6 haitham haitham PROPN iajs-2429 233 7 jour jour X iajs-2429 233 8 . . PROPN iajs-2429 233 9 for for ADP iajs-2429 233 10 pure pure ADJ iajs-2429 233 11 & & CCONJ iajs-2429 233 12 appl appl PROPN iajs-2429 233 13 . . PUNCT iajs-2429 234 1 sci sci PROPN iajs-2429 234 2 . . PROPN iajs-2429 235 1 33 33 NUM iajs-2429 235 2 ( ( PUNCT iajs-2429 235 3 2 2 NUM iajs-2429 235 4 ) ) PUNCT iajs-2429 235 5 2020 2020 NUM iajs-2429 235 6 corollary corollary ADJ iajs-2429 235 7 ( ( PUNCT iajs-2429 235 8 4.5 4.5 NUM iajs-2429 235 9 ): ): PUNCT iajs-2429 235 10 every every DET iajs-2429 235 11 nonzero nonzero ADJ iajs-2429 235 12 semisimple semisimple NOUN iajs-2429 235 13 module module NOUN iajs-2429 235 14 over over ADP iajs-2429 235 15 commutative commutative ADJ iajs-2429 235 16 ring ring NOUN iajs-2429 235 17 is be AUX iajs-2429 235 18 semisecond semisecond ADJ iajs-2429 235 19 . . PUNCT iajs-2429 236 1 corollary corollary ADJ iajs-2429 236 2 ( ( PUNCT iajs-2429 236 3 4.6 4.6 NUM iajs-2429 236 4 ): ): PUNCT iajs-2429 236 5 every every DET iajs-2429 236 6 submodule submodule NOUN iajs-2429 236 7 of of ADP iajs-2429 236 8 a a DET iajs-2429 236 9 semisimple semisimple NOUN iajs-2429 236 10 module module NOUN iajs-2429 236 11 over over ADP iajs-2429 236 12 commutative commutative ADJ iajs-2429 236 13 ring ring NOUN iajs-2429 236 14 is be AUX iajs-2429 236 15 semisecond semisecond ADJ iajs-2429 236 16 . . PUNCT iajs-2429 237 1 theorem theorem NOUN iajs-2429 237 2 ( ( PUNCT iajs-2429 237 3 4.7 4.7 NUM iajs-2429 237 4 ): ): PUNCT iajs-2429 237 5 the the DET iajs-2429 237 6 von von PROPN iajs-2429 237 7 neumann neumann PROPN iajs-2429 237 8 regular regular ADJ iajs-2429 237 9 and and CCONJ iajs-2429 237 10 semisecond semisecond ADJ iajs-2429 237 11 notions notion NOUN iajs-2429 237 12 in in ADP iajs-2429 237 13 the the DET iajs-2429 237 14 commutative commutative ADJ iajs-2429 237 15 rings ring NOUN iajs-2429 237 16 are be AUX iajs-2429 237 17 the the DET iajs-2429 237 18 same same ADJ iajs-2429 237 19 . . PUNCT iajs-2429 238 1 proof proof NOUN iajs-2429 238 2 . . PUNCT iajs-2429 239 1 it it PRON iajs-2429 239 2 is be AUX iajs-2429 239 3 clear clear ADJ iajs-2429 239 4 by by ADP iajs-2429 239 5 definitions definition NOUN iajs-2429 239 6 both both DET iajs-2429 239 7 notions notion NOUN iajs-2429 239 8 . . PUNCT iajs-2429 240 1 examples example NOUN iajs-2429 240 2 ( ( PUNCT iajs-2429 240 3 4.8 4.8 NUM iajs-2429 240 4 ): ): PUNCT iajs-2429 240 5 ( ( PUNCT iajs-2429 240 6 1 1 X iajs-2429 240 7 ) ) PUNCT iajs-2429 240 8 the the DET iajs-2429 240 9 commutativity commutativity NOUN iajs-2429 240 10 condition condition NOUN iajs-2429 240 11 in in ADP iajs-2429 240 12 theorem theorem NOUN iajs-2429 240 13 and and CCONJ iajs-2429 240 14 theorem theorem NOUN iajs-2429 240 15 can can AUX iajs-2429 240 16 not not PART iajs-2429 240 17 be be AUX iajs-2429 240 18 dropped drop VERB iajs-2429 240 19 . . PUNCT iajs-2429 241 1 consider consider VERB iajs-2429 241 2 the the DET iajs-2429 241 3 ring ring NOUN iajs-2429 241 4 𝑅 𝑅 PROPN iajs-2429 241 5 ℤ ℤ PROPN iajs-2429 241 6 ℤ ℤ PROPN iajs-2429 241 7 ℤ ℤ PROPN iajs-2429 241 8 ℤ ℤ PROPN iajs-2429 241 9 as as ADP iajs-2429 241 10 a a DET iajs-2429 241 11 right right ADJ iajs-2429 241 12 𝑅-module 𝑅-module NOUN iajs-2429 241 13 . . PUNCT iajs-2429 242 1 by by ADP iajs-2429 242 2 simple simple ADJ iajs-2429 242 3 calculation calculation NOUN iajs-2429 242 4 , , PUNCT iajs-2429 242 5 we we PRON iajs-2429 242 6 see see VERB iajs-2429 242 7 that that SCONJ iajs-2429 242 8 𝑅 𝑅 PROPN iajs-2429 242 9 is be AUX iajs-2429 242 10 von von PROPN iajs-2429 242 11 neumann neumann PROPN iajs-2429 242 12 regular regular PROPN iajs-2429 242 13 and and CCONJ iajs-2429 242 14 𝑅 𝑅 PROPN iajs-2429 242 15 is be AUX iajs-2429 242 16 not not PART iajs-2429 242 17 commutative commutative ADJ iajs-2429 242 18 . . PUNCT iajs-2429 243 1 on on ADP iajs-2429 243 2 the the DET iajs-2429 243 3 other other ADJ iajs-2429 243 4 hand hand NOUN iajs-2429 243 5 , , PUNCT iajs-2429 243 6 if if SCONJ iajs-2429 243 7 we we PRON iajs-2429 243 8 take take VERB iajs-2429 243 9 𝑎 𝑎 ADJ iajs-2429 243 10 0 0 NUM iajs-2429 243 11 1 1 NUM iajs-2429 243 12 0 0 NUM iajs-2429 243 13 0 0 NUM iajs-2429 243 14 ∈ ∈ PROPN iajs-2429 243 15 𝑅 𝑅 PROPN iajs-2429 243 16 implies imply VERB iajs-2429 243 17 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 243 18 𝑎 𝑎 PROPN iajs-2429 243 19 𝑅 𝑅 PROPN iajs-2429 243 20 0 0 NUM iajs-2429 243 21 0 0 NUM iajs-2429 243 22 0 0 NUM iajs-2429 243 23 0 0 NUM iajs-2429 243 24 , , PUNCT iajs-2429 243 25 what what PRON iajs-2429 243 26 follows follow VERB iajs-2429 243 27 𝑅 𝑅 PROPN iajs-2429 243 28 is be AUX iajs-2429 243 29 not not PART iajs-2429 243 30 a a DET iajs-2429 243 31 semisecond semisecond ADJ iajs-2429 243 32 ring ring NOUN iajs-2429 243 33 . . PUNCT iajs-2429 244 1 ( ( PUNCT iajs-2429 244 2 2 2 X iajs-2429 244 3 ) ) PUNCT iajs-2429 244 4 consider consider VERB iajs-2429 244 5 the the DET iajs-2429 244 6 ring ring NOUN iajs-2429 244 7 𝑅 𝑅 PROPN iajs-2429 244 8 ℤ ℤ PROPN iajs-2429 244 9 ℤ ℤ PROPN iajs-2429 244 10 0 0 NUM iajs-2429 244 11 0 0 NUM iajs-2429 244 12 0 0 NUM iajs-2429 244 13 0 0 NUM iajs-2429 244 14 0 0 NUM iajs-2429 244 15 0 0 NUM iajs-2429 244 16 0 0 NUM iajs-2429 244 17 1 1 NUM iajs-2429 244 18 0 0 NUM iajs-2429 244 19 0 0 NUM iajs-2429 244 20 1 1 NUM iajs-2429 244 21 0 0 NUM iajs-2429 244 22 0 0 NUM iajs-2429 244 23 0 0 NUM iajs-2429 244 24 1 1 NUM iajs-2429 244 25 1 1 NUM iajs-2429 244 26 0 0 NUM iajs-2429 244 27 0 0 NUM iajs-2429 244 28 as as ADP iajs-2429 244 29 a a DET iajs-2429 244 30 right right NOUN iajs-2429 244 31 𝑅module 𝑅module PROPN iajs-2429 244 32 where where SCONJ iajs-2429 244 33 𝑅 𝑅 PROPN iajs-2429 244 34 is be AUX iajs-2429 244 35 not not PART iajs-2429 244 36 commutative commutative ADJ iajs-2429 244 37 . . PUNCT iajs-2429 245 1 by by ADP iajs-2429 245 2 simple simple ADJ iajs-2429 245 3 steps step NOUN iajs-2429 245 4 , , PUNCT iajs-2429 245 5 we we PRON iajs-2429 245 6 have have VERB iajs-2429 245 7 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 245 8 𝑎 𝑎 PROPN iajs-2429 245 9 𝑅 𝑅 PROPN iajs-2429 245 10 for for ADP iajs-2429 245 11 each each DET iajs-2429 245 12 𝑎 𝑎 PRON iajs-2429 245 13 ∈ ∈ PROPN iajs-2429 245 14 𝑅 𝑅 PROPN iajs-2429 245 15 it it PRON iajs-2429 245 16 follows follow VERB iajs-2429 245 17 that that SCONJ iajs-2429 245 18 𝑅 𝑅 PROPN iajs-2429 245 19 is be AUX iajs-2429 245 20 semisecond semisecond ADJ iajs-2429 245 21 but but CCONJ iajs-2429 245 22 𝑅 𝑅 PROPN iajs-2429 245 23 is be AUX iajs-2429 245 24 not not PART iajs-2429 245 25 von von PROPN iajs-2429 245 26 neumann neumann PROPN iajs-2429 245 27 regular regular PROPN iajs-2429 245 28 since since SCONJ iajs-2429 245 29 0 0 NUM iajs-2429 245 30 1 1 NUM iajs-2429 245 31 0 0 NUM iajs-2429 245 32 0 0 NUM iajs-2429 245 33 0 0 NUM iajs-2429 245 34 1 1 NUM iajs-2429 245 35 0 0 NUM iajs-2429 245 36 0 0 NUM iajs-2429 245 37 𝑏 𝑏 NOUN iajs-2429 245 38 0 0 NUM iajs-2429 245 39 1 1 NUM iajs-2429 245 40 0 0 NUM iajs-2429 245 41 0 0 NUM iajs-2429 245 42 for for ADP iajs-2429 245 43 each each DET iajs-2429 245 44 𝑏 𝑏 PROPN iajs-2429 245 45 ∈ ∈ NOUN iajs-2429 245 46 𝑅. 𝑅. NOUN iajs-2429 245 47 ( ( PUNCT iajs-2429 245 48 3 3 X iajs-2429 245 49 ) ) PUNCT iajs-2429 245 50 semisecond semisecond NOUN iajs-2429 245 51 modules module NOUN iajs-2429 245 52 may may AUX iajs-2429 245 53 not not PART iajs-2429 245 54 be be AUX iajs-2429 245 55 semisimple semisimple ADJ iajs-2429 245 56 . . PUNCT iajs-2429 246 1 consider consider VERB iajs-2429 246 2 𝑅 𝑅 PROPN iajs-2429 246 3 ∏ ∏ PROPN iajs-2429 246 4 𝔽∈∧ 𝔽∈∧ NOUN iajs-2429 246 5 is be AUX iajs-2429 246 6 commutative commutative ADJ iajs-2429 246 7 von von PROPN iajs-2429 246 8 neumann neumann PROPN iajs-2429 246 9 regular regular PROPN iajs-2429 246 10 ring ring NOUN iajs-2429 246 11 ( ( PUNCT iajs-2429 246 12 𝑅 𝑅 NOUN iajs-2429 246 13 is be AUX iajs-2429 246 14 a a DET iajs-2429 246 15 regular regular ADJ iajs-2429 246 16 as as ADP iajs-2429 246 17 𝑅-module 𝑅-module PROPN iajs-2429 246 18 ) ) PUNCT iajs-2429 246 19 and and CCONJ iajs-2429 246 20 hence hence ADV iajs-2429 246 21 𝑅 𝑅 PROPN iajs-2429 246 22 is be AUX iajs-2429 246 23 semisecond semisecond ADJ iajs-2429 246 24 but but CCONJ iajs-2429 246 25 𝑅 𝑅 NOUN iajs-2429 246 26 is be AUX iajs-2429 246 27 not not PART iajs-2429 246 28 semisimple semisimple NOUN iajs-2429 246 29 since since SCONJ iajs-2429 246 30 the the DET iajs-2429 246 31 submodule submodule PROPN iajs-2429 246 32 𝑅 𝑅 PROPN iajs-2429 246 33 ⊕ ⊕ PROPN iajs-2429 246 34 ∈∧ ∈∧ PROPN iajs-2429 246 35 𝔽 𝔽 PROPN iajs-2429 246 36 is be AUX iajs-2429 246 37 not not PART iajs-2429 246 38 a a DET iajs-2429 246 39 direct direct ADJ iajs-2429 246 40 summand summand NOUN iajs-2429 246 41 of of ADP iajs-2429 246 42 𝑅. 𝑅. NOUN iajs-2429 246 43 proposition proposition NOUN iajs-2429 246 44 ( ( PUNCT iajs-2429 246 45 4.9 4.9 NUM iajs-2429 246 46 ): ): PUNCT iajs-2429 246 47 let let VERB iajs-2429 246 48 𝑅 𝑅 PROPN iajs-2429 246 49 be be AUX iajs-2429 246 50 a a DET iajs-2429 246 51 commutative commutative ADJ iajs-2429 246 52 ring ring NOUN iajs-2429 246 53 then then ADV iajs-2429 246 54 we we PRON iajs-2429 246 55 have have VERB iajs-2429 246 56 the the DET iajs-2429 246 57 equivalent equivalent ADJ iajs-2429 246 58 ( ( PUNCT iajs-2429 246 59 1 1 NUM iajs-2429 246 60 ) ) PUNCT iajs-2429 246 61 𝑅 𝑅 PROPN iajs-2429 246 62 is be AUX iajs-2429 246 63 von von PROPN iajs-2429 246 64 neumann neumann PROPN iajs-2429 246 65 regular regular PROPN iajs-2429 246 66 . . PUNCT iajs-2429 247 1 ( ( PUNCT iajs-2429 247 2 2 2 X iajs-2429 247 3 ) ) PUNCT iajs-2429 247 4 𝑅 𝑅 NOUN iajs-2429 247 5 is be AUX iajs-2429 247 6 fully fully ADV iajs-2429 247 7 semiprime semiprime NOUN iajs-2429 247 8 . . PUNCT iajs-2429 248 1 ( ( PUNCT iajs-2429 248 2 3 3 X iajs-2429 248 3 ) ) PUNCT iajs-2429 248 4 𝑅 𝑅 NOUN iajs-2429 248 5 is be AUX iajs-2429 248 6 fully fully ADV iajs-2429 248 7 idempotent idempotent ADJ iajs-2429 248 8 . . PUNCT iajs-2429 249 1 ( ( PUNCT iajs-2429 249 2 4 4 X iajs-2429 249 3 ) ) PUNCT iajs-2429 249 4 𝑅 𝑅 NOUN iajs-2429 249 5 is be AUX iajs-2429 249 6 a a DET iajs-2429 249 7 dual dual ADJ iajs-2429 249 8 rickart rickart NOUN iajs-2429 249 9 as as ADP iajs-2429 249 10 𝑅-module 𝑅-module PROPN iajs-2429 249 11 . . PUNCT iajs-2429 250 1 ( ( PUNCT iajs-2429 250 2 5 5 X iajs-2429 250 3 ) ) PUNCT iajs-2429 250 4 𝑅 𝑅 PROPN iajs-2429 250 5 is be AUX iajs-2429 250 6 semisecond semisecond ADJ iajs-2429 250 7 ( ( PUNCT iajs-2429 250 8 6 6 NUM iajs-2429 250 9 ) ) PUNCT iajs-2429 250 10 𝑅 𝑅 PROPN iajs-2429 250 11 is be AUX iajs-2429 250 12 cosemisimple cosemisimple NOUN iajs-2429 250 13 . . PUNCT iajs-2429 251 1 proof proof NOUN iajs-2429 251 2 . . PUNCT iajs-2429 252 1 ( ( PUNCT iajs-2429 252 2 1 1 X iajs-2429 252 3 ) ) PUNCT iajs-2429 252 4   NOUN iajs-2429 252 5 ( ( PUNCT iajs-2429 252 6 2 2 X iajs-2429 252 7 ) ) PUNCT iajs-2429 252 8   NOUN iajs-2429 252 9 ( ( PUNCT iajs-2429 252 10 3 3 X iajs-2429 252 11 ) ) PUNCT iajs-2429 252 12   NOUN iajs-2429 252 13 ( ( PUNCT iajs-2429 252 14 4 4 NUM iajs-2429 252 15 ) ) PUNCT iajs-2429 252 16 as as SCONJ iajs-2429 252 17 we we PRON iajs-2429 252 18 mentioned mention VERB iajs-2429 252 19 before before ADP iajs-2429 252 20 where where SCONJ iajs-2429 252 21 the the DET iajs-2429 252 22 commutativity commutativity NOUN iajs-2429 252 23 condition condition NOUN iajs-2429 252 24 is be AUX iajs-2429 252 25 not not PART iajs-2429 252 26 necessary necessary ADJ iajs-2429 252 27 , , PUNCT iajs-2429 252 28 ( ( PUNCT iajs-2429 252 29 1 1 X iajs-2429 252 30 ) ) PUNCT iajs-2429 252 31   NOUN iajs-2429 252 32 ( ( PUNCT iajs-2429 252 33 5 5 NUM iajs-2429 252 34 ) ) PUNCT iajs-2429 252 35 by by ADP iajs-2429 252 36 theorem theorem NOUN iajs-2429 252 37 4.4 4.4 NUM iajs-2429 252 38 and and CCONJ iajs-2429 252 39 ( ( PUNCT iajs-2429 252 40 1 1 X iajs-2429 252 41 ) ) PUNCT iajs-2429 252 42   NOUN iajs-2429 252 43 ( ( PUNCT iajs-2429 252 44 6 6 NUM iajs-2429 252 45 ) ) PUNCT iajs-2429 252 46 via via ADP iajs-2429 252 47 [ [ X iajs-2429 252 48 7 7 NUM iajs-2429 252 49 ] ] PUNCT iajs-2429 252 50 . . PUNCT iajs-2429 253 1 proposition proposition NOUN iajs-2429 253 2 ( ( PUNCT iajs-2429 253 3 4.10 4.10 NUM iajs-2429 253 4 ): ): PUNCT iajs-2429 253 5 every every DET iajs-2429 253 6 nonzero nonzero NOUN iajs-2429 253 7 module module NOUN iajs-2429 253 8 over over ADP iajs-2429 253 9 semisecond semisecond ADJ iajs-2429 253 10 ring ring NOUN iajs-2429 253 11 is be AUX iajs-2429 253 12 semisecond semisecond ADJ iajs-2429 253 13 . . PUNCT iajs-2429 254 1 proof proof NOUN iajs-2429 254 2 . . PUNCT iajs-2429 255 1 let let VERB iajs-2429 255 2 0 0 NUM iajs-2429 255 3 𝑀 𝑀 PROPN iajs-2429 255 4 be be AUX iajs-2429 255 5 a a DET iajs-2429 255 6 module module NOUN iajs-2429 255 7 over over ADP iajs-2429 255 8 a a DET iajs-2429 255 9 semisecond semisecond ADJ iajs-2429 255 10 ring ring NOUN iajs-2429 255 11 𝑅 𝑅 PROPN iajs-2429 255 12 implies imply VERB iajs-2429 255 13 𝑅𝑎 𝑅𝑎 PROPN iajs-2429 255 14 𝑅𝑎 𝑅𝑎 PROPN iajs-2429 255 15 and and CCONJ iajs-2429 255 16 thus thus ADV iajs-2429 255 17 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 255 18 𝑀𝑎. 𝑀𝑎. PROPN iajs-2429 255 19 example example NOUN iajs-2429 255 20 ( ( PUNCT iajs-2429 255 21 4.11 4.11 NUM iajs-2429 255 22 ): ): PUNCT iajs-2429 255 23 let let VERB iajs-2429 255 24 𝑅 𝑅 PROPN iajs-2429 255 25 ℤ ℤ PROPN iajs-2429 255 26 ℤ ℤ PROPN iajs-2429 255 27 0 0 NUM iajs-2429 255 28 ℤ ℤ PROPN iajs-2429 255 29 and and CCONJ iajs-2429 255 30 𝑀 𝑀 PROPN iajs-2429 255 31 0 0 PUNCT iajs-2429 256 1 ℤ ℤ NOUN iajs-2429 256 2 0 0 X iajs-2429 256 3 ℤ ℤ NOUN iajs-2429 256 4 be be AUX iajs-2429 256 5 considered consider VERB iajs-2429 256 6 as as ADP iajs-2429 256 7 a a DET iajs-2429 256 8 right right ADJ iajs-2429 256 9 𝑅module 𝑅module PROPN iajs-2429 256 10 . . PUNCT iajs-2429 256 11 by by ADP iajs-2429 256 12 simple simple ADJ iajs-2429 256 13 steps step NOUN iajs-2429 256 14 , , PUNCT iajs-2429 256 15 we we PRON iajs-2429 256 16 see see VERB iajs-2429 256 17 that that SCONJ iajs-2429 256 18 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 256 19 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 256 20 for for ADP iajs-2429 256 21 each each DET iajs-2429 256 22 𝑎 𝑎 PRON iajs-2429 256 23 ∈ ∈ PROPN iajs-2429 256 24 𝑅 𝑅 PROPN iajs-2429 256 25 that that SCONJ iajs-2429 256 26 𝑀 𝑀 PROPN iajs-2429 256 27 is be AUX iajs-2429 256 28 semisecond semisecond ADJ iajs-2429 256 29 but but CCONJ iajs-2429 256 30 𝑅 𝑅 NOUN iajs-2429 256 31 is be AUX iajs-2429 256 32 not not PART iajs-2429 256 33 semisecond semisecond ADJ iajs-2429 256 34 since since SCONJ iajs-2429 256 35 if if SCONJ iajs-2429 256 36 we we PRON iajs-2429 256 37 take take VERB iajs-2429 256 38 𝑎 𝑎 ADJ iajs-2429 256 39 0 0 NUM iajs-2429 256 40 1 1 NUM iajs-2429 256 41 0 0 NUM iajs-2429 256 42 0 0 NUM iajs-2429 257 1 we we PRON iajs-2429 257 2 have have VERB iajs-2429 257 3 𝑅𝑎 𝑅𝑎 PROPN iajs-2429 257 4 𝑅𝑎 𝑅𝑎 PROPN iajs-2429 257 5 . . PUNCT iajs-2429 258 1 in in ADP iajs-2429 258 2 fact fact NOUN iajs-2429 258 3 if if SCONJ iajs-2429 258 4 𝑅 𝑅 PROPN iajs-2429 258 5 is be AUX iajs-2429 258 6 semisecond semisecond ADJ iajs-2429 258 7 , , PUNCT iajs-2429 258 8 then then ADV iajs-2429 258 9 𝑀 𝑀 PROPN iajs-2429 258 10 is be AUX iajs-2429 258 11 semisecond semisecond ADJ iajs-2429 258 12 which which PRON iajs-2429 258 13 is be AUX iajs-2429 258 14 a a DET iajs-2429 258 15 contradiction contradiction NOUN iajs-2429 258 16 by by ADP iajs-2429 258 17 proposition proposition NOUN iajs-2429 258 18 4.3 4.3 NUM iajs-2429 258 19 . . PUNCT iajs-2429 259 1 moreover moreover ADV iajs-2429 259 2 , , PUNCT iajs-2429 259 3 𝑀 𝑀 PROPN iajs-2429 259 4 is be AUX iajs-2429 259 5 not not PART iajs-2429 259 6 semisimple semisimple NOUN iajs-2429 259 7 since since SCONJ iajs-2429 259 8 0 0 NUM iajs-2429 259 9 ℤ ℤ NOUN iajs-2429 259 10 0 0 NUM iajs-2429 259 11 0 0 NUM iajs-2429 259 12 is be AUX iajs-2429 259 13 a a DET iajs-2429 259 14 cyclic cyclic ADJ iajs-2429 259 15 submodule submodule NOUN iajs-2429 259 16 of of ADP iajs-2429 259 17 𝑀 𝑀 PROPN iajs-2429 259 18 which which PRON iajs-2429 259 19 is be AUX iajs-2429 259 20 not not PART iajs-2429 259 21 a a DET iajs-2429 259 22 direct direct ADJ iajs-2429 259 23 summand summand NOUN iajs-2429 259 24     SPACE iajs-2429 259 25 89 89 NUM iajs-2429 259 26     SPACE iajs-2429 259 27 ibn ibn PROPN iajs-2429 259 28 al al PROPN iajs-2429 259 29 - - PUNCT iajs-2429 259 30 haitham haitham PROPN iajs-2429 259 31 jour jour X iajs-2429 259 32 . . PROPN iajs-2429 260 1 for for ADP iajs-2429 260 2 pure pure ADJ iajs-2429 260 3 & & CCONJ iajs-2429 260 4 appl appl PROPN iajs-2429 260 5 . . PUNCT iajs-2429 261 1 sci sci PROPN iajs-2429 261 2 . . PROPN iajs-2429 262 1 33 33 NUM iajs-2429 262 2 ( ( PUNCT iajs-2429 262 3 2 2 NUM iajs-2429 262 4 ) ) PUNCT iajs-2429 262 5 2020 2020 NUM iajs-2429 262 6 of of ADP iajs-2429 262 7 𝑀. 𝑀. PROPN iajs-2429 262 8 also also ADV iajs-2429 262 9 , , PUNCT iajs-2429 262 10 𝑀 𝑀 PROPN iajs-2429 262 11 is be AUX iajs-2429 262 12 not not PART iajs-2429 262 13 regular regular ADJ iajs-2429 262 14 since since SCONJ iajs-2429 262 15 0 0 NUM iajs-2429 262 16 ℤ ℤ NOUN iajs-2429 262 17 0 0 NUM iajs-2429 262 18 ℤ ℤ NOUN iajs-2429 262 19 0 0 NUM iajs-2429 262 20 0 0 NUM iajs-2429 262 21 0 0 NUM iajs-2429 262 22 1 1 NUM iajs-2429 262 23 ∩ ∩ NOUN iajs-2429 262 24 0 0 X iajs-2429 263 1 ℤ ℤ NOUN iajs-2429 263 2 0 0 NUM iajs-2429 263 3 0 0 NUM iajs-2429 263 4 0 0 NUM iajs-2429 264 1 ℤ ℤ NOUN iajs-2429 264 2 0 0 NUM iajs-2429 264 3 0 0 NUM iajs-2429 264 4 0 0 NUM iajs-2429 265 1 ℤ ℤ NOUN iajs-2429 265 2 0 0 NUM iajs-2429 265 3 0 0 NUM iajs-2429 265 4 0 0 NUM iajs-2429 265 5 0 0 NUM iajs-2429 265 6 0 0 NUM iajs-2429 265 7 1 1 NUM iajs-2429 265 8 0 0 NUM iajs-2429 265 9 0 0 NUM iajs-2429 265 10 0 0 NUM iajs-2429 265 11 0 0 NUM iajs-2429 265 12 thus thus ADV iajs-2429 265 13 0 0 X iajs-2429 265 14 ℤ ℤ NOUN iajs-2429 265 15 0 0 NUM iajs-2429 265 16 0 0 NUM iajs-2429 265 17 is be AUX iajs-2429 265 18 not not PART iajs-2429 265 19 a a DET iajs-2429 265 20 pure pure ADJ iajs-2429 265 21 submodule submodule NOUN iajs-2429 265 22 of of ADP iajs-2429 265 23 𝑀. 𝑀. PROPN iajs-2429 265 24 corollary corollary NOUN iajs-2429 265 25 ( ( PUNCT iajs-2429 265 26 4.12 4.12 NUM iajs-2429 265 27 ): ): PUNCT iajs-2429 265 28 let let VERB iajs-2429 265 29 𝑀 𝑀 PRON iajs-2429 265 30 be be AUX iajs-2429 265 31 an an DET iajs-2429 265 32 𝑅-module 𝑅-module PROPN iajs-2429 265 33 and and CCONJ iajs-2429 265 34 𝐼 𝐼 PROPN iajs-2429 265 35 be be VERB iajs-2429 265 36 an an DET iajs-2429 265 37 ideal ideal NOUN iajs-2429 265 38 of of ADP iajs-2429 265 39 𝑅 𝑅 PROPN iajs-2429 265 40 such such ADJ iajs-2429 265 41 𝐼 𝐼 PROPN iajs-2429 265 42 ⊆ ⊆ NUM iajs-2429 265 43 𝑎𝑛𝑛 𝑎𝑛𝑛 X iajs-2429 265 44 𝑀 𝑀 PROPN iajs-2429 265 45 . . PUNCT iajs-2429 266 1 if if SCONJ iajs-2429 266 2 is be AUX iajs-2429 266 3 a a DET iajs-2429 266 4 semisecond semisecond ADJ iajs-2429 266 5 ring ring NOUN iajs-2429 266 6 then then ADV iajs-2429 266 7 𝑁 𝑁 PROPN iajs-2429 266 8 is be AUX iajs-2429 266 9 semisecond semisecond ADJ iajs-2429 266 10 . . PUNCT iajs-2429 267 1 proof proof NOUN iajs-2429 267 2 . . PUNCT iajs-2429 268 1 since since SCONJ iajs-2429 268 2 𝑁 𝑁 PROPN iajs-2429 268 3 is be AUX iajs-2429 268 4 considered consider VERB iajs-2429 268 5 as as ADP iajs-2429 268 6 – – PUNCT iajs-2429 268 7 module module NOUN iajs-2429 268 8 so so ADV iajs-2429 268 9 by by ADP iajs-2429 268 10 proposition proposition NOUN iajs-2429 268 11 4.10 4.10 NUM iajs-2429 268 12 , , PUNCT iajs-2429 268 13 the the DET iajs-2429 268 14 result result NOUN iajs-2429 268 15 is be AUX iajs-2429 268 16 obtained obtain VERB iajs-2429 268 17 . . PUNCT iajs-2429 269 1 proposition proposition NOUN iajs-2429 269 2 ( ( PUNCT iajs-2429 269 3 4.13 4.13 NUM iajs-2429 269 4 ): ): PUNCT iajs-2429 269 5 let let VERB iajs-2429 269 6 𝑀 𝑀 PRON iajs-2429 269 7 be be AUX iajs-2429 269 8 an an DET iajs-2429 269 9 𝑅-module 𝑅-module PROPN iajs-2429 269 10 and and CCONJ iajs-2429 269 11 𝐼 𝐼 PROPN iajs-2429 269 12 be be VERB iajs-2429 269 13 an an DET iajs-2429 269 14 ideal ideal NOUN iajs-2429 269 15 of of ADP iajs-2429 269 16 𝑅 𝑅 PROPN iajs-2429 269 17 such such ADJ iajs-2429 269 18 that that SCONJ iajs-2429 269 19 𝐼 𝐼 PROPN iajs-2429 269 20 ⊆ ⊆ NUM iajs-2429 269 21 𝑎𝑛𝑛 𝑎𝑛𝑛 X iajs-2429 269 22 𝑀 𝑀 PROPN iajs-2429 269 23 . . PUNCT iajs-2429 270 1 then then ADV iajs-2429 270 2 𝑀 𝑀 PROPN iajs-2429 270 3 is be AUX iajs-2429 270 4 a a DET iajs-2429 270 5 semisecond semisecond ADJ iajs-2429 270 6 𝑅-module 𝑅-module PROPN iajs-2429 270 7 if if SCONJ iajs-2429 270 8 and and CCONJ iajs-2429 270 9 only only ADV iajs-2429 270 10 if if SCONJ iajs-2429 270 11 𝑀 𝑀 PROPN iajs-2429 270 12 is be AUX iajs-2429 270 13 a a DET iajs-2429 270 14 semisecond semisecond ADJ iajs-2429 270 15 -module -module NOUN iajs-2429 270 16 . . PUNCT iajs-2429 271 1 proof proof NOUN iajs-2429 271 2 . . PUNCT iajs-2429 272 1 it it PRON iajs-2429 272 2 is be AUX iajs-2429 272 3 clear clear ADJ iajs-2429 272 4 . . PUNCT iajs-2429 273 1 examples example NOUN iajs-2429 273 2 ( ( PUNCT iajs-2429 273 3 4.14 4.14 NUM iajs-2429 273 4 ): ): PUNCT iajs-2429 273 5 ( ( PUNCT iajs-2429 273 6 1 1 X iajs-2429 273 7 ) ) PUNCT iajs-2429 273 8 ℤ ℤ PROPN iajs-2429 273 9 and and CCONJ iajs-2429 273 10 ℚ ℚ PROPN iajs-2429 273 11 as as SCONJ iajs-2429 273 12 ℤ-modules ℤ-modules PROPN iajs-2429 273 13 are be AUX iajs-2429 273 14 semisecond semisecond ADJ iajs-2429 273 15 but but CCONJ iajs-2429 273 16 ℤ ℤ PROPN iajs-2429 273 17 ℤ ℤ PROPN iajs-2429 273 18 ℚ ℚ PROPN iajs-2429 273 19 ≅ ≅ NOUN iajs-2429 273 20 ℤ ℤ PROPN iajs-2429 273 21 ≅ ≅ PROPN iajs-2429 273 22 ℤ ℤ PROPN iajs-2429 273 23 ℤ ℤ PROPN iajs-2429 273 24 ℤ ℤ PROPN iajs-2429 273 25 is be AUX iajs-2429 273 26 not not PART iajs-2429 273 27 semisecond semisecond ADJ iajs-2429 273 28 . . PUNCT iajs-2429 274 1 ( ( PUNCT iajs-2429 274 2 2 2 X iajs-2429 274 3 ) ) PUNCT iajs-2429 274 4 consider consider VERB iajs-2429 274 5 ℤ ℤ PROPN iajs-2429 274 6 as as ADP iajs-2429 274 7 ℤ-module ℤ-module PROPN iajs-2429 274 8 implies imply VERB iajs-2429 274 9 ℤ ℤ PROPN iajs-2429 274 10 ℤ ℤ PROPN iajs-2429 274 11 ℤ ℤ PROPN iajs-2429 274 12 ℤ ℤ PROPN iajs-2429 274 13 is be AUX iajs-2429 274 14 semisecond semisecond ADJ iajs-2429 274 15 but but CCONJ iajs-2429 274 16 𝑅 𝑅 PROPN iajs-2429 274 17 ℤ ℤ PROPN iajs-2429 274 18 is be AUX iajs-2429 274 19 not not PART iajs-2429 274 20 semisecond semisecond ADJ iajs-2429 274 21 . . PUNCT iajs-2429 275 1 proposition proposition NOUN iajs-2429 275 2 ( ( PUNCT iajs-2429 275 3 4.15 4.15 NUM iajs-2429 275 4 ): ): PUNCT iajs-2429 275 5 if if SCONJ iajs-2429 275 6 𝑁 𝑁 PROPN iajs-2429 275 7 is be AUX iajs-2429 275 8 a a DET iajs-2429 275 9 cancellation cancellation NOUN iajs-2429 275 10 semisecond semisecond NOUN iajs-2429 275 11 submodule submodule NOUN iajs-2429 275 12 of of ADP iajs-2429 275 13 an an DET iajs-2429 275 14 𝑅-module 𝑅-module PROPN iajs-2429 275 15 𝑀 𝑀 PROPN iajs-2429 275 16 then then ADV iajs-2429 275 17 𝑅 𝑅 PROPN iajs-2429 275 18 is be AUX iajs-2429 275 19 semisecond semisecond ADJ iajs-2429 275 20 . . PUNCT iajs-2429 276 1 proof proof NOUN iajs-2429 276 2 . . PUNCT iajs-2429 277 1 for for ADP iajs-2429 277 2 each each DET iajs-2429 277 3 𝑎 𝑎 PROPN iajs-2429 277 4 ∈ ∈ PROPN iajs-2429 277 5 𝑅 𝑅 PROPN iajs-2429 277 6 , , PUNCT iajs-2429 277 7 we we PRON iajs-2429 277 8 have have VERB iajs-2429 277 9 𝑎 𝑎 ADJ iajs-2429 277 10 𝑁 𝑁 PROPN iajs-2429 277 11 𝑎𝑁 𝑎𝑁 NOUN iajs-2429 277 12 , , PUNCT iajs-2429 277 13 then then ADV iajs-2429 277 14 𝑎 𝑎 PROPN iajs-2429 277 15 𝑅 𝑅 PROPN iajs-2429 277 16 𝑁 𝑁 PROPN iajs-2429 277 17 𝑎𝑅 𝑎𝑅 NOUN iajs-2429 277 18 𝑁 𝑁 PROPN iajs-2429 277 19 and and CCONJ iajs-2429 277 20 since since SCONJ iajs-2429 277 21 𝑁 𝑁 PROPN iajs-2429 277 22 is be AUX iajs-2429 277 23 cancellation cancellation NOUN iajs-2429 277 24 implies imply VERB iajs-2429 277 25 𝑎 𝑎 PROPN iajs-2429 277 26 𝑅 𝑅 PROPN iajs-2429 277 27 𝑎𝑅 𝑎𝑅 PROPN iajs-2429 277 28 as as SCONJ iajs-2429 277 29 desired desire VERB iajs-2429 277 30 . . PUNCT iajs-2429 278 1 corollary corollary ADJ iajs-2429 278 2 ( ( PUNCT iajs-2429 278 3 4.16 4.16 NUM iajs-2429 278 4 ): ): PUNCT iajs-2429 278 5 if if SCONJ iajs-2429 278 6 𝑀 𝑀 PROPN iajs-2429 278 7 is be AUX iajs-2429 278 8 a a DET iajs-2429 278 9 finitely finitely ADV iajs-2429 278 10 generated generate VERB iajs-2429 278 11 faithful faithful ADJ iajs-2429 278 12 multiplication multiplication NOUN iajs-2429 278 13 semisecond semisecond ADJ iajs-2429 278 14 𝑅-module 𝑅-module PROPN iajs-2429 278 15 then then ADV iajs-2429 278 16 𝑅 𝑅 PROPN iajs-2429 278 17 is be AUX iajs-2429 278 18 fully fully ADV iajs-2429 278 19 idempotent idempotent ADJ iajs-2429 278 20 ( ( PUNCT iajs-2429 278 21 and and CCONJ iajs-2429 278 22 hence hence ADV iajs-2429 278 23 semisecond semisecond NOUN iajs-2429 278 24 ) ) PUNCT iajs-2429 278 25 . . PUNCT iajs-2429 279 1 proof proof NOUN iajs-2429 279 2 . . PUNCT iajs-2429 280 1 let let VERB iajs-2429 280 2 𝑀 𝑀 PRON iajs-2429 280 3 be be AUX iajs-2429 280 4 a a DET iajs-2429 280 5 semisecond semisecond ADJ iajs-2429 280 6 𝑅-module 𝑅-module PROPN iajs-2429 280 7 then then ADV iajs-2429 280 8 𝐼 𝐼 PROPN iajs-2429 280 9 𝑀 𝑀 PROPN iajs-2429 280 10 𝐼𝑀 𝐼𝑀 PROPN iajs-2429 280 11 for for ADP iajs-2429 280 12 each each DET iajs-2429 280 13 ideal ideal NOUN iajs-2429 280 14 𝐼 𝐼 ADP iajs-2429 280 15 of of ADP iajs-2429 280 16 𝑅. 𝑅. NOUN iajs-2429 280 17 since since SCONJ iajs-2429 280 18 𝑀 𝑀 PROPN iajs-2429 280 19 is be AUX iajs-2429 280 20 a a DET iajs-2429 280 21 finitely finitely ADV iajs-2429 280 22 generated generate VERB iajs-2429 280 23 faithful faithful ADJ iajs-2429 280 24 multiplication multiplication NOUN iajs-2429 280 25 so so ADV iajs-2429 280 26 by by ADP iajs-2429 280 27 [ [ PUNCT iajs-2429 280 28 13 13 NUM iajs-2429 280 29 ] ] PUNCT iajs-2429 280 30 , , PUNCT iajs-2429 280 31 𝑀 𝑀 PROPN iajs-2429 280 32 is be AUX iajs-2429 280 33 cancellation cancellation NOUN iajs-2429 280 34 then then ADV iajs-2429 280 35 𝐼 𝐼 ADP iajs-2429 280 36 𝐼 𝐼 PROPN iajs-2429 280 37 thus thus ADV iajs-2429 280 38 𝑅 𝑅 PROPN iajs-2429 280 39 is be AUX iajs-2429 280 40 fully fully ADV iajs-2429 280 41 idempotent idempotent ADJ iajs-2429 280 42 . . PUNCT iajs-2429 281 1 corollary corollary ADJ iajs-2429 281 2 ( ( PUNCT iajs-2429 281 3 4.17 4.17 NUM iajs-2429 281 4 ): ): PUNCT iajs-2429 281 5 if if SCONJ iajs-2429 281 6 𝑀 𝑀 PROPN iajs-2429 281 7 is be AUX iajs-2429 281 8 a a DET iajs-2429 281 9 cancellation cancellation NOUN iajs-2429 281 10 ( ( PUNCT iajs-2429 281 11 or or CCONJ iajs-2429 281 12 finitely finitely ADV iajs-2429 281 13 generated generate VERB iajs-2429 281 14 faithful faithful ADJ iajs-2429 281 15 multiplication multiplication NOUN iajs-2429 281 16 ) ) PUNCT iajs-2429 281 17 semisecond semisecond NOUN iajs-2429 281 18 𝑅-module 𝑅-module PROPN iajs-2429 281 19 such such ADJ iajs-2429 281 20 that that SCONJ iajs-2429 281 21 the the DET iajs-2429 281 22 set set NOUN iajs-2429 281 23 of of ADP iajs-2429 281 24 ideals ideal NOUN iajs-2429 281 25 of of ADP iajs-2429 281 26 𝑅 𝑅 PROPN iajs-2429 281 27 is be AUX iajs-2429 281 28 totally totally ADV iajs-2429 281 29 ordered order VERB iajs-2429 281 30 under under ADP iajs-2429 281 31 inclusion inclusion NOUN iajs-2429 281 32 then then ADV iajs-2429 281 33 𝑅 𝑅 PROPN iajs-2429 281 34 is be AUX iajs-2429 281 35 fully fully ADV iajs-2429 281 36 prime prime ADJ iajs-2429 281 37 . . PUNCT iajs-2429 282 1 proof proof NOUN iajs-2429 282 2 . . PUNCT iajs-2429 283 1 by by ADP iajs-2429 283 2 corollary corollary ADJ iajs-2429 283 3 4.15 4.15 NUM iajs-2429 283 4 , , PUNCT iajs-2429 283 5 𝑅 𝑅 PROPN iajs-2429 283 6 is be AUX iajs-2429 283 7 fully fully ADV iajs-2429 283 8 idempotent idempotent ADJ iajs-2429 283 9 so so ADV iajs-2429 283 10 by by ADP iajs-2429 283 11 [ [ PUNCT iajs-2429 283 12 11 11 NUM iajs-2429 283 13 ] ] PUNCT iajs-2429 283 14 . . PUNCT iajs-2429 284 1 𝑅 𝑅 NOUN iajs-2429 284 2 is be AUX iajs-2429 284 3 fully fully ADV iajs-2429 284 4 prime prime ADJ iajs-2429 284 5 . . PUNCT iajs-2429 285 1 theorem theorem NOUN iajs-2429 285 2 ( ( PUNCT iajs-2429 285 3 4.18 4.18 NUM iajs-2429 285 4 ): ): PUNCT iajs-2429 285 5 let let VERB iajs-2429 285 6 𝑀 𝑀 PRON iajs-2429 285 7 be be AUX iajs-2429 285 8 a a DET iajs-2429 285 9 multiplication multiplication NOUN iajs-2429 285 10 𝑅-module 𝑅-module NOUN iajs-2429 285 11 . . PUNCT iajs-2429 286 1 if if SCONJ iajs-2429 286 2 𝑁 𝑁 PROPN iajs-2429 286 3 : : PUNCT iajs-2429 286 4 𝑀 𝑀 PROPN iajs-2429 286 5 is be AUX iajs-2429 286 6 a a DET iajs-2429 286 7 semisecond semisecond ADJ iajs-2429 286 8 ideal ideal NOUN iajs-2429 286 9 of of ADP iajs-2429 286 10 𝑅 𝑅 PROPN iajs-2429 286 11 then then ADV iajs-2429 286 12 𝑁 𝑁 PROPN iajs-2429 286 13 is be AUX iajs-2429 286 14 a a DET iajs-2429 286 15 semisecond semisecond ADJ iajs-2429 286 16 submodule submodule NOUN iajs-2429 286 17 of of ADP iajs-2429 286 18 𝑀. 𝑀. NOUN iajs-2429 286 19 proof proof NOUN iajs-2429 286 20 . . PUNCT iajs-2429 287 1 by by ADP iajs-2429 287 2 hypothesis hypothesis NOUN iajs-2429 287 3 , , PUNCT iajs-2429 287 4 𝑁 𝑁 PROPN iajs-2429 287 5 : : PUNCT iajs-2429 287 6 𝑀 𝑀 PROPN iajs-2429 287 7 𝐼 𝐼 PROPN iajs-2429 287 8 𝑁 𝑁 PROPN iajs-2429 287 9 : : PUNCT iajs-2429 287 10 𝑀 𝑀 PROPN iajs-2429 287 11 𝐼 𝐼 PROPN iajs-2429 287 12 for for ADP iajs-2429 287 13 each each DET iajs-2429 287 14 ideal ideal ADJ iajs-2429 287 15 𝐼 𝐼 PROPN iajs-2429 287 16 of of ADP iajs-2429 287 17 𝑅 𝑅 PROPN iajs-2429 287 18 then then ADV iajs-2429 287 19 𝑀 𝑀 PROPN iajs-2429 287 20 𝑁 𝑁 PROPN iajs-2429 287 21 : : PUNCT iajs-2429 287 22 𝑀 𝑀 PROPN iajs-2429 287 23 𝐼 𝐼 PROPN iajs-2429 287 24 𝑀 𝑀 PROPN iajs-2429 287 25 𝑁 𝑁 PROPN iajs-2429 287 26 : : PUNCT iajs-2429 287 27 𝑀 𝑀 PROPN iajs-2429 287 28 𝐼 𝐼 PROPN iajs-2429 287 29 . . PUNCT iajs-2429 288 1 by by ADP iajs-2429 288 2 hypothesis hypothesis NOUN iajs-2429 288 3 𝑀 𝑀 PROPN iajs-2429 288 4 is be AUX iajs-2429 288 5 multiplication multiplication NOUN iajs-2429 288 6 thus thus ADV iajs-2429 288 7 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 288 8 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 288 9 so so ADV iajs-2429 288 10 𝑁 𝑁 PROPN iajs-2429 288 11 is be AUX iajs-2429 288 12 semisecond semisecond ADJ iajs-2429 288 13 . . PUNCT iajs-2429 289 1 theorem theorem NOUN iajs-2429 289 2 ( ( PUNCT iajs-2429 289 3 4.19 4.19 NUM iajs-2429 289 4 ): ): PUNCT iajs-2429 289 5 let let VERB iajs-2429 289 6 𝑀 𝑀 PRON iajs-2429 289 7 be be AUX iajs-2429 289 8 a a DET iajs-2429 289 9 finitely finitely ADV iajs-2429 289 10 generated generate VERB iajs-2429 289 11 faithful faithful ADJ iajs-2429 289 12 multiplication multiplication NOUN iajs-2429 289 13 𝑅-module 𝑅-module PROPN iajs-2429 289 14 . . PUNCT iajs-2429 290 1 if if SCONJ iajs-2429 290 2 𝑁 𝑁 PROPN iajs-2429 290 3 is be AUX iajs-2429 290 4 a a DET iajs-2429 290 5 semisecond semisecond ADJ iajs-2429 290 6 submodule submodule NOUN iajs-2429 290 7 of of ADP iajs-2429 290 8 𝑀 𝑀 PROPN iajs-2429 290 9 then then ADV iajs-2429 290 10 𝑁 𝑁 PROPN iajs-2429 290 11 : : PUNCT iajs-2429 290 12 𝑀 𝑀 PROPN iajs-2429 290 13 is be AUX iajs-2429 290 14 a a DET iajs-2429 290 15 semisecond semisecond ADJ iajs-2429 290 16 ideal ideal NOUN iajs-2429 290 17 of of ADP iajs-2429 290 18 𝑅. 𝑅. ADJ iajs-2429 290 19 proof proof NOUN iajs-2429 290 20 . . PUNCT iajs-2429 291 1 since since SCONJ iajs-2429 291 2 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 291 3 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 291 4 for for ADP iajs-2429 291 5 each each DET iajs-2429 291 6 ideal ideal ADJ iajs-2429 291 7 𝐼 𝐼 PROPN iajs-2429 291 8 of of ADP iajs-2429 291 9 𝑅 𝑅 PROPN iajs-2429 291 10 then then ADV iajs-2429 291 11 𝑀 𝑀 PROPN iajs-2429 291 12 𝑁 𝑁 PROPN iajs-2429 291 13 : : PUNCT iajs-2429 291 14 𝑀 𝑀 PROPN iajs-2429 291 15 𝐼 𝐼 PROPN iajs-2429 291 16 𝑀 𝑀 PROPN iajs-2429 291 17 𝑁 𝑁 PROPN iajs-2429 291 18 : : PUNCT iajs-2429 291 19 𝑀 𝑀 PROPN iajs-2429 291 20 𝐼 𝐼 PROPN iajs-2429 291 21 because because SCONJ iajs-2429 291 22 𝑀 𝑀 PROPN iajs-2429 291 23 is be AUX iajs-2429 291 24 multiplication multiplication NOUN iajs-2429 291 25 . . PUNCT iajs-2429 292 1 but but CCONJ iajs-2429 292 2 𝑀 𝑀 PROPN iajs-2429 292 3 is be AUX iajs-2429 292 4 finitely finitely ADV iajs-2429 292 5 generated generate VERB iajs-2429 292 6 faithful faithful ADJ iajs-2429 292 7 implies implie NOUN iajs-2429 292 8 𝑀 𝑀 PROPN iajs-2429 292 9 is be AUX iajs-2429 292 10 cancellation cancellation NOUN iajs-2429 292 11 and and CCONJ iajs-2429 292 12 hence hence ADV iajs-2429 292 13 𝑁 𝑁 PROPN iajs-2429 292 14 : : PUNCT iajs-2429 292 15 𝑀 𝑀 PROPN iajs-2429 292 16 𝐼 𝐼 PROPN iajs-2429 292 17 𝑁 𝑁 PROPN iajs-2429 292 18 : : PUNCT iajs-2429 292 19 𝑀 𝑀 PROPN iajs-2429 292 20 𝐼 𝐼 PROPN iajs-2429 292 21 thus thus ADV iajs-2429 292 22 𝑁 𝑁 PROPN iajs-2429 292 23 : : PUNCT iajs-2429 292 24 𝑀 𝑀 PROPN iajs-2429 292 25 is be AUX iajs-2429 292 26 semisecond semisecond ADJ iajs-2429 292 27 . . PUNCT iajs-2429 293 1 remark remark NOUN iajs-2429 293 2 ( ( PUNCT iajs-2429 293 3 4.20 4.20 NUM iajs-2429 293 4 ): ): PUNCT iajs-2429 293 5 if if SCONJ iajs-2429 293 6 𝐼 𝐼 PROPN iajs-2429 293 7 is be AUX iajs-2429 293 8 a a DET iajs-2429 293 9 semisecond semisecond ADJ iajs-2429 293 10 ideal ideal NOUN iajs-2429 293 11 of of ADP iajs-2429 293 12 𝑅 𝑅 PROPN iajs-2429 293 13 then then ADV iajs-2429 293 14 𝐼 𝐼 ADP iajs-2429 293 15 𝐼 𝐼 PROPN iajs-2429 293 16 . . PUNCT iajs-2429 294 1 proof proof NOUN iajs-2429 294 2 . . PUNCT iajs-2429 295 1 since since SCONJ iajs-2429 295 2 𝐼𝐽 𝐼𝐽 NOUN iajs-2429 295 3 𝐼𝐽 𝐼𝐽 NOUN iajs-2429 295 4 for for ADP iajs-2429 295 5 each each DET iajs-2429 295 6 ideal ideal NOUN iajs-2429 295 7 𝐽of 𝐽of PROPN iajs-2429 295 8 𝑅 𝑅 PROPN iajs-2429 295 9 so so ADV iajs-2429 295 10 if if SCONJ iajs-2429 295 11 we we PRON iajs-2429 295 12 choose choose VERB iajs-2429 295 13 𝐽 𝐽 PROPN iajs-2429 295 14 𝐼 𝐼 PROPN iajs-2429 295 15 implies imply VERB iajs-2429 295 16 𝐼 𝐼 ADP iajs-2429 295 17 𝐼 𝐼 PROPN iajs-2429 295 18 . . PUNCT iajs-2429 296 1 proposition proposition NOUN iajs-2429 296 2 ( ( PUNCT iajs-2429 296 3 4.21 4.21 NUM iajs-2429 296 4 ): ): PUNCT iajs-2429 296 5 every every DET iajs-2429 296 6 nonzero nonzero ADJ iajs-2429 296 7 pure pure ADJ iajs-2429 296 8 submodule submodule NOUN iajs-2429 296 9 of of ADP iajs-2429 296 10 a a DET iajs-2429 296 11 semisecond semisecond ADJ iajs-2429 296 12 module module NOUN iajs-2429 296 13 is be AUX iajs-2429 296 14 semisecond semisecond ADJ iajs-2429 296 15 . . PUNCT iajs-2429 296 16     SPACE iajs-2429 297 1 90 90 NUM iajs-2429 297 2     SPACE iajs-2429 297 3 ibn ibn PROPN iajs-2429 297 4 al al PROPN iajs-2429 297 5 - - PUNCT iajs-2429 297 6 haitham haitham PROPN iajs-2429 297 7 jour jour X iajs-2429 297 8 . . PROPN iajs-2429 297 9 for for ADP iajs-2429 297 10 pure pure ADJ iajs-2429 297 11 & & CCONJ iajs-2429 297 12 appl appl PROPN iajs-2429 297 13 . . PUNCT iajs-2429 298 1 sci sci PROPN iajs-2429 298 2 . . PROPN iajs-2429 299 1 33 33 NUM iajs-2429 299 2 ( ( PUNCT iajs-2429 299 3 2 2 NUM iajs-2429 299 4 ) ) PUNCT iajs-2429 299 5 2020 2020 NUM iajs-2429 299 6 proof proof NOUN iajs-2429 299 7 . . PUNCT iajs-2429 300 1 let let VERB iajs-2429 300 2 𝑁 𝑁 PROPN iajs-2429 300 3 be be AUX iajs-2429 300 4 a a DET iajs-2429 300 5 nonzero nonzero ADJ iajs-2429 300 6 pure pure ADJ iajs-2429 300 7 submodule submodule NOUN iajs-2429 300 8 of of ADP iajs-2429 300 9 a a DET iajs-2429 300 10 semisecond semisecond ADJ iajs-2429 300 11 𝑅-module 𝑅-module PROPN iajs-2429 300 12 𝑀. 𝑀. PROPN iajs-2429 300 13 then then ADV iajs-2429 300 14 for for SCONJ iajs-2429 300 15 each each DET iajs-2429 300 16 ideal ideal ADJ iajs-2429 300 17 𝐼 𝐼 PROPN iajs-2429 300 18 of of ADP iajs-2429 300 19 𝑅 𝑅 PROPN iajs-2429 300 20 implies imply VERB iajs-2429 300 21 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 300 22 𝑁 𝑁 PROPN iajs-2429 300 23 ∩ ∩ ADJ iajs-2429 300 24 𝑀𝐼 𝑀𝐼 PROPN iajs-2429 300 25 𝑁 𝑁 PROPN iajs-2429 300 26 ∩ ∩ ADJ iajs-2429 300 27 𝑀𝐼 𝑀𝐼 PROPN iajs-2429 300 28 𝑁𝐼 𝑁𝐼 PROPN iajs-2429 300 29 as as SCONJ iajs-2429 300 30 desired desire VERB iajs-2429 300 31 . . PUNCT iajs-2429 301 1 the the DET iajs-2429 301 2 following following ADJ iajs-2429 301 3 result result NOUN iajs-2429 301 4 is be AUX iajs-2429 301 5 appeared appear VERB iajs-2429 301 6 in in ADP iajs-2429 301 7 [ [ X iajs-2429 301 8 2 2 NUM iajs-2429 301 9 ] ] PUNCT iajs-2429 301 10 . . PUNCT iajs-2429 302 1 without without ADP iajs-2429 302 2 proof proof NOUN iajs-2429 302 3 proposition proposition NOUN iajs-2429 302 4 ( ( PUNCT iajs-2429 302 5 4.22 4.22 NUM iajs-2429 302 6 ): ): PUNCT iajs-2429 302 7 every every DET iajs-2429 302 8 sum sum NOUN iajs-2429 302 9 of of ADP iajs-2429 302 10 second second ADJ iajs-2429 302 11 submodules submodule NOUN iajs-2429 302 12 is be AUX iajs-2429 302 13 semisecond semisecond ADJ iajs-2429 302 14 . . PUNCT iajs-2429 303 1 proof proof NOUN iajs-2429 303 2 . . PUNCT iajs-2429 304 1 let let VERB iajs-2429 304 2 𝑎 𝑎 PROPN iajs-2429 304 3 ∈ ∈ PROPN iajs-2429 304 4 𝑅 𝑅 PROPN iajs-2429 304 5 , , PUNCT iajs-2429 304 6 𝑁 𝑁 PROPN iajs-2429 304 7 and and CCONJ iajs-2429 304 8 𝐻 𝐻 PROPN iajs-2429 304 9 be be VERB iajs-2429 304 10 second second ADJ iajs-2429 304 11 submodules submodule NOUN iajs-2429 304 12 of of ADP iajs-2429 304 13 an an DET iajs-2429 304 14 𝑅-module 𝑅-module PROPN iajs-2429 304 15 𝑀 𝑀 PROPN iajs-2429 304 16 implies imply VERB iajs-2429 304 17 either either CCONJ iajs-2429 304 18 𝑁 𝑁 PROPN iajs-2429 304 19 𝐻 𝐻 PROPN iajs-2429 304 20 𝑎 𝑎 X iajs-2429 304 21 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 304 22 𝐻𝑎 𝐻𝑎 PROPN iajs-2429 304 23 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 304 24 𝐻𝑎 𝐻𝑎 PROPN iajs-2429 304 25 𝑁 𝑁 NOUN iajs-2429 304 26 𝐻 𝐻 PROPN iajs-2429 304 27 𝑎 𝑎 NOUN iajs-2429 304 28 or or CCONJ iajs-2429 304 29 𝑁 𝑁 ADP iajs-2429 304 30 𝐻 𝐻 PROPN iajs-2429 304 31 𝑎 𝑎 X iajs-2429 304 32 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 304 33 𝐻𝑎 𝐻𝑎 PROPN iajs-2429 304 34 0 0 NUM iajs-2429 305 1 𝐻𝑎 𝐻𝑎 PROPN iajs-2429 305 2 𝐻𝑎 𝐻𝑎 PROPN iajs-2429 305 3 ⊆ ⊆ NUM iajs-2429 305 4 𝑁 𝑁 PROPN iajs-2429 305 5 𝐻 𝐻 PROPN iajs-2429 305 6 𝑎 𝑎 NOUN iajs-2429 305 7 or or CCONJ iajs-2429 305 8 𝑁 𝑁 ADP iajs-2429 305 9 𝐻 𝐻 PROPN iajs-2429 305 10 𝑎 𝑎 X iajs-2429 305 11 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 305 12 𝐻𝑎 𝐻𝑎 PROPN iajs-2429 305 13 0 0 NUM iajs-2429 305 14 0 0 NUM iajs-2429 305 15 0 0 NUM iajs-2429 305 16 ⊆ ⊆ NUM iajs-2429 305 17 𝑁 𝑁 PROPN iajs-2429 305 18 𝐻 𝐻 PROPN iajs-2429 305 19 𝑎 𝑎 NOUN iajs-2429 305 20 and and CCONJ iajs-2429 305 21 hence hence ADV iajs-2429 305 22 𝑁 𝑁 PROPN iajs-2429 305 23 𝐻 𝐻 PROPN iajs-2429 305 24 𝑎 𝑎 VERB iajs-2429 305 25 𝑁 𝑁 NOUN iajs-2429 305 26 𝐻 𝐻 PROPN iajs-2429 305 27 𝑎 𝑎 NOUN iajs-2429 305 28 . . PUNCT iajs-2429 306 1 example example NOUN iajs-2429 306 2 ( ( PUNCT iajs-2429 306 3 4.23 4.23 NUM iajs-2429 306 4 ): ): PUNCT iajs-2429 306 5 the the DET iajs-2429 306 6 sum sum NOUN iajs-2429 306 7 of of ADP iajs-2429 306 8 second second ADJ iajs-2429 306 9 submodules submodule NOUN iajs-2429 306 10 may may AUX iajs-2429 306 11 not not PART iajs-2429 306 12 be be AUX iajs-2429 306 13 second second ADJ iajs-2429 306 14 . . PUNCT iajs-2429 307 1 the the DET iajs-2429 307 2 submodules submodule NOUN iajs-2429 307 3 ℤ ℤ PROPN iajs-2429 307 4 . . PUNCT iajs-2429 307 5 2 2 NUM iajs-2429 307 6 and and CCONJ iajs-2429 307 7 ℤ ℤ PROPN iajs-2429 307 8 . . PUNCT iajs-2429 307 9 3 3 NUM iajs-2429 307 10 are be AUX iajs-2429 307 11 simple simple ADJ iajs-2429 307 12 and and CCONJ iajs-2429 307 13 hence hence ADV iajs-2429 307 14 second second ADJ iajs-2429 307 15 of of ADP iajs-2429 307 16 ℤ ℤ PROPN iajs-2429 307 17 as as ADP iajs-2429 307 18 ℤ-module ℤ-module PROPN iajs-2429 307 19 while while SCONJ iajs-2429 307 20 ℤ ℤ PROPN iajs-2429 307 21 . . NOUN iajs-2429 307 22 2 2 NUM iajs-2429 307 23 ℤ ℤ NOUN iajs-2429 307 24 . . PUNCT iajs-2429 308 1 3 3 NUM iajs-2429 308 2 ℤ ℤ PROPN iajs-2429 308 3 is be AUX iajs-2429 308 4 semisecond semisecond ADJ iajs-2429 308 5 but but CCONJ iajs-2429 308 6 not not PART iajs-2429 308 7 second second ADJ iajs-2429 308 8 . . PUNCT iajs-2429 309 1 proposition proposition NOUN iajs-2429 309 2 ( ( PUNCT iajs-2429 309 3 4.24 4.24 NUM iajs-2429 309 4 ): ): PUNCT iajs-2429 309 5 every every DET iajs-2429 309 6 semisecond semisecond ADJ iajs-2429 309 7 submodule submodule NOUN iajs-2429 309 8 of of ADP iajs-2429 309 9 prime prime ADJ iajs-2429 309 10 module module NOUN iajs-2429 309 11 is be AUX iajs-2429 309 12 second second ADJ iajs-2429 309 13 . . PUNCT iajs-2429 310 1 proof proof NOUN iajs-2429 310 2 . . PUNCT iajs-2429 311 1 let let VERB iajs-2429 311 2 𝑎 𝑎 PRON iajs-2429 311 3 ∈ ∈ PROPN iajs-2429 311 4 𝑅 𝑅 PROPN iajs-2429 311 5 and and CCONJ iajs-2429 311 6 𝑁 𝑁 PROPN iajs-2429 311 7 be be VERB iajs-2429 311 8 a a DET iajs-2429 311 9 semisecond semisecond ADJ iajs-2429 311 10 submodule submodule NOUN iajs-2429 311 11 of of ADP iajs-2429 311 12 a a DET iajs-2429 311 13 prime prime ADJ iajs-2429 311 14 𝑅-module 𝑅-module PROPN iajs-2429 311 15 𝑀 𝑀 PROPN iajs-2429 311 16 implies imply VERB iajs-2429 311 17 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 311 18 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 311 19 then then ADV iajs-2429 311 20 for for ADP iajs-2429 311 21 each each DET iajs-2429 311 22 𝑛 𝑛 PRON iajs-2429 311 23 ∈ ∈ NOUN iajs-2429 311 24 𝑁 𝑁 NOUN iajs-2429 311 25 we we PRON iajs-2429 311 26 have have VERB iajs-2429 311 27 𝑛𝑎 𝑛𝑎 PRON iajs-2429 311 28 𝑚𝑎 𝑚𝑎 ADP iajs-2429 311 29 for for ADP iajs-2429 311 30 some some DET iajs-2429 311 31 𝑚 𝑚 NOUN iajs-2429 311 32 ∈ ∈ NOUN iajs-2429 311 33 𝑁 𝑁 PROPN iajs-2429 311 34 implies imply VERB iajs-2429 311 35 𝑛 𝑛 PRON iajs-2429 311 36 𝑚𝑎 𝑚𝑎 ADP iajs-2429 311 37 𝑎 𝑎 PROPN iajs-2429 311 38 0 0 NUM iajs-2429 311 39 . . PUNCT iajs-2429 312 1 but but CCONJ iajs-2429 312 2 0 0 NUM iajs-2429 312 3 is be AUX iajs-2429 312 4 a a DET iajs-2429 312 5 prime prime ADJ iajs-2429 312 6 submodule submodule NOUN iajs-2429 312 7 in in ADV iajs-2429 312 8 , , PUNCT iajs-2429 312 9 it it PRON iajs-2429 312 10 follows follow VERB iajs-2429 312 11 either either CCONJ iajs-2429 312 12 𝑛 𝑛 DET iajs-2429 312 13 𝑚𝑎 𝑚𝑎 ADP iajs-2429 312 14 ∈ ∈ PROPN iajs-2429 312 15 0 0 PUNCT iajs-2429 312 16 implies imply VERB iajs-2429 312 17 𝑛 𝑛 PRON iajs-2429 312 18 𝑚𝑎 𝑚𝑎 ADV iajs-2429 312 19 and and CCONJ iajs-2429 312 20 hence hence ADV iajs-2429 312 21 𝑁 𝑁 PROPN iajs-2429 312 22 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 312 23 or or CCONJ iajs-2429 312 24 𝑎 𝑎 PROPN iajs-2429 312 25 ∈ ∈ NOUN iajs-2429 312 26 0 0 NUM iajs-2429 312 27 : : PUNCT iajs-2429 312 28 𝑀 𝑀 PROPN iajs-2429 312 29 ⊆ ⊆ NUM iajs-2429 312 30 0 0 NUM iajs-2429 312 31 : : PUNCT iajs-2429 312 32 𝑁 𝑁 PROPN iajs-2429 312 33 implies imply VERB iajs-2429 312 34 𝑁𝑎 𝑁𝑎 PROPN iajs-2429 312 35 0 0 NUM iajs-2429 312 36 as as SCONJ iajs-2429 312 37 desired desire VERB iajs-2429 312 38 . . PUNCT iajs-2429 313 1 proposition proposition NOUN iajs-2429 313 2 ( ( PUNCT iajs-2429 313 3 4.25 4.25 NUM iajs-2429 313 4 ): ): PUNCT iajs-2429 313 5 every every DET iajs-2429 313 6 semisecond semisecond ADJ iajs-2429 313 7 submodule submodule NOUN iajs-2429 313 8 of of ADP iajs-2429 313 9 primary primary ADJ iajs-2429 313 10 module module NOUN iajs-2429 313 11 is be AUX iajs-2429 313 12 secondary secondary ADJ iajs-2429 313 13 . . PUNCT iajs-2429 314 1 proof proof NOUN iajs-2429 314 2 . . PUNCT iajs-2429 315 1 similarly similarly ADV iajs-2429 315 2 of of ADP iajs-2429 315 3 proposition proposition NOUN iajs-2429 315 4 4.24 4.24 NUM iajs-2429 315 5 . . PUNCT iajs-2429 316 1 5 5 NUM iajs-2429 316 2 . . X iajs-2429 317 1 𝑺-semisecond 𝑺-semisecond ADJ iajs-2429 317 2 modules module NOUN iajs-2429 317 3 definition definition NOUN iajs-2429 317 4 ( ( PUNCT iajs-2429 317 5 5.1 5.1 NUM iajs-2429 317 6 ): ): PUNCT iajs-2429 317 7 a a DET iajs-2429 317 8 nonzero nonzero ADJ iajs-2429 317 9 𝑅-module 𝑅-module PROPN iajs-2429 317 10 𝑀 𝑀 PROPN iajs-2429 317 11 is be AUX iajs-2429 317 12 called call VERB iajs-2429 317 13 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 317 14 whenever whenever SCONJ iajs-2429 317 15 𝑓 𝑓 PRON iajs-2429 317 16 𝑀 𝑀 PROPN iajs-2429 317 17 ⊆ ⊆ PROPN iajs-2429 317 18 𝐾 𝐾 PROPN iajs-2429 317 19 , , PUNCT iajs-2429 317 20 where where SCONJ iajs-2429 317 21 𝑓 𝑓 DET iajs-2429 317 22 ∈ ∈ PROPN iajs-2429 317 23 𝑆 𝑆 PROPN iajs-2429 317 24 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2429 317 25 𝑀 𝑀 PROPN iajs-2429 317 26 and and CCONJ iajs-2429 317 27 𝐾 𝐾 PROPN iajs-2429 317 28 a a DET iajs-2429 317 29 submodule submodule NOUN iajs-2429 317 30 of of ADP iajs-2429 317 31 𝑀 𝑀 PROPN iajs-2429 317 32 implies imply VERB iajs-2429 317 33 𝑓 𝑓 PRON iajs-2429 317 34 𝑀 𝑀 PROPN iajs-2429 317 35 ⊆ ⊆ NUM iajs-2429 317 36 𝐾. 𝐾. PROPN iajs-2429 317 37 theorem theorem NOUN iajs-2429 317 38 ( ( PUNCT iajs-2429 317 39 5.2 5.2 NUM iajs-2429 317 40 ): ): PUNCT iajs-2429 317 41 the the DET iajs-2429 317 42 following following NOUN iajs-2429 317 43 are be AUX iajs-2429 317 44 equivalent equivalent ADJ iajs-2429 317 45 ( ( PUNCT iajs-2429 317 46 1 1 NUM iajs-2429 317 47 ) ) PUNCT iajs-2429 317 48 𝑀 𝑀 PROPN iajs-2429 317 49 is be AUX iajs-2429 317 50 a a DET iajs-2429 317 51 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 317 52 𝑅-module 𝑅-module PROPN iajs-2429 317 53 . . PUNCT iajs-2429 318 1 ( ( PUNCT iajs-2429 318 2 2 2 X iajs-2429 318 3 ) ) PUNCT iajs-2429 318 4 𝑀 𝑀 PROPN iajs-2429 318 5 0 0 PUNCT iajs-2429 319 1 and and CCONJ iajs-2429 319 2 𝑓 𝑓 DET iajs-2429 319 3 𝑀 𝑀 PROPN iajs-2429 319 4 𝑓 𝑓 PROPN iajs-2429 319 5 𝑀 𝑀 PROPN iajs-2429 319 6 for for ADP iajs-2429 319 7 each each DET iajs-2429 319 8 𝑓 𝑓 DET iajs-2429 319 9 ∈ ∈ NOUN iajs-2429 319 10 𝑆. 𝑆. NOUN iajs-2429 319 11 proof proof NOUN iajs-2429 319 12 . . PUNCT iajs-2429 320 1 ( ( PUNCT iajs-2429 320 2 1 1 X iajs-2429 320 3 ) ) PUNCT iajs-2429 320 4   NOUN iajs-2429 320 5 ( ( PUNCT iajs-2429 320 6 2 2 X iajs-2429 320 7 ) ) PUNCT iajs-2429 320 8 assume assume VERB iajs-2429 320 9 𝑀 𝑀 PROPN iajs-2429 320 10 is be AUX iajs-2429 320 11 an an DET iajs-2429 320 12 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 320 13 𝑅-module 𝑅-module PROPN iajs-2429 320 14 implies imply VERB iajs-2429 320 15 𝑀 𝑀 PROPN iajs-2429 320 16 0 0 NUM iajs-2429 320 17 . . PUNCT iajs-2429 321 1 since since SCONJ iajs-2429 321 2 𝑓 𝑓 PRON iajs-2429 321 3 𝑀 𝑀 PROPN iajs-2429 321 4 ⊆ ⊆ PROPN iajs-2429 321 5 𝑓 𝑓 DET iajs-2429 321 6 𝑀 𝑀 PROPN iajs-2429 321 7 implies imply VERB iajs-2429 321 8 𝑓 𝑓 DET iajs-2429 321 9 𝑀 𝑀 PROPN iajs-2429 321 10 ⊆ ⊆ PROPN iajs-2429 321 11 𝑓 𝑓 DET iajs-2429 321 12 𝑀 𝑀 PROPN iajs-2429 321 13 and and CCONJ iajs-2429 321 14 hence hence ADV iajs-2429 321 15 𝑓 𝑓 DET iajs-2429 321 16 𝑀 𝑀 PROPN iajs-2429 321 17 𝑓 𝑓 PROPN iajs-2429 321 18 𝑀 𝑀 PROPN iajs-2429 321 19 for for ADP iajs-2429 321 20 each each DET iajs-2429 321 21 𝑓 𝑓 PRON iajs-2429 321 22 ∈ ∈ PROPN iajs-2429 321 23 𝑆 𝑆 PROPN iajs-2429 321 24 as as SCONJ iajs-2429 321 25 desired desire VERB iajs-2429 321 26 . . PUNCT iajs-2429 322 1 as as SCONJ iajs-2429 322 2 desired desire VERB iajs-2429 322 3 . . PUNCT iajs-2429 323 1 ( ( PUNCT iajs-2429 323 2 2 2 X iajs-2429 323 3 ) ) PUNCT iajs-2429 323 4   NOUN iajs-2429 323 5 ( ( PUNCT iajs-2429 323 6 1 1 X iajs-2429 323 7 ) ) PUNCT iajs-2429 323 8 assume assume VERB iajs-2429 323 9 𝑓 𝑓 PRON iajs-2429 323 10 𝑀 𝑀 PROPN iajs-2429 323 11 ⊆ ⊆ PROPN iajs-2429 323 12 𝐾 𝐾 PROPN iajs-2429 323 13 , , PUNCT iajs-2429 323 14 where where SCONJ iajs-2429 323 15 𝑓 𝑓 DET iajs-2429 323 16 ∈ ∈ PROPN iajs-2429 323 17 𝑆 𝑆 PROPN iajs-2429 323 18 and and CCONJ iajs-2429 323 19 𝐾 𝐾 PROPN iajs-2429 323 20 a a DET iajs-2429 323 21 submodule submodule NOUN iajs-2429 323 22 of of ADP iajs-2429 323 23 𝑀 𝑀 PROPN iajs-2429 323 24 implies imply VERB iajs-2429 323 25 𝑓 𝑓 DET iajs-2429 323 26 𝑀 𝑀 PROPN iajs-2429 323 27 𝑓 𝑓 PROPN iajs-2429 323 28 𝑀 𝑀 PROPN iajs-2429 323 29 ⊆ ⊆ NUM iajs-2429 323 30 𝐾 𝐾 PROPN iajs-2429 323 31 as as SCONJ iajs-2429 323 32 required require VERB iajs-2429 323 33 . . PUNCT iajs-2429 324 1 proposition proposition NOUN iajs-2429 324 2 ( ( PUNCT iajs-2429 324 3 5.3 5.3 NUM iajs-2429 324 4 ): ): PUNCT iajs-2429 324 5 every every DET iajs-2429 324 6 semisecond semisecond ADJ iajs-2429 324 7 multiplication multiplication NOUN iajs-2429 324 8 module module NOUN iajs-2429 324 9 is be AUX iajs-2429 324 10 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 324 11 . . PUNCT iajs-2429 324 12 proof proof NOUN iajs-2429 324 13 . . PUNCT iajs-2429 325 1 let let VERB iajs-2429 325 2 𝑀 𝑀 PRON iajs-2429 325 3 be be AUX iajs-2429 325 4 a a DET iajs-2429 325 5 semisecond semisecond ADJ iajs-2429 325 6 multiplication multiplication NOUN iajs-2429 325 7 𝑅-module 𝑅-module PROPN iajs-2429 325 8 and and CCONJ iajs-2429 325 9 𝑓 𝑓 DET iajs-2429 325 10 ∈ ∈ PROPN iajs-2429 325 11 𝑆 𝑆 PROPN iajs-2429 325 12 with with ADP iajs-2429 325 13 𝑓 𝑓 DET iajs-2429 325 14 𝑀 𝑀 PROPN iajs-2429 325 15 ⊆ ⊆ NUM iajs-2429 325 16 𝐾 𝐾 PROPN iajs-2429 325 17 for for ADP iajs-2429 325 18 some some DET iajs-2429 325 19 𝐾 𝐾 PROPN iajs-2429 325 20 a a DET iajs-2429 325 21 submodule submodule NOUN iajs-2429 325 22 of of ADP iajs-2429 325 23 𝑀. 𝑀. PROPN iajs-2429 325 24 since since SCONJ iajs-2429 325 25 𝑀 𝑀 PROPN iajs-2429 325 26 is be AUX iajs-2429 325 27 multiplication multiplication NOUN iajs-2429 325 28 then then ADV iajs-2429 325 29 𝑓 𝑓 PRON iajs-2429 325 30 𝑀 𝑀 PROPN iajs-2429 325 31 𝑓 𝑓 DET iajs-2429 325 32 𝐼𝑀 𝐼𝑀 PROPN iajs-2429 326 1 𝐼𝑓 𝐼𝑓 PROPN iajs-2429 326 2 𝑀 𝑀 PROPN iajs-2429 326 3 𝐼𝐼𝑀 𝐼𝐼𝑀 PROPN iajs-2429 326 4 for for ADP iajs-2429 326 5 some some DET iajs-2429 326 6 ideal ideal ADJ iajs-2429 326 7 𝐼 𝐼 PROPN iajs-2429 326 8 of of ADP iajs-2429 326 9 𝑅 𝑅 PROPN iajs-2429 326 10 and and CCONJ iajs-2429 326 11 hence hence ADV iajs-2429 326 12 𝐼 𝐼 ADP iajs-2429 326 13 𝑀 𝑀 PROPN iajs-2429 326 14 ⊆ ⊆ NUM iajs-2429 326 15 𝐾. 𝐾. PROPN iajs-2429 326 16 by by ADP iajs-2429 326 17 theorem theorem NOUN iajs-2429 326 18 3.6 3.6 NUM iajs-2429 326 19 , , PUNCT iajs-2429 326 20 we we PRON iajs-2429 326 21 have have VERB iajs-2429 326 22 𝐼𝑀 𝐼𝑀 PROPN iajs-2429 326 23 ⊆ ⊆ NUM iajs-2429 326 24 𝐾 𝐾 PROPN iajs-2429 326 25 it it PRON iajs-2429 326 26 follows follow VERB iajs-2429 326 27 𝑓 𝑓 DET iajs-2429 326 28 𝑀 𝑀 PROPN iajs-2429 326 29 ⊆ ⊆ PROPN iajs-2429 326 30 𝐾 𝐾 PROPN iajs-2429 326 31 that that PRON iajs-2429 326 32 is be AUX iajs-2429 326 33 𝑀 𝑀 PROPN iajs-2429 326 34 is be AUX iajs-2429 326 35 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 326 36 . . PUNCT iajs-2429 326 37 corollary corollary NOUN iajs-2429 326 38 ( ( PUNCT iajs-2429 326 39 5.4 5.4 NUM iajs-2429 326 40 ): ): PUNCT iajs-2429 326 41 every every DET iajs-2429 326 42 semisecond semisecond ADJ iajs-2429 326 43 cyclic cyclic NOUN iajs-2429 326 44 module module NOUN iajs-2429 326 45 is be AUX iajs-2429 326 46 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 326 47 . . PUNCT iajs-2429 327 1 remarks remark NOUN iajs-2429 327 2 and and CCONJ iajs-2429 327 3 examples example NOUN iajs-2429 327 4 ( ( PUNCT iajs-2429 327 5 5.5 5.5 NUM iajs-2429 327 6 ): ): PUNCT iajs-2429 327 7 ( ( PUNCT iajs-2429 327 8 1 1 X iajs-2429 327 9 ) ) PUNCT iajs-2429 327 10 every every DET iajs-2429 327 11 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 327 12 module module NOUN iajs-2429 327 13 is be AUX iajs-2429 327 14 semisecond semisecond ADJ iajs-2429 327 15 . . PUNCT iajs-2429 328 1 proof proof NOUN iajs-2429 328 2 . . PUNCT iajs-2429 329 1 let let VERB iajs-2429 329 2 𝑀 𝑀 PROPN iajs-2429 329 3 be be AUX iajs-2429 329 4 an an DET iajs-2429 329 5 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 329 6 𝑅-module 𝑅-module PROPN iajs-2429 329 7 , , PUNCT iajs-2429 329 8 then then ADV iajs-2429 329 9 𝑀 𝑀 PROPN iajs-2429 329 10 0 0 NUM iajs-2429 329 11 . . PUNCT iajs-2429 330 1 let let VERB iajs-2429 330 2 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 330 3 ⊆ ⊆ NUM iajs-2429 330 4 𝐾 𝐾 PROPN iajs-2429 330 5 for for ADP iajs-2429 330 6 some some DET iajs-2429 330 7 𝑎 𝑎 PRON iajs-2429 330 8 ∈ ∈ PROPN iajs-2429 330 9 𝑅 𝑅 PROPN iajs-2429 330 10 and and CCONJ iajs-2429 330 11 𝐾 𝐾 PROPN iajs-2429 330 12 a a DET iajs-2429 330 13 submodule submodule NOUN iajs-2429 330 14 of of ADP iajs-2429 330 15 𝑀. 𝑀. PROPN iajs-2429 330 16 define define VERB iajs-2429 330 17 the the DET iajs-2429 330 18 endomorphisms endomorphism NOUN iajs-2429 330 19 𝑓 𝑓 X iajs-2429 330 20 : : PUNCT iajs-2429 330 21 𝑀 𝑀 PROPN iajs-2429 330 22 → → SYM iajs-2429 330 23 𝑀 𝑀 PROPN iajs-2429 330 24 by by ADP iajs-2429 330 25 𝑓 𝑓 DET iajs-2429 330 26 𝑚 𝑚 NOUN iajs-2429 330 27 𝑚𝑎 𝑚𝑎 ADV iajs-2429 330 28 for for ADP iajs-2429 330 29 each each DET iajs-2429 330 30 𝑚 𝑚 PROPN iajs-2429 330 31 ∈ ∈ PROPN iajs-2429 330 32 𝑀. 𝑀. PROPN iajs-2429 330 33 then then ADV iajs-2429 330 34 , , PUNCT iajs-2429 330 35 𝑓 𝑓 DET iajs-2429 330 36 𝑀 𝑀 PROPN iajs-2429 330 37 𝑓 𝑓 PROPN iajs-2429 330 38 𝑓 𝑓 PROPN iajs-2429 330 39 𝑀 𝑀 PROPN iajs-2429 330 40 𝑓 𝑓 PROPN iajs-2429 330 41 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 330 42 𝑓 𝑓 DET iajs-2429 330 43 𝑀 𝑀 PROPN iajs-2429 330 44 𝑎 𝑎 PROPN iajs-2429 330 45 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 330 46 ⊆ ⊆ NUM iajs-2429 330 47 𝐾. 𝐾. PROPN iajs-2429 330 48 by by ADP iajs-2429 330 49 hypothesis hypothesis NOUN iajs-2429 330 50 , , PUNCT iajs-2429 330 51 we we PRON iajs-2429 330 52 have have VERB iajs-2429 330 53 𝑓 𝑓 DET iajs-2429 330 54 𝑀 𝑀 PROPN iajs-2429 330 55 ⊆ ⊆ NUM iajs-2429 330 56 𝐾 𝐾 PROPN iajs-2429 330 57 that that PRON iajs-2429 330 58 is be AUX iajs-2429 330 59 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 330 60 ⊆ ⊆ NUM iajs-2429 330 61 𝐾 𝐾 PROPN iajs-2429 330 62 as as SCONJ iajs-2429 330 63 desired desire VERB iajs-2429 330 64 . . PUNCT iajs-2429 331 1 ( ( PUNCT iajs-2429 331 2 2 2 X iajs-2429 331 3 ) ) PUNCT iajs-2429 331 4 the the DET iajs-2429 331 5 converse converse NOUN iajs-2429 331 6 of of ADP iajs-2429 331 7 ( ( PUNCT iajs-2429 331 8 1 1 X iajs-2429 331 9 ) ) PUNCT iajs-2429 331 10 is be AUX iajs-2429 331 11 not not PART iajs-2429 331 12 true true ADJ iajs-2429 331 13 in in ADP iajs-2429 331 14 general general ADJ iajs-2429 331 15 . . PUNCT iajs-2429 332 1 for for ADP iajs-2429 332 2 example example NOUN iajs-2429 332 3 , , PUNCT iajs-2429 332 4 𝑀 𝑀 PROPN iajs-2429 332 5 ℤ ℤ PROPN iajs-2429 332 6 ⊕ ⊕ PROPN iajs-2429 332 7 ℤ ℤ PROPN iajs-2429 332 8 as as SCONJ iajs-2429 332 9 ℤmodule ℤmodule PROPN iajs-2429 332 10 is be AUX iajs-2429 332 11 semisecond semisecond ADJ iajs-2429 332 12 where where SCONJ iajs-2429 332 13     SPACE iajs-2429 332 14 91 91 NUM iajs-2429 332 15     SPACE iajs-2429 332 16 ibn ibn PROPN iajs-2429 332 17 al al PROPN iajs-2429 332 18 - - PUNCT iajs-2429 332 19 haitham haitham PROPN iajs-2429 332 20 jour jour X iajs-2429 332 21 . . PROPN iajs-2429 333 1 for for ADP iajs-2429 333 2 pure pure ADJ iajs-2429 333 3 & & CCONJ iajs-2429 333 4 appl appl PROPN iajs-2429 333 5 . . PUNCT iajs-2429 334 1 sci sci PROPN iajs-2429 334 2 . . PROPN iajs-2429 335 1 33 33 NUM iajs-2429 335 2 ( ( PUNCT iajs-2429 335 3 2 2 NUM iajs-2429 335 4 ) ) PUNCT iajs-2429 335 5 2020 2020 NUM iajs-2429 335 6 ( ( PUNCT iajs-2429 335 7 3 3 X iajs-2429 335 8 ) ) PUNCT iajs-2429 335 9 𝑆 𝑆 PROPN iajs-2429 335 10 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 335 11 ℤ ℤ PROPN iajs-2429 335 12 ⊕ ⊕ PROPN iajs-2429 335 13 ℤ ℤ NOUN iajs-2429 335 14 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 335 15 ℤ ℤ PROPN iajs-2429 335 16 𝐻𝑜𝑚ℤ 𝐻𝑜𝑚ℤ NOUN iajs-2429 335 17 ℤ ℤ PROPN iajs-2429 335 18 , , PUNCT iajs-2429 335 19 ℤ ℤ PROPN iajs-2429 335 20 𝐻𝑜𝑚ℤ 𝐻𝑜𝑚ℤ NOUN iajs-2429 335 21 ℤ ℤ PROPN iajs-2429 335 22 , , PUNCT iajs-2429 335 23 ℤ ℤ PROPN iajs-2429 335 24 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 335 25 ℤ ℤ PROPN iajs-2429 335 26 ≅ ≅ PROPN iajs-2429 335 27 𝑀𝑎𝑡 𝑀𝑎𝑡 PROPN iajs-2429 335 28 ℤ ℤ PROPN iajs-2429 335 29 ℤ ℤ NOUN iajs-2429 335 30 ℤ ℤ NOUN iajs-2429 335 31 ℤ ℤ NOUN iajs-2429 335 32 ℤ ℤ PROPN iajs-2429 335 33 is be AUX iajs-2429 335 34 not not PART iajs-2429 335 35 semisecond semisecond ADJ iajs-2429 335 36 ring ring NOUN iajs-2429 335 37 by by ADP iajs-2429 335 38 example example NOUN iajs-2429 335 39 4.8(1 4.8(1 NUM iajs-2429 335 40 ) ) PUNCT iajs-2429 335 41 and and CCONJ iajs-2429 335 42 ℤ ℤ PROPN iajs-2429 335 43 ⊕ ⊕ PROPN iajs-2429 335 44 ℤ ℤ PROPN iajs-2429 335 45 ≅ ≅ PROPN iajs-2429 335 46 , , PUNCT iajs-2429 335 47 , , PUNCT iajs-2429 335 48 , , PUNCT iajs-2429 335 49 so so CCONJ iajs-2429 335 50 if if SCONJ iajs-2429 335 51 we we PRON iajs-2429 335 52 take take VERB iajs-2429 335 53 𝑓 𝑓 PRON iajs-2429 335 54 0 0 NUM iajs-2429 335 55 1 1 NUM iajs-2429 335 56 0 0 NUM iajs-2429 335 57 0 0 NUM iajs-2429 335 58 ∈ ∈ PROPN iajs-2429 335 59 𝑆 𝑆 PROPN iajs-2429 335 60 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 335 61 ℤ ℤ PROPN iajs-2429 335 62 ⊕ ⊕ PROPN iajs-2429 335 63 ℤ ℤ PROPN iajs-2429 335 64 implies imply VERB iajs-2429 335 65 𝑓 𝑓 DET iajs-2429 335 66 𝑀 𝑀 PROPN iajs-2429 335 67 0 0 NUM iajs-2429 335 68 1 1 NUM iajs-2429 335 69 0 0 NUM iajs-2429 335 70 0 0 NUM iajs-2429 335 71 , , PUNCT iajs-2429 335 72 𝑥 𝑥 X iajs-2429 335 73 , , PUNCT iajs-2429 335 74 𝑦 𝑦 NOUN iajs-2429 335 75 ∈ ∈ NOUN iajs-2429 335 76 ℤ ℤ PROPN iajs-2429 335 77 , , PUNCT iajs-2429 335 78 𝑓 𝑓 DET iajs-2429 335 79 𝑀 𝑀 PROPN iajs-2429 335 80 0 0 NUM iajs-2429 335 81 0 0 NUM iajs-2429 335 82 0 0 NUM iajs-2429 335 83 0 0 NUM iajs-2429 335 84 , , PUNCT iajs-2429 335 85 𝑥 𝑥 X iajs-2429 335 86 , , PUNCT iajs-2429 335 87 𝑦 𝑦 NOUN iajs-2429 335 88 ∈ ∈ NOUN iajs-2429 336 1 ℤ ℤ NOUN iajs-2429 336 2 it it PRON iajs-2429 336 3 follows follow VERB iajs-2429 336 4 that that SCONJ iajs-2429 336 5 𝑀 𝑀 PROPN iajs-2429 336 6 is be AUX iajs-2429 336 7 not not PART iajs-2429 336 8 semisecond semisecond ADJ iajs-2429 336 9 as as ADP iajs-2429 336 10 𝑆-module 𝑆-module PROPN iajs-2429 336 11 that that PRON iajs-2429 336 12 is be AUX iajs-2429 336 13 , , PUNCT iajs-2429 336 14 𝑀 𝑀 PROPN iajs-2429 336 15 is be AUX iajs-2429 336 16 not not PART iajs-2429 336 17 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 336 18 as as ADP iajs-2429 336 19 ℤ-module ℤ-module PROPN iajs-2429 336 20 . . PUNCT iajs-2429 337 1 ( ( PUNCT iajs-2429 337 2 4 4 X iajs-2429 337 3 ) ) PUNCT iajs-2429 337 4 if if SCONJ iajs-2429 337 5 0 0 NUM iajs-2429 337 6 𝑀 𝑀 PROPN iajs-2429 337 7 is be AUX iajs-2429 337 8 not not PART iajs-2429 337 9 a a DET iajs-2429 337 10 divisible divisible ADJ iajs-2429 337 11 ℤ-module ℤ-module NOUN iajs-2429 337 12 , , PUNCT iajs-2429 337 13 then then ADV iajs-2429 337 14 𝑀 𝑀 PROPN iajs-2429 337 15 ⊕ ⊕ PROPN iajs-2429 337 16 𝑀 𝑀 PROPN iajs-2429 337 17 can can AUX iajs-2429 337 18 not not PART iajs-2429 337 19 be be AUX iajs-2429 337 20 an an DET iajs-2429 337 21 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 337 22 ℤmodule ℤmodule NOUN iajs-2429 337 23 . . PUNCT iajs-2429 338 1 proof proof NOUN iajs-2429 338 2 . . PUNCT iajs-2429 339 1 let let VERB iajs-2429 339 2 𝑀 𝑀 PRON iajs-2429 339 3 be be AUX iajs-2429 339 4 a a DET iajs-2429 339 5 not not PART iajs-2429 339 6 divisible divisible ADJ iajs-2429 339 7 ℤ-module ℤ-module PROPN iajs-2429 339 8 . . PROPN iajs-2429 339 9 suppose suppose VERB iajs-2429 339 10 that that SCONJ iajs-2429 339 11 𝑀 𝑀 PROPN iajs-2429 339 12 ⊕ ⊕ PROPN iajs-2429 339 13 𝑀 𝑀 PROPN iajs-2429 339 14 is be AUX iajs-2429 339 15 an an DET iajs-2429 339 16 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 339 17 ℤ-module ℤ-module PROPN iajs-2429 339 18 . . PUNCT iajs-2429 340 1 we we PRON iajs-2429 340 2 can can AUX iajs-2429 340 3 define define VERB iajs-2429 340 4 the the DET iajs-2429 340 5 maps map NOUN iajs-2429 340 6 𝑓 𝑓 X iajs-2429 340 7 : : PUNCT iajs-2429 340 8 𝑀 𝑀 PROPN iajs-2429 340 9 ⊕ ⊕ PROPN iajs-2429 340 10 𝑀 𝑀 PROPN iajs-2429 340 11 → → SYM iajs-2429 340 12 𝑀 𝑀 PROPN iajs-2429 340 13 ⊕ ⊕ PROPN iajs-2429 340 14 𝑀 𝑀 PROPN iajs-2429 340 15 𝑓 𝑓 PRON iajs-2429 340 16 𝑥 𝑥 PROPN iajs-2429 340 17 , , PUNCT iajs-2429 340 18 𝑦 𝑦 PROPN iajs-2429 340 19 𝑦2 𝑦2 PROPN iajs-2429 340 20 , , PUNCT iajs-2429 340 21 𝑥 𝑥 PROPN iajs-2429 340 22 for for ADP iajs-2429 340 23 each each DET iajs-2429 340 24 𝑥 𝑥 PROPN iajs-2429 340 25 , , PUNCT iajs-2429 340 26 𝑦 𝑦 PRON iajs-2429 340 27 ∈ ∈ NOUN iajs-2429 340 28 𝑀. 𝑀. NOUN iajs-2429 340 29 it it PRON iajs-2429 340 30 is be AUX iajs-2429 340 31 clear clear ADJ iajs-2429 340 32 that that SCONJ iajs-2429 340 33 𝑓 𝑓 DET iajs-2429 340 34 ∈ ∈ PROPN iajs-2429 340 35 𝑆 𝑆 PROPN iajs-2429 340 36 implies imply VERB iajs-2429 340 37 𝑓 𝑓 DET iajs-2429 340 38 𝑀 𝑀 PROPN iajs-2429 340 39 ⊕ ⊕ PROPN iajs-2429 340 40 𝑀 𝑀 PROPN iajs-2429 340 41 𝑓𝑓 𝑓𝑓 VERB iajs-2429 340 42 𝑀 𝑀 PROPN iajs-2429 340 43 ⊕ ⊕ PROPN iajs-2429 340 44 𝑀 𝑀 PROPN iajs-2429 340 45 𝑓 𝑓 DET iajs-2429 340 46 𝑀2 𝑀2 PROPN iajs-2429 340 47 ⊕ ⊕ PROPN iajs-2429 340 48 𝑀 𝑀 PROPN iajs-2429 340 49 𝑀2 𝑀2 PROPN iajs-2429 340 50 ⊕ ⊕ PROPN iajs-2429 340 51 𝑀2 𝑀2 PROPN iajs-2429 340 52 𝑓 𝑓 DET iajs-2429 340 53 𝑀 𝑀 PROPN iajs-2429 340 54 ⊕ ⊕ PROPN iajs-2429 340 55 𝑀 𝑀 PROPN iajs-2429 340 56 𝑀2 𝑀2 PROPN iajs-2429 340 57 ⊕ ⊕ PROPN iajs-2429 340 58 𝑀 𝑀 PROPN iajs-2429 340 59 which which PRON iajs-2429 340 60 is be AUX iajs-2429 340 61 a a DET iajs-2429 340 62 contradiction contradiction NOUN iajs-2429 340 63 . . PUNCT iajs-2429 341 1 ( ( PUNCT iajs-2429 341 2 5 5 NUM iajs-2429 341 3 ) ) PUNCT iajs-2429 341 4 as as ADP iajs-2429 341 5 another another DET iajs-2429 341 6 example example NOUN iajs-2429 341 7 of of ADP iajs-2429 341 8 the the DET iajs-2429 341 9 converse converse NOUN iajs-2429 341 10 of of ADP iajs-2429 341 11 ( ( PUNCT iajs-2429 341 12 1 1 NUM iajs-2429 341 13 ) ) PUNCT iajs-2429 341 14 , , PUNCT iajs-2429 341 15 we we PRON iajs-2429 341 16 have have VERB iajs-2429 341 17 ℤ ℤ PROPN iajs-2429 341 18 ⊕ ⊕ PROPN iajs-2429 341 19 ℤ ℤ PROPN iajs-2429 341 20 , , PUNCT iajs-2429 341 21 ℤ ℤ PROPN iajs-2429 341 22 ⊕ ⊕ PROPN iajs-2429 341 23 ℤ ℤ PROPN iajs-2429 341 24 as as SCONJ iajs-2429 341 25 ℤ-modules ℤ-modules PROPN iajs-2429 341 26 are be AUX iajs-2429 341 27 semisecond semisecond ADJ iajs-2429 341 28 but but CCONJ iajs-2429 341 29 they they PRON iajs-2429 341 30 are be AUX iajs-2429 341 31 not not PART iajs-2429 341 32 𝑆-semisecond 𝑆-semisecond AUX iajs-2429 341 33 by by ADP iajs-2429 341 34 ( ( PUNCT iajs-2429 341 35 2 2 NUM iajs-2429 341 36 ) ) PUNCT iajs-2429 341 37 . . PUNCT iajs-2429 342 1 in in ADP iajs-2429 342 2 fact fact NOUN iajs-2429 342 3 , , PUNCT iajs-2429 342 4 𝑆 𝑆 PROPN iajs-2429 342 5 is be AUX iajs-2429 342 6 not not PART iajs-2429 342 7 semisecond semisecond ADJ iajs-2429 342 8 ring ring NOUN iajs-2429 342 9 so so SCONJ iajs-2429 342 10 any any DET iajs-2429 342 11 module module NOUN iajs-2429 342 12 over over ADP iajs-2429 342 13 𝑆 𝑆 PROPN iajs-2429 342 14 can can AUX iajs-2429 342 15 not not PART iajs-2429 342 16 be be AUX iajs-2429 342 17 semisecond semisecond ADJ iajs-2429 342 18 by by ADP iajs-2429 342 19 proposition proposition NOUN iajs-2429 342 20 4.10 4.10 NUM iajs-2429 342 21 as as SCONJ iajs-2429 342 22 we we PRON iajs-2429 342 23 mentioned mention VERB iajs-2429 342 24 in in ADP iajs-2429 342 25 proposition proposition NOUN iajs-2429 342 26 . . PUNCT iajs-2429 343 1 ( ( PUNCT iajs-2429 343 2 6 6 NUM iajs-2429 343 3 ) ) PUNCT iajs-2429 343 4 the the DET iajs-2429 343 5 direct direct ADJ iajs-2429 343 6 sum sum NOUN iajs-2429 343 7 of of ADP iajs-2429 343 8 𝑆-semisecond 𝑆-semisecond NOUN iajs-2429 343 9 modules module NOUN iajs-2429 343 10 needs needs AUX iajs-2429 343 11 not not PART iajs-2429 343 12 be be AUX iajs-2429 343 13 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 343 14 . . PUNCT iajs-2429 344 1 for for ADP iajs-2429 344 2 example example NOUN iajs-2429 344 3 , , PUNCT iajs-2429 344 4 ℤ ℤ PROPN iajs-2429 344 5 ⊕ ⊕ PROPN iajs-2429 344 6 ℤ ℤ PROPN iajs-2429 344 7 , , PUNCT iajs-2429 344 8 ℤ ℤ PROPN iajs-2429 344 9 ⊕ ⊕ PROPN iajs-2429 344 10 ℤ ℤ PROPN iajs-2429 344 11 are be AUX iajs-2429 344 12 not not PART iajs-2429 344 13 𝑆-semisecond 𝑆-semisecond NOUN iajs-2429 344 14 as as ADP iajs-2429 344 15 ℤ-modules ℤ-modules PROPN iajs-2429 344 16 ( ( PUNCT iajs-2429 344 17 7 7 NUM iajs-2429 344 18 ) ) PUNCT iajs-2429 344 19 it it PRON iajs-2429 344 20 is be AUX iajs-2429 344 21 clear clear ADJ iajs-2429 344 22 every every PRON iajs-2429 344 23 that that SCONJ iajs-2429 344 24 𝑆-weakly 𝑆-weakly ADJ iajs-2429 344 25 second second ADJ iajs-2429 344 26 module module NOUN iajs-2429 344 27 is be AUX iajs-2429 344 28 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 344 29 . . PUNCT iajs-2429 345 1 the the DET iajs-2429 345 2 converse converse NOUN iajs-2429 345 3 is be AUX iajs-2429 345 4 not not PART iajs-2429 345 5 hold hold NOUN iajs-2429 345 6 in in ADP iajs-2429 345 7 general general ADJ iajs-2429 345 8 , , PUNCT iajs-2429 345 9 ℤ ℤ PROPN iajs-2429 345 10 as as SCONJ iajs-2429 345 11 ℤ-module ℤ-module PROPN iajs-2429 345 12 is be AUX iajs-2429 345 13 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 345 14 since since SCONJ iajs-2429 345 15 ℤ ℤ PROPN iajs-2429 345 16 is be AUX iajs-2429 345 17 multiplication multiplication NOUN iajs-2429 345 18 and and CCONJ iajs-2429 345 19 semisecond semisecond ADJ iajs-2429 345 20 and and CCONJ iajs-2429 345 21 hence hence ADV iajs-2429 345 22 it it PRON iajs-2429 345 23 is be AUX iajs-2429 345 24 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 345 25 but but CCONJ iajs-2429 345 26 not not PART iajs-2429 345 27 weakly weakly ADV iajs-2429 345 28 second second ADJ iajs-2429 345 29 and and CCONJ iajs-2429 345 30 hence hence ADV iajs-2429 345 31 not not PART iajs-2429 345 32 𝑆weakly 𝑆weakly PROPN iajs-2429 345 33 second second NOUN iajs-2429 345 34 . . PUNCT iajs-2429 346 1 ( ( PUNCT iajs-2429 346 2 8) 8) NUM iajs-2429 346 3 as as ADP iajs-2429 346 4 another another DET iajs-2429 346 5 example example NOUN iajs-2429 346 6 of of ADP iajs-2429 346 7 ( ( PUNCT iajs-2429 346 8 6 6 NUM iajs-2429 346 9 ) ) PUNCT iajs-2429 346 10 , , PUNCT iajs-2429 346 11 consider consider VERB iajs-2429 346 12 𝑀 𝑀 PROPN iajs-2429 346 13 ℚ ℚ PROPN iajs-2429 346 14 ⊕ ⊕ PROPN iajs-2429 346 15 ℤ ℤ PROPN iajs-2429 346 16 as as ADP iajs-2429 346 17 ℤ-module ℤ-module PROPN iajs-2429 346 18 . . PUNCT iajs-2429 347 1 then then ADV iajs-2429 347 2 𝑆 𝑆 PROPN iajs-2429 347 3 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 347 4 ℚ ℚ PROPN iajs-2429 347 5 ⊕ ⊕ PROPN iajs-2429 347 6 ℤ ℤ PROPN iajs-2429 347 7 ≅ ≅ PROPN iajs-2429 347 8 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 347 9 ℚ ℚ PROPN iajs-2429 347 10 𝐻𝑜𝑚ℤ 𝐻𝑜𝑚ℤ PROPN iajs-2429 347 11 ℤ ℤ PROPN iajs-2429 347 12 , , PUNCT iajs-2429 347 13 ℚ ℚ PROPN iajs-2429 347 14 𝐻𝑜𝑚ℤ 𝐻𝑜𝑚ℤ PROPN iajs-2429 347 15 ℚ ℚ PROPN iajs-2429 347 16 , , PUNCT iajs-2429 347 17 ℤ ℤ PROPN iajs-2429 347 18 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 347 19 ℤ ℤ PROPN iajs-2429 347 20 ℚ ℚ PROPN iajs-2429 347 21 0 0 NUM iajs-2429 347 22 0 0 NUM iajs-2429 348 1 ℤ ℤ NOUN iajs-2429 348 2 is be AUX iajs-2429 348 3 a a DET iajs-2429 348 4 commutative commutative ADJ iajs-2429 348 5 von von PROPN iajs-2429 348 6 neumann neumann PROPN iajs-2429 348 7 regular regular ADJ iajs-2429 348 8 ring ring NOUN iajs-2429 348 9 and and CCONJ iajs-2429 348 10 hence hence ADV iajs-2429 348 11 𝑆 𝑆 PROPN iajs-2429 348 12 is be AUX iajs-2429 348 13 semisecond semisecond ADJ iajs-2429 348 14 so so ADV iajs-2429 348 15 by by ADP iajs-2429 348 16 proposition proposition NOUN iajs-2429 348 17 4.10 4.10 NUM iajs-2429 348 18 , , PUNCT iajs-2429 348 19 ℚ ℚ PROPN iajs-2429 348 20 ⊕ ⊕ PROPN iajs-2429 348 21 ℤ ℤ PROPN iajs-2429 348 22 is be AUX iajs-2429 348 23 semisecond semisecond ADJ iajs-2429 348 24 as as ADP iajs-2429 348 25 𝑆-module 𝑆-module PROPN iajs-2429 348 26 ; ; PUNCT iajs-2429 348 27 that that PRON iajs-2429 348 28 is is ADV iajs-2429 348 29 , , PUNCT iajs-2429 348 30 ℚ ℚ PROPN iajs-2429 348 31 ⊕ ⊕ PROPN iajs-2429 348 32 ℤ ℤ PROPN iajs-2429 348 33 is be AUX iajs-2429 348 34 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 348 35 as as ADP iajs-2429 348 36 ℤ-module ℤ-module PROPN iajs-2429 348 37 . . PUNCT iajs-2429 349 1 but but CCONJ iajs-2429 349 2 ℚ ℚ PROPN iajs-2429 349 3 ⊕ ⊕ PROPN iajs-2429 349 4 ℤ ℤ PROPN iajs-2429 349 5 is be AUX iajs-2429 349 6 not not PART iajs-2429 349 7 𝑆-weakly 𝑆-weakly ADV iajs-2429 349 8 second second ADJ iajs-2429 349 9 as as ADP iajs-2429 349 10 ℤ-module ℤ-module PROPN iajs-2429 349 11 since since SCONJ iajs-2429 349 12 if if SCONJ iajs-2429 349 13 we we PRON iajs-2429 349 14 take take VERB iajs-2429 349 15 𝑓 𝑓 PRON iajs-2429 349 16 1 1 NUM iajs-2429 349 17 0 0 NUM iajs-2429 349 18 0 0 NUM iajs-2429 349 19 0 0 NUM iajs-2429 349 20 , , PUNCT iajs-2429 349 21 𝑔 𝑔 PROPN iajs-2429 349 22 0 0 NUM iajs-2429 349 23 0 0 NUM iajs-2429 349 24 0 0 NUM iajs-2429 349 25 1 1 NUM iajs-2429 350 1 then then ADV iajs-2429 350 2 0 0 NUM iajs-2429 350 3 ⊕ ⊕ PROPN iajs-2429 350 4 ℤ ℤ PROPN iajs-2429 350 5 𝑔 𝑔 PROPN iajs-2429 350 6 𝑀 𝑀 PROPN iajs-2429 350 7 𝑓𝑔 𝑓𝑔 VERB iajs-2429 350 8 𝑀 𝑀 PROPN iajs-2429 350 9 1 1 NUM iajs-2429 350 10 0 0 NUM iajs-2429 350 11 0 0 NUM iajs-2429 350 12 0 0 NUM iajs-2429 350 13 0 0 NUM iajs-2429 350 14 0 0 NUM iajs-2429 350 15 0 0 NUM iajs-2429 350 16 1 1 NUM iajs-2429 350 17 𝑥 𝑥 PROPN iajs-2429 350 18 𝑦 𝑦 NUM iajs-2429 350 19   NOUN iajs-2429 350 20 𝑥 𝑥 PRON iajs-2429 350 21 ∈ ∈ PROPN iajs-2429 350 22 ℚ ℚ PROPN iajs-2429 350 23 , , PUNCT iajs-2429 350 24 𝑦 𝑦 NOUN iajs-2429 350 25 ∈ ∈ NOUN iajs-2429 350 26 ℤ ℤ NOUN iajs-2429 350 27 0 0 NUM iajs-2429 350 28 0 0 NUM iajs-2429 350 29 0 0 NUM iajs-2429 350 30 0 0 NUM iajs-2429 351 1 𝑓 𝑓 DET iajs-2429 351 2 𝑀 𝑀 PROPN iajs-2429 351 3 ℚ ℚ PROPN iajs-2429 351 4 ⊕ ⊕ PROPN iajs-2429 351 5 0 0 PUNCT iajs-2429 352 1 ( ( PUNCT iajs-2429 352 2 9 9 X iajs-2429 352 3 ) ) PUNCT iajs-2429 352 4 we we PRON iajs-2429 352 5 have have VERB iajs-2429 352 6 the the DET iajs-2429 352 7 implication implication NOUN iajs-2429 352 8 coquasi coquasi NOUN iajs-2429 352 9 - - PUNCT iajs-2429 352 10 dedekind dedekind NOUN iajs-2429 352 11 modules module NOUN iajs-2429 352 12   PUNCT iajs-2429 353 1 𝑆-second 𝑆-second NOUN iajs-2429 353 2 modules module NOUN iajs-2429 353 3   PUNCT iajs-2429 354 1 𝑆-weakly 𝑆-weakly ADJ iajs-2429 354 2 second second ADJ iajs-2429 354 3 modules module NOUN iajs-2429 354 4   PUNCT iajs-2429 355 1 𝑆-semisecond 𝑆-semisecond NOUN iajs-2429 355 2 modules module NOUN iajs-2429 355 3 . . PUNCT iajs-2429 356 1 proposition proposition NOUN iajs-2429 356 2 ( ( PUNCT iajs-2429 356 3 5.6 5.6 NUM iajs-2429 356 4 ): ): PUNCT iajs-2429 356 5 every every DET iajs-2429 356 6 semisecond semisecond ADJ iajs-2429 356 7 scalar scalar ADJ iajs-2429 356 8 module module NOUN iajs-2429 356 9 is be AUX iajs-2429 356 10 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 356 11 . . PUNCT iajs-2429 356 12 proof proof NOUN iajs-2429 356 13 . . PUNCT iajs-2429 357 1 let let VERB iajs-2429 357 2 𝑀 𝑀 PRON iajs-2429 357 3 be be AUX iajs-2429 357 4 a a DET iajs-2429 357 5 semisecond semisecond ADJ iajs-2429 357 6 scalar scalar NOUN iajs-2429 357 7 𝑅-module 𝑅-module PROPN iajs-2429 357 8 and and CCONJ iajs-2429 357 9 𝑓 𝑓 DET iajs-2429 357 10 ∈ ∈ PROPN iajs-2429 357 11 𝑆 𝑆 PROPN iajs-2429 357 12 with with ADP iajs-2429 357 13 𝑓 𝑓 DET iajs-2429 357 14 𝑀 𝑀 PROPN iajs-2429 357 15 ⊆ ⊆ NUM iajs-2429 357 16 𝐾 𝐾 PROPN iajs-2429 357 17 for for ADP iajs-2429 357 18 some some DET iajs-2429 357 19 𝐾 𝐾 PROPN iajs-2429 357 20 a a DET iajs-2429 357 21 submodule submodule NOUN iajs-2429 357 22 of of ADP iajs-2429 357 23 𝑀. 𝑀. PROPN iajs-2429 357 24 since since SCONJ iajs-2429 357 25 𝑀 𝑀 PROPN iajs-2429 357 26 is be AUX iajs-2429 357 27 scalar scalar ADJ iajs-2429 357 28 , , PUNCT iajs-2429 357 29 then then ADV iajs-2429 357 30 there there PRON iajs-2429 357 31 exist exist VERB iajs-2429 357 32 𝑎 𝑎 PRON iajs-2429 357 33 ∈ ∈ PROPN iajs-2429 357 34 𝑅 𝑅 PROPN iajs-2429 357 35 such such ADJ iajs-2429 357 36 that that SCONJ iajs-2429 357 37 𝑓 𝑓 DET iajs-2429 357 38 𝑚 𝑚 NOUN iajs-2429 357 39 𝑚𝑎 𝑚𝑎 ADV iajs-2429 357 40 for for ADP iajs-2429 357 41 all all DET iajs-2429 357 42 𝑚 𝑚 ADP iajs-2429 357 43 ∈ ∈ PROPN iajs-2429 357 44 𝑀. 𝑀. PROPN iajs-2429 357 45 then then ADV iajs-2429 358 1 𝑓 𝑓 PRON iajs-2429 358 2 𝑀 𝑀 PROPN iajs-2429 358 3 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 358 4 implies imply VERB iajs-2429 358 5 𝑀𝑎 𝑀𝑎 PROPN iajs-2429 358 6 ⊆ ⊆ NUM iajs-2429 358 7 𝐾 𝐾 PROPN iajs-2429 358 8 and and CCONJ iajs-2429 358 9 hence hence ADV iajs-2429 358 10 𝑓 𝑓 PRON iajs-2429 358 11 𝑀 𝑀 PROPN iajs-2429 358 12 ⊆ ⊆ NUM iajs-2429 358 13 𝐾 𝐾 PROPN iajs-2429 358 14 as as SCONJ iajs-2429 358 15 desired desire VERB iajs-2429 358 16 . . PUNCT iajs-2429 359 1 theorem theorem NOUN iajs-2429 359 2 ( ( PUNCT iajs-2429 359 3 5.7 5.7 NUM iajs-2429 359 4 ): ): PUNCT iajs-2429 359 5 let let VERB iajs-2429 359 6 0 0 NUM iajs-2429 359 7 𝑀 𝑀 PROPN iajs-2429 359 8 be be VERB iajs-2429 359 9 an an DET iajs-2429 359 10 𝑅-module 𝑅-module PROPN iajs-2429 359 11 such such ADJ iajs-2429 359 12 that that SCONJ iajs-2429 359 13 𝑆 𝑆 PROPN iajs-2429 359 14 is be AUX iajs-2429 359 15 commutative commutative ADJ iajs-2429 359 16 . . PUNCT iajs-2429 360 1 if if SCONJ iajs-2429 360 2 𝑀 𝑀 PROPN iajs-2429 360 3 is be AUX iajs-2429 360 4 a a DET iajs-2429 360 5 regular regular ADJ iajs-2429 360 6 𝑆module 𝑆module PROPN iajs-2429 360 7 then then ADV iajs-2429 360 8 𝑀 𝑀 PROPN iajs-2429 360 9 is be AUX iajs-2429 360 10 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 360 11 . . PUNCT iajs-2429 360 12 proof proof NOUN iajs-2429 360 13 . . PUNCT iajs-2429 361 1 similarly similarly ADV iajs-2429 361 2 proof proof NOUN iajs-2429 361 3 of of ADP iajs-2429 361 4 theorem theorem ADJ iajs-2429 361 5 4.1 4.1 NUM iajs-2429 361 6 . . PUNCT iajs-2429 362 1 corollary corollary ADJ iajs-2429 362 2 ( ( PUNCT iajs-2429 362 3 5.8 5.8 NUM iajs-2429 362 4 ): ): PUNCT iajs-2429 362 5 every every DET iajs-2429 362 6 rickart rickart NOUN iajs-2429 362 7 and and CCONJ iajs-2429 362 8 dual dual ADJ iajs-2429 362 9 rickart rickart NOUN iajs-2429 362 10 module module NOUN iajs-2429 362 11 has have VERB iajs-2429 362 12 a a DET iajs-2429 362 13 commutative commutative ADJ iajs-2429 362 14 endomorphism endomorphism NOUN iajs-2429 362 15 ring ring NOUN iajs-2429 362 16 is be AUX iajs-2429 362 17 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 362 18 . . PUNCT iajs-2429 362 19     SPACE iajs-2429 362 20 92 92 NUM iajs-2429 362 21     SPACE iajs-2429 362 22 ibn ibn PROPN iajs-2429 362 23 al al PROPN iajs-2429 362 24 - - PUNCT iajs-2429 362 25 haitham haitham PROPN iajs-2429 362 26 jour jour X iajs-2429 362 27 . . PROPN iajs-2429 363 1 for for ADP iajs-2429 363 2 pure pure ADJ iajs-2429 363 3 & & CCONJ iajs-2429 363 4 appl appl PROPN iajs-2429 363 5 . . PUNCT iajs-2429 364 1 sci sci PROPN iajs-2429 364 2 . . PROPN iajs-2429 365 1 33 33 NUM iajs-2429 365 2 ( ( PUNCT iajs-2429 365 3 2 2 NUM iajs-2429 365 4 ) ) PUNCT iajs-2429 365 5 2020 2020 NUM iajs-2429 365 6 proof proof NOUN iajs-2429 365 7 . . PUNCT iajs-2429 366 1 by by ADP iajs-2429 366 2 [ [ X iajs-2429 366 3 16 16 NUM iajs-2429 366 4 ] ] PUNCT iajs-2429 366 5 . . PUNCT iajs-2429 367 1 the the DET iajs-2429 367 2 endomorphism endomorphism PROPN iajs-2429 367 3 ring ring NOUN iajs-2429 367 4 of of ADP iajs-2429 367 5 rickart rickart NOUN iajs-2429 367 6 and and CCONJ iajs-2429 367 7 dual dual ADJ iajs-2429 367 8 rickart rickart NOUN iajs-2429 367 9 modules module NOUN iajs-2429 367 10 is be AUX iajs-2429 367 11 von von PROPN iajs-2429 367 12 neumann neumann PROPN iajs-2429 367 13 regular regular PROPN iajs-2429 367 14 so so ADV iajs-2429 367 15 by by ADP iajs-2429 367 16 theorem theorem NOUN iajs-2429 367 17 5.7 5.7 NUM iajs-2429 367 18 , , PUNCT iajs-2429 367 19 the the DET iajs-2429 367 20 result result NOUN iajs-2429 367 21 is be AUX iajs-2429 367 22 obtained obtain VERB iajs-2429 367 23 . . PUNCT iajs-2429 368 1 remark remark NOUN iajs-2429 368 2 ( ( PUNCT iajs-2429 368 3 5.9 5.9 NUM iajs-2429 368 4 ): ): PUNCT iajs-2429 368 5 the the DET iajs-2429 368 6 commutativity commutativity NOUN iajs-2429 368 7 condition condition NOUN iajs-2429 368 8 in in ADP iajs-2429 368 9 theorem theorem ADJ iajs-2429 368 10 5.7 5.7 NUM iajs-2429 368 11 or or CCONJ iajs-2429 368 12 corollary corollary ADJ iajs-2429 368 13 5.8 5.8 NUM iajs-2429 368 14 can can AUX iajs-2429 368 15 not not PART iajs-2429 368 16 ( ( PUNCT iajs-2429 368 17 can can AUX iajs-2429 368 18 not not PART iajs-2429 368 19 ) ) PUNCT iajs-2429 368 20 ( ( PUNCT iajs-2429 368 21 be be AUX iajs-2429 368 22 ) ) PUNCT iajs-2429 368 23 dropped drop VERB iajs-2429 368 24 as as SCONJ iajs-2429 368 25 follows follow VERB iajs-2429 368 26 , , PUNCT iajs-2429 368 27 𝑀 𝑀 PROPN iajs-2429 368 28 ℤ ℤ PROPN iajs-2429 368 29 ⊕ ⊕ PROPN iajs-2429 368 30 ℤ ℤ PROPN iajs-2429 368 31 as as SCONJ iajs-2429 368 32 ℤ-module ℤ-module PROPN iajs-2429 368 33 is be AUX iajs-2429 368 34 rickart rickart NOUN iajs-2429 368 35 and and CCONJ iajs-2429 368 36 dual dual ADJ iajs-2429 368 37 rickart rickart NOUN iajs-2429 368 38 and and CCONJ iajs-2429 369 1 hence hence ADV iajs-2429 369 2 𝑆 𝑆 PROPN iajs-2429 369 3 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 369 4 ℤ ℤ PROPN iajs-2429 369 5 ⊕ ⊕ PROPN iajs-2429 369 6 ℤ ℤ NOUN iajs-2429 369 7 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 369 8 ℤ ℤ PROPN iajs-2429 369 9 𝐻𝑜𝑚ℤ 𝐻𝑜𝑚ℤ NOUN iajs-2429 369 10 ℤ ℤ PROPN iajs-2429 369 11 , , PUNCT iajs-2429 369 12 ℤ ℤ PROPN iajs-2429 369 13 𝐻𝑜𝑚ℤ 𝐻𝑜𝑚ℤ NOUN iajs-2429 369 14 ℤ ℤ PROPN iajs-2429 369 15 , , PUNCT iajs-2429 369 16 ℤ ℤ PROPN iajs-2429 369 17 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 369 18 ℤ ℤ PROPN iajs-2429 369 19 ≅ ≅ PROPN iajs-2429 369 20 𝑀𝑎𝑡 𝑀𝑎𝑡 PROPN iajs-2429 369 21 ℤ ℤ PROPN iajs-2429 369 22 ℤ ℤ NOUN iajs-2429 369 23 ℤ ℤ NOUN iajs-2429 369 24 ℤ ℤ NOUN iajs-2429 369 25 ℤ ℤ PROPN iajs-2429 369 26 is be AUX iajs-2429 369 27 von von PROPN iajs-2429 369 28 neumann neumann PROPN iajs-2429 369 29 regular regular PROPN iajs-2429 369 30 , , PUNCT iajs-2429 369 31 but but CCONJ iajs-2429 369 32 𝑀𝑎𝑡 𝑀𝑎𝑡 VERB iajs-2429 369 33 ℤ ℤ PROPN iajs-2429 369 34 not not PART iajs-2429 369 35 commutative commutative ADJ iajs-2429 369 36 ring ring NOUN iajs-2429 369 37 . . PUNCT iajs-2429 370 1 on on ADP iajs-2429 370 2 the the DET iajs-2429 370 3 other other ADJ iajs-2429 370 4 hand hand NOUN iajs-2429 370 5 , , PUNCT iajs-2429 370 6 ℤ ℤ PROPN iajs-2429 370 7 ⊕ ⊕ PROPN iajs-2429 370 8 ℤ ℤ PROPN iajs-2429 370 9 ≅ ≅ PROPN iajs-2429 370 10 , , PUNCT iajs-2429 370 11 , , PUNCT iajs-2429 370 12 , , PUNCT iajs-2429 370 13 , , PUNCT iajs-2429 370 14 so so CCONJ iajs-2429 370 15 if if SCONJ iajs-2429 370 16 we we PRON iajs-2429 370 17 take take VERB iajs-2429 370 18 𝑓 𝑓 PRON iajs-2429 370 19 0 0 NUM iajs-2429 370 20 1 1 NUM iajs-2429 370 21 0 0 NUM iajs-2429 370 22 0 0 NUM iajs-2429 370 23 ∈ ∈ PROPN iajs-2429 370 24 𝑆 𝑆 PROPN iajs-2429 370 25 𝐸𝑛𝑑ℤ 𝐸𝑛𝑑ℤ PROPN iajs-2429 370 26 ℤ ℤ PROPN iajs-2429 370 27 ⊕ ⊕ PROPN iajs-2429 370 28 ℤ ℤ PROPN iajs-2429 370 29 implies imply VERB iajs-2429 370 30 𝑓 𝑓 DET iajs-2429 370 31 𝑀 𝑀 PROPN iajs-2429 370 32 0 0 NUM iajs-2429 370 33 1 1 NUM iajs-2429 370 34 0 0 NUM iajs-2429 370 35 0 0 NUM iajs-2429 370 36 , , PUNCT iajs-2429 370 37 𝑥 𝑥 X iajs-2429 370 38 , , PUNCT iajs-2429 370 39 𝑦 𝑦 NOUN iajs-2429 370 40 ∈ ∈ NOUN iajs-2429 370 41 ℤ ℤ PROPN iajs-2429 370 42 , , PUNCT iajs-2429 370 43 𝑓 𝑓 DET iajs-2429 370 44 𝑀 𝑀 PROPN iajs-2429 370 45 0 0 NUM iajs-2429 370 46 0 0 NUM iajs-2429 370 47 0 0 NUM iajs-2429 370 48 0 0 NUM iajs-2429 370 49 , , PUNCT iajs-2429 370 50 𝑥 𝑥 X iajs-2429 370 51 , , PUNCT iajs-2429 370 52 𝑦 𝑦 NOUN iajs-2429 370 53 ∈ ∈ NOUN iajs-2429 371 1 ℤ ℤ NOUN iajs-2429 371 2 it it PRON iajs-2429 371 3 follows follow VERB iajs-2429 371 4 that that SCONJ iajs-2429 371 5 𝑀 𝑀 PROPN iajs-2429 371 6 is be AUX iajs-2429 371 7 not not PART iajs-2429 371 8 semisecond semisecond ADJ iajs-2429 371 9 as as ADP iajs-2429 371 10 𝑆-module 𝑆-module PROPN iajs-2429 371 11 that that PRON iajs-2429 371 12 is be AUX iajs-2429 371 13 , , PUNCT iajs-2429 371 14 𝑀 𝑀 PROPN iajs-2429 371 15 is be AUX iajs-2429 371 16 not not PART iajs-2429 371 17 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 371 18 as as ADP iajs-2429 371 19 ℤ-module ℤ-module PROPN iajs-2429 371 20 . . PUNCT iajs-2429 371 21 proposition proposition NOUN iajs-2429 371 22 ( ( PUNCT iajs-2429 371 23 5.10 5.10 NUM iajs-2429 371 24 ): ): PUNCT iajs-2429 371 25 every every DET iajs-2429 371 26 non non ADJ iajs-2429 371 27 - - ADJ iajs-2429 371 28 zero zero ADJ iajs-2429 371 29 direct direct ADJ iajs-2429 371 30 summand summand NOUN iajs-2429 371 31 of of ADP iajs-2429 371 32 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 371 33 module module NOUN iajs-2429 371 34 is be AUX iajs-2429 371 35 𝑆semisecond 𝑆semisecond PROPN iajs-2429 371 36 . . PUNCT iajs-2429 371 37 proof proof NOUN iajs-2429 371 38 . . PUNCT iajs-2429 372 1 let let VERB iajs-2429 372 2 𝑁 𝑁 PROPN iajs-2429 372 3 be be AUX iajs-2429 372 4 a a DET iajs-2429 372 5 direct direct ADJ iajs-2429 372 6 summand summand NOUN iajs-2429 372 7 of of ADP iajs-2429 372 8 an an DET iajs-2429 372 9 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 372 10 𝑅-module 𝑅-module PROPN iajs-2429 372 11 𝑀 𝑀 PROPN iajs-2429 372 12 then then ADV iajs-2429 372 13 𝑀 𝑀 PROPN iajs-2429 372 14 𝑁 𝑁 PROPN iajs-2429 372 15 ⊕ ⊕ PROPN iajs-2429 372 16 𝐻 𝐻 NOUN iajs-2429 372 17 for for ADP iajs-2429 372 18 some some DET iajs-2429 372 19 submodule submodule NOUN iajs-2429 372 20 𝐻 𝐻 PROPN iajs-2429 372 21 of of ADP iajs-2429 372 22 𝑀. 𝑀. PROPN iajs-2429 372 23 let let VERB iajs-2429 372 24 𝑓 𝑓 DET iajs-2429 372 25 ∈ ∈ NOUN iajs-2429 372 26 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-2429 372 27 𝑁 𝑁 PROPN iajs-2429 372 28 with with ADP iajs-2429 372 29 𝑓 𝑓 DET iajs-2429 372 30 𝑁 𝑁 PROPN iajs-2429 372 31 ⊆ ⊆ NUM iajs-2429 372 32 𝐾 𝐾 PROPN iajs-2429 372 33 for for ADP iajs-2429 372 34 some some DET iajs-2429 372 35 𝐾 𝐾 PROPN iajs-2429 372 36 a a DET iajs-2429 372 37 submodule submodule NOUN iajs-2429 372 38 of of ADP iajs-2429 372 39 𝑁. 𝑁. PROPN iajs-2429 372 40 we we PRON iajs-2429 372 41 can can AUX iajs-2429 372 42 define define VERB iajs-2429 372 43 𝛼 𝛼 DET iajs-2429 372 44 𝑛 𝑛 ADP iajs-2429 372 45 ℎ ℎ X iajs-2429 372 46 𝑓 𝑓 ADP iajs-2429 372 47 𝑛 𝑛 ADJ iajs-2429 372 48 where where SCONJ iajs-2429 372 49 𝑛 𝑛 DET iajs-2429 372 50 ∈ ∈ PROPN iajs-2429 372 51 𝑁 𝑁 PROPN iajs-2429 372 52 and and CCONJ iajs-2429 372 53 ℎ ℎ ADP iajs-2429 372 54 ∈ ∈ PROPN iajs-2429 372 55 𝐻. 𝐻. PROPN iajs-2429 372 56 it it PRON iajs-2429 372 57 is be AUX iajs-2429 372 58 easy easy ADJ iajs-2429 372 59 to to PART iajs-2429 372 60 see see VERB iajs-2429 372 61 that that SCONJ iajs-2429 372 62 𝛼 𝛼 PROPN iajs-2429 372 63 ∈ ∈ PROPN iajs-2429 372 64 𝑆 𝑆 PROPN iajs-2429 372 65 , , PUNCT iajs-2429 372 66 𝛼 𝛼 VERB iajs-2429 372 67 𝑀 𝑀 PROPN iajs-2429 372 68 𝑓 𝑓 PRON iajs-2429 372 69 𝑁 𝑁 PROPN iajs-2429 372 70 implies imply VERB iajs-2429 372 71 𝛼 𝛼 PRON iajs-2429 372 72 𝑀 𝑀 PROPN iajs-2429 372 73 𝑓 𝑓 DET iajs-2429 372 74 𝑁 𝑁 PROPN iajs-2429 372 75 ⊆ ⊆ NUM iajs-2429 372 76 𝐾. 𝐾. PROPN iajs-2429 372 77 it it PRON iajs-2429 372 78 follows follow VERB iajs-2429 372 79 𝛼 𝛼 PRON iajs-2429 372 80 𝑀 𝑀 PROPN iajs-2429 372 81 ⊆ ⊆ PROPN iajs-2429 372 82 𝐾 𝐾 PROPN iajs-2429 372 83 implies imply VERB iajs-2429 372 84 𝛼 𝛼 PROPN iajs-2429 372 85 𝑀 𝑀 PROPN iajs-2429 372 86 ⊆ ⊆ NUM iajs-2429 372 87 𝐾 𝐾 PROPN iajs-2429 372 88 and and CCONJ iajs-2429 372 89 hence hence ADV iajs-2429 372 90 𝑓 𝑓 DET iajs-2429 372 91 𝑁 𝑁 PROPN iajs-2429 372 92 ⊆ ⊆ NUM iajs-2429 372 93 𝐾 𝐾 PROPN iajs-2429 372 94 as as SCONJ iajs-2429 372 95 desired desire VERB iajs-2429 372 96 . . PUNCT iajs-2429 373 1 theorem theorem NOUN iajs-2429 373 2 ( ( PUNCT iajs-2429 373 3 5.11 5.11 NUM iajs-2429 373 4 ): ): PUNCT iajs-2429 373 5 the the DET iajs-2429 373 6 following following ADJ iajs-2429 373 7 statements statement NOUN iajs-2429 373 8 are be AUX iajs-2429 373 9     SPACE iajs-2429 373 10 equivalent equivalent ADJ iajs-2429 373 11 ( ( PUNCT iajs-2429 373 12 1 1 NUM iajs-2429 373 13 ) ) PUNCT iajs-2429 373 14 𝑀 𝑀 PROPN iajs-2429 373 15 is be AUX iajs-2429 373 16 a a DET iajs-2429 373 17 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 373 18 𝑅-module 𝑅-module PROPN iajs-2429 373 19 . . PUNCT iajs-2429 374 1 ( ( PUNCT iajs-2429 374 2 2 2 X iajs-2429 374 3 ) ) PUNCT iajs-2429 374 4 𝑀 𝑀 PROPN iajs-2429 374 5 0 0 PUNCT iajs-2429 374 6 and and CCONJ iajs-2429 374 7 𝐾 𝐾 PROPN iajs-2429 374 8 : : PUNCT iajs-2429 374 9 𝑀 𝑀 PROPN iajs-2429 374 10 is be AUX iajs-2429 374 11 a a DET iajs-2429 374 12 semiprime semiprime NOUN iajs-2429 374 13 ideal ideal NOUN iajs-2429 374 14 of of ADP iajs-2429 374 15 𝑆 𝑆 PROPN iajs-2429 374 16 for for ADP iajs-2429 374 17 each each DET iajs-2429 374 18 proper proper ADJ iajs-2429 374 19 submodule submodule NOUN iajs-2429 374 20 𝐾 𝐾 PROPN iajs-2429 374 21 of of ADP iajs-2429 374 22 𝑀. 𝑀. PROPN iajs-2429 374 23 proof proof NOUN iajs-2429 374 24 . . PUNCT iajs-2429 375 1 similarly similarly ADV iajs-2429 375 2 , , PUNCT iajs-2429 375 3 proof proof NOUN iajs-2429 375 4 of of ADP iajs-2429 375 5 theorem theorem ADJ iajs-2429 375 6 3.1 3.1 NUM iajs-2429 375 7 . . PUNCT iajs-2429 375 8 corollary corollary NOUN iajs-2429 375 9 ( ( PUNCT iajs-2429 375 10 5.12 5.12 NUM iajs-2429 375 11 ): ): PUNCT iajs-2429 375 12 if if SCONJ iajs-2429 375 13 𝑀 𝑀 PROPN iajs-2429 375 14 is be AUX iajs-2429 375 15 an an DET iajs-2429 375 16 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 375 17 𝑅-module 𝑅-module PROPN iajs-2429 375 18 𝑀 𝑀 PROPN iajs-2429 375 19 then then ADV iajs-2429 375 20 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 375 21 𝑀 𝑀 PROPN iajs-2429 375 22 𝑓 𝑓 PRON iajs-2429 375 23 ∈ ∈ PROPN iajs-2429 375 24 𝑆 𝑆 PROPN iajs-2429 375 25 : : PUNCT iajs-2429 375 26 𝑓 𝑓 PROPN iajs-2429 375 27 𝑀 𝑀 PROPN iajs-2429 375 28 0 0 NUM iajs-2429 375 29 is be AUX iajs-2429 375 30 a a DET iajs-2429 375 31 semiprime semiprime NOUN iajs-2429 375 32 ideal ideal NOUN iajs-2429 375 33 of of ADP iajs-2429 375 34 𝑆. 𝑆. PROPN iajs-2429 375 35 proof proof NOUN iajs-2429 375 36 . . PUNCT iajs-2429 376 1 directly directly ADV iajs-2429 376 2 by by ADP iajs-2429 376 3 theorem theorem NOUN iajs-2429 376 4 5.11 5.11 NUM iajs-2429 376 5 . . PUNCT iajs-2429 376 6 examples example NOUN iajs-2429 376 7 ( ( PUNCT iajs-2429 376 8 5.13 5.13 NUM iajs-2429 376 9 ): ): PUNCT iajs-2429 376 10 the the DET iajs-2429 376 11 opposite opposite ADJ iajs-2429 376 12 result result NOUN iajs-2429 376 13 is be AUX iajs-2429 376 14 not not PART iajs-2429 376 15 held hold VERB iajs-2429 376 16 in in ADP iajs-2429 376 17 general general ADJ iajs-2429 376 18 for for ADP iajs-2429 376 19 example example NOUN iajs-2429 376 20 ℤ ℤ PROPN iajs-2429 376 21 is be AUX iajs-2429 376 22 not not PART iajs-2429 376 23 semisecond semisecond ADJ iajs-2429 376 24 and and CCONJ iajs-2429 377 1 hence hence ADV iajs-2429 377 2 not not PART iajs-2429 377 3 𝑆-semisecond 𝑆-semisecond NOUN iajs-2429 377 4 while while SCONJ iajs-2429 377 5 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 377 6 ℤ ℤ PROPN iajs-2429 377 7 0 0 NUM iajs-2429 377 8 is be AUX iajs-2429 377 9 a a DET iajs-2429 377 10 semiprime semiprime NOUN iajs-2429 377 11 ideal ideal NOUN iajs-2429 377 12 of of ADP iajs-2429 377 13 𝑆. 𝑆. PROPN iajs-2429 377 14 corollary corollary NOUN iajs-2429 377 15 ( ( PUNCT iajs-2429 377 16 5.14 5.14 NUM iajs-2429 377 17 ): ): PUNCT iajs-2429 378 1 if if SCONJ iajs-2429 378 2 𝑀 𝑀 PROPN iajs-2429 378 3 is be AUX iajs-2429 378 4 an an DET iajs-2429 378 5 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 378 6 𝑅-module 𝑅-module PROPN iajs-2429 378 7 then then ADV iajs-2429 378 8 for for ADP iajs-2429 378 9 every every DET iajs-2429 378 10 proper proper ADJ iajs-2429 378 11 submodule submodule NOUN iajs-2429 378 12 𝐾 𝐾 PROPN iajs-2429 378 13 of of ADP iajs-2429 378 14 𝑀 𝑀 PROPN iajs-2429 378 15 we we PRON iajs-2429 378 16 have have VERB iajs-2429 378 17 𝐾 𝐾 NOUN iajs-2429 378 18 : : PUNCT iajs-2429 378 19 𝑀 𝑀 PROPN iajs-2429 378 20 𝐾 𝐾 PROPN iajs-2429 378 21 : : PUNCT iajs-2429 378 22 𝑔 𝑔 PROPN iajs-2429 378 23 𝑀 𝑀 PROPN iajs-2429 378 24 for for ADP iajs-2429 378 25 each each DET iajs-2429 378 26 𝑔 𝑔 PROPN iajs-2429 378 27 ∈ ∈ PROPN iajs-2429 378 28 𝑆. 𝑆. NOUN iajs-2429 378 29 proof proof NOUN iajs-2429 378 30 . . PUNCT iajs-2429 379 1 similarly similarly ADV iajs-2429 379 2 , , PUNCT iajs-2429 379 3 proof proof NOUN iajs-2429 379 4 of of ADP iajs-2429 379 5 corollary corollary ADJ iajs-2429 379 6 3.5 3.5 NUM iajs-2429 379 7 . . PUNCT iajs-2429 380 1 corollary corollary NOUN iajs-2429 380 2 ( ( PUNCT iajs-2429 380 3 5.15 5.15 NUM iajs-2429 380 4 ): ): PUNCT iajs-2429 380 5 if if SCONJ iajs-2429 380 6 𝑀 𝑀 PROPN iajs-2429 380 7 is be AUX iajs-2429 380 8 an an DET iajs-2429 380 9 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 380 10 𝑅-module 𝑅-module PROPN iajs-2429 380 11 then then ADV iajs-2429 380 12 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2429 380 13 𝑀 𝑀 PROPN iajs-2429 380 14 𝑎𝑛𝑛 𝑎𝑛𝑛 ADV iajs-2429 380 15 𝑔𝑀 𝑔𝑀 VERB iajs-2429 380 16 for for ADP iajs-2429 380 17 each each DET iajs-2429 380 18 𝑔 𝑔 PROPN iajs-2429 380 19 ∈ ∈ PROPN iajs-2429 380 20 𝑆. 𝑆. NOUN iajs-2429 380 21 proof proof NOUN iajs-2429 380 22 . . PUNCT iajs-2429 381 1 directly directly ADV iajs-2429 381 2 by by ADP iajs-2429 381 3 corollary corollary ADJ iajs-2429 381 4 5.14 5.14 NUM iajs-2429 381 5 . . PUNCT iajs-2429 382 1 theorem theorem NOUN iajs-2429 382 2 ( ( PUNCT iajs-2429 382 3 5.16 5.16 NUM iajs-2429 382 4 ): ): PUNCT iajs-2429 382 5 the the DET iajs-2429 382 6 following following ADJ iajs-2429 382 7 statements statement NOUN iajs-2429 382 8 are be AUX iajs-2429 382 9 equivalent equivalent ADJ iajs-2429 382 10 ( ( PUNCT iajs-2429 382 11 1 1 NUM iajs-2429 382 12 ) ) PUNCT iajs-2429 382 13 𝑀 𝑀 PROPN iajs-2429 382 14 is be AUX iajs-2429 382 15 an an DET iajs-2429 382 16 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 382 17 𝑅-module 𝑅-module PROPN iajs-2429 382 18 . . PUNCT iajs-2429 383 1 ( ( PUNCT iajs-2429 383 2 2 2 X iajs-2429 383 3 ) ) PUNCT iajs-2429 383 4 𝑀 𝑀 PROPN iajs-2429 383 5 0 0 PUNCT iajs-2429 384 1 and and CCONJ iajs-2429 384 2 for for ADP iajs-2429 384 3 each each DET iajs-2429 384 4 ideals ideal NOUN iajs-2429 384 5 𝐼 𝐼 ADP iajs-2429 384 6 of of ADP iajs-2429 384 7 𝑆 𝑆 PROPN iajs-2429 384 8 and and CCONJ iajs-2429 384 9 𝐾 𝐾 PROPN iajs-2429 384 10 a a DET iajs-2429 384 11 submodule submodule NOUN iajs-2429 384 12 of of ADP iajs-2429 384 13 𝑀 𝑀 PROPN iajs-2429 384 14 such such ADJ iajs-2429 384 15 that that SCONJ iajs-2429 384 16 𝐼 𝐼 ADP iajs-2429 384 17 𝑀 𝑀 PROPN iajs-2429 384 18 ⊆ ⊆ NUM iajs-2429 384 19 𝐾 𝐾 PROPN iajs-2429 384 20 implies imply VERB iajs-2429 384 21 𝐼𝑀 𝐼𝑀 PROPN iajs-2429 384 22 ⊆ ⊆ NUM iajs-2429 384 23 𝐾. 𝐾. PROPN iajs-2429 384 24 proof proof NOUN iajs-2429 384 25 . . PUNCT iajs-2429 385 1 similarly similarly ADV iajs-2429 385 2 , , PUNCT iajs-2429 385 3 proof proof NOUN iajs-2429 385 4 of of ADP iajs-2429 385 5 theorem theorem ADJ iajs-2429 385 6 3.7 3.7 NUM iajs-2429 385 7 . . PUNCT iajs-2429 386 1 corollary corollary ADJ iajs-2429 386 2 ( ( PUNCT iajs-2429 386 3 5.17 5.17 NUM iajs-2429 386 4 ): ): PUNCT iajs-2429 386 5 the the DET iajs-2429 386 6 following following ADJ iajs-2429 386 7 statements statement NOUN iajs-2429 386 8 are be AUX iajs-2429 386 9     SPACE iajs-2429 386 10 equivalent equivalent ADJ iajs-2429 386 11 ( ( PUNCT iajs-2429 386 12 1 1 NUM iajs-2429 386 13 ) ) PUNCT iajs-2429 386 14 𝑀 𝑀 PROPN iajs-2429 386 15 is be AUX iajs-2429 386 16 an an DET iajs-2429 386 17 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 386 18 𝑅-module 𝑅-module PROPN iajs-2429 386 19 . . PUNCT iajs-2429 387 1 ( ( PUNCT iajs-2429 387 2 2 2 X iajs-2429 387 3 ) ) PUNCT iajs-2429 387 4 𝑀 𝑀 PROPN iajs-2429 387 5 0 0 PUNCT iajs-2429 388 1 and and CCONJ iajs-2429 388 2 for for ADP iajs-2429 388 3 each each DET iajs-2429 388 4 ideals ideal NOUN iajs-2429 388 5 𝐼 𝐼 ADP iajs-2429 388 6 of of ADP iajs-2429 388 7 𝑆 𝑆 PROPN iajs-2429 388 8 and and CCONJ iajs-2429 388 9 𝐾 𝐾 PROPN iajs-2429 388 10 a a DET iajs-2429 388 11 proper proper ADJ iajs-2429 388 12 submodule submodule NOUN iajs-2429 388 13 of of ADP iajs-2429 388 14 𝑀 𝑀 PROPN iajs-2429 388 15 and and CCONJ iajs-2429 388 16 𝐼 𝐼 PROPN iajs-2429 388 17 ⊆ ⊆ PROPN iajs-2429 388 18 𝐾 𝐾 PROPN iajs-2429 388 19 : : PUNCT iajs-2429 388 20 𝑀 𝑀 PROPN iajs-2429 388 21 implies imply VERB iajs-2429 388 22 𝐼 𝐼 ADP iajs-2429 388 23 ⊆ ⊆ NUM iajs-2429 388 24 𝐾 𝐾 PROPN iajs-2429 388 25 : : PUNCT iajs-2429 388 26 𝑀 𝑀 PROPN iajs-2429 388 27 . . PUNCT iajs-2429 389 1 proof proof NOUN iajs-2429 389 2 . . PUNCT iajs-2429 390 1 directly directly ADV iajs-2429 390 2 via via ADP iajs-2429 390 3 theorem theorem NOUN iajs-2429 390 4 5.16 5.16 NUM iajs-2429 390 5 . . PUNCT iajs-2429 390 6 proposition proposition NOUN iajs-2429 390 7 ( ( PUNCT iajs-2429 390 8 5.18 5.18 NUM iajs-2429 390 9 ): ): PUNCT iajs-2429 390 10 the the DET iajs-2429 390 11 following following ADJ iajs-2429 390 12 statements statement NOUN iajs-2429 390 13 are be AUX iajs-2429 390 14     SPACE iajs-2429 390 15 equivalent equivalent ADJ iajs-2429 390 16     SPACE iajs-2429 390 17 93 93 NUM iajs-2429 390 18     SPACE iajs-2429 390 19 ibn ibn PROPN iajs-2429 390 20 al al PROPN iajs-2429 390 21 - - PUNCT iajs-2429 390 22 haitham haitham PROPN iajs-2429 390 23 jour jour X iajs-2429 390 24 . . PROPN iajs-2429 391 1 for for ADP iajs-2429 391 2 pure pure ADJ iajs-2429 391 3 & & CCONJ iajs-2429 391 4 appl appl PROPN iajs-2429 391 5 . . PUNCT iajs-2429 392 1 sci sci PROPN iajs-2429 392 2 . . PROPN iajs-2429 393 1 33 33 NUM iajs-2429 393 2 ( ( PUNCT iajs-2429 393 3 2 2 NUM iajs-2429 393 4 ) ) PUNCT iajs-2429 393 5 2020 2020 NUM iajs-2429 393 6 ( ( PUNCT iajs-2429 393 7 1 1 X iajs-2429 393 8 ) ) PUNCT iajs-2429 393 9 𝑀 𝑀 PROPN iajs-2429 393 10 is be AUX iajs-2429 393 11 an an DET iajs-2429 393 12 𝑆-semisecond 𝑆-semisecond PROPN iajs-2429 393 13 𝑅-module 𝑅-module PROPN iajs-2429 393 14 . . PUNCT iajs-2429 394 1 ( ( PUNCT iajs-2429 394 2 2 2 X iajs-2429 394 3 ) ) PUNCT iajs-2429 394 4 𝑀 𝑀 PROPN iajs-2429 394 5 0 0 NUM iajs-2429 395 1 and and CCONJ iajs-2429 395 2 for for ADP iajs-2429 395 3 each each DET iajs-2429 395 4 ideal ideal ADJ iajs-2429 395 5 𝐼 𝐼 PROPN iajs-2429 395 6 of of ADP iajs-2429 395 7 𝑆 𝑆 PROPN iajs-2429 395 8 implies imply VERB iajs-2429 395 9 𝐼 𝐼 ADP iajs-2429 395 10 𝑀 𝑀 PROPN iajs-2429 395 11 𝐼𝑀. 𝐼𝑀. VERB iajs-2429 395 12 proof proof ADJ iajs-2429 395 13 . . PUNCT iajs-2429 396 1 by by ADP iajs-2429 396 2 using use VERB iajs-2429 396 3 theorem theorem ADJ iajs-2429 396 4 5.10 5.10 NUM iajs-2429 396 5 and and CCONJ iajs-2429 396 6 theorem theorem VERB iajs-2429 396 7 5.2 5.2 NUM iajs-2429 396 8 . . NOUN iajs-2429 397 1 6 6 NUM iajs-2429 397 2 . . X iajs-2429 397 3 conclusion conclusion NOUN iajs-2429 397 4 in in ADP iajs-2429 397 5 this this DET iajs-2429 397 6 research research NOUN iajs-2429 397 7 we we PRON iajs-2429 397 8 present present VERB iajs-2429 397 9 comprehensive comprehensive ADJ iajs-2429 397 10 study study NOUN iajs-2429 397 11 of of ADP iajs-2429 397 12 semisecond semisecond ADJ iajs-2429 397 13 submodules submodule NOUN iajs-2429 397 14 . . PUNCT iajs-2429 398 1 we we PRON iajs-2429 398 2 show show VERB iajs-2429 398 3 that that SCONJ iajs-2429 398 4 every every DET iajs-2429 398 5 regular regular ADJ iajs-2429 398 6 module module NOUN iajs-2429 398 7 is be AUX iajs-2429 398 8 semisecond semisecond ADJ iajs-2429 398 9 , , PUNCT iajs-2429 398 10 and and CCONJ iajs-2429 398 11 the the DET iajs-2429 398 12 semisecond semisecond ADJ iajs-2429 398 13 and and CCONJ iajs-2429 398 14 regular regular ADJ iajs-2429 398 15 concepts concept NOUN iajs-2429 398 16 in in ADP iajs-2429 398 17 the the DET iajs-2429 398 18 commutative commutative ADJ iajs-2429 398 19 rings ring NOUN iajs-2429 398 20 are be AUX iajs-2429 398 21 the the DET iajs-2429 398 22 same same ADJ iajs-2429 398 23 . . PUNCT iajs-2429 399 1 comprehensive comprehensive ADJ iajs-2429 399 2 study study NOUN iajs-2429 399 3 in in ADP iajs-2429 399 4 this this DET iajs-2429 399 5 type type NOUN iajs-2429 399 6 of of ADP iajs-2429 399 7 modules module NOUN iajs-2429 399 8 is be AUX iajs-2429 399 9 introduced introduce VERB iajs-2429 399 10 and and CCONJ iajs-2429 399 11 numerous numerous ADJ iajs-2429 399 12 examples example NOUN iajs-2429 399 13 and and CCONJ iajs-2429 399 14 basic basic ADJ iajs-2429 399 15 properties property NOUN iajs-2429 399 16 are be AUX iajs-2429 399 17 provided provide VERB iajs-2429 399 18 . . PUNCT iajs-2429 400 1 references reference NOUN iajs-2429 400 2 1 1 NUM iajs-2429 400 3 . . PUNCT iajs-2429 400 4 yassemi yassemi NOUN iajs-2429 400 5 , , PUNCT iajs-2429 400 6 s. s. PROPN iajs-2429 400 7 the the DET iajs-2429 400 8 dual dual ADJ iajs-2429 400 9 notion notion NOUN iajs-2429 400 10 of of ADP iajs-2429 400 11 prime prime ADJ iajs-2429 400 12 submodules submodule NOUN iajs-2429 400 13 . . PUNCT iajs-2429 401 1 arch arch NOUN iajs-2429 401 2 . . PUNCT iajs-2429 402 1 math math NOUN iajs-2429 402 2 . . PUNCT iajs-2429 403 1 ( ( PUNCT iajs-2429 403 2 brno).2001 brno).2001 PROPN iajs-2429 403 3 , , PUNCT iajs-2429 403 4 37 37 NUM iajs-2429 403 5 , , PUNCT iajs-2429 403 6 273 273 NUM iajs-2429 403 7 – – PUNCT iajs-2429 403 8 278 278 NUM iajs-2429 403 9 . . X iajs-2429 404 1 2 2 NUM iajs-2429 404 2 . . X iajs-2429 404 3 ansari ansari ADJ iajs-2429 404 4 - - PUNCT iajs-2429 404 5 toroghy toroghy ADJ iajs-2429 404 6 , , PUNCT iajs-2429 404 7 h. h. PROPN iajs-2429 404 8 ; ; PUNCT iajs-2429 404 9 farshadifar farshadifar PROPN iajs-2429 404 10 , , PUNCT iajs-2429 404 11 f. f. PROPN iajs-2429 404 12 the the DET iajs-2429 404 13 dual dual ADJ iajs-2429 404 14 notions notion NOUN iajs-2429 404 15 of of ADP iajs-2429 404 16 some some DET iajs-2429 404 17 generalizations generalization NOUN iajs-2429 404 18 of of ADP iajs-2429 404 19 prime prime ADJ iajs-2429 404 20 submodules submodule NOUN iajs-2429 404 21 , , PUNCT iajs-2429 404 22 comm comm NOUN iajs-2429 404 23 . . PUNCT iajs-2429 405 1 algebra.2011 algebra.2011 PROPN iajs-2429 405 2 , , PUNCT iajs-2429 405 3 39 39 NUM iajs-2429 405 4 , , PUNCT iajs-2429 405 5 7 7 NUM iajs-2429 405 6 , , PUNCT iajs-2429 405 7 2396 2396 NUM iajs-2429 405 8 - - SYM iajs-2429 405 9 2416 2416 NUM iajs-2429 405 10 . . PUNCT iajs-2429 406 1 3 3 X iajs-2429 406 2 . . X iajs-2429 406 3 saadi saadi PROPN iajs-2429 406 4 , , PUNCT iajs-2429 406 5 z. z. PROPN iajs-2429 406 6 ; ; PUNCT iajs-2429 406 7 ahmed ahmed PROPN iajs-2429 406 8 , , PUNCT iajs-2429 406 9 g. g. 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PUNCT iajs-2429 411 1 journal journal PROPN iajs-2429 411 2 of of ADP iajs-2429 411 3 al al PROPN iajs-2429 411 4 - - PUNCT iajs-2429 411 5 qadisiyah qadisiyah NOUN iajs-2429 411 6 for for ADP iajs-2429 411 7 computer computer NOUN iajs-2429 411 8 science science NOUN iajs-2429 411 9 and and CCONJ iajs-2429 411 10 mathematics.2019 mathematics.2019 PROPN iajs-2429 411 11 , , PUNCT iajs-2429 411 12 11 11 NUM iajs-2429 411 13 , , PUNCT iajs-2429 411 14 3 3 NUM iajs-2429 411 15 , , PUNCT iajs-2429 411 16 50 50 NUM iajs-2429 411 17 . . NOUN iajs-2429 411 18 6 6 NUM iajs-2429 411 19 . . X iajs-2429 411 20 baziar baziar ADJ iajs-2429 411 21 , , PUNCT iajs-2429 411 22 m. m. NOUN iajs-2429 411 23 ; ; PUNCT iajs-2429 411 24 behboodi behboodi NOUN iajs-2429 411 25 , , PUNCT iajs-2429 411 26 m. m. NOUN iajs-2429 411 27 classical classical ADJ iajs-2429 411 28 primary primary ADJ iajs-2429 411 29 submodules submodule NOUN iajs-2429 411 30 and and CCONJ iajs-2429 411 31 decomposition decomposition NOUN iajs-2429 411 32 theory theory NOUN iajs-2429 411 33 of of ADP iajs-2429 411 34 modules module NOUN iajs-2429 411 35 , , PUNCT iajs-2429 411 36 j. j. PROPN iajs-2429 411 37 algebra algebra PROPN iajs-2429 411 38 appl.2009 appl.2009 PROPN iajs-2429 411 39 , , PUNCT iajs-2429 411 40 8 8 NUM iajs-2429 411 41 , , PUNCT iajs-2429 411 42 3 3 NUM iajs-2429 411 43 , , PUNCT iajs-2429 411 44 351 351 NUM iajs-2429 411 45 - - SYM iajs-2429 411 46 362 362 NUM iajs-2429 411 47 . . PUNCT iajs-2429 412 1 7 7 NUM iajs-2429 412 2 . . NOUN iajs-2429 412 3 wisbauer wisbauer NOUN iajs-2429 412 4 , , PUNCT iajs-2429 412 5 r. r. 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PUNCT iajs-2429 415 1 ph.d ph.d PROPN iajs-2429 415 2 thesis thesis NOUN iajs-2429 415 3 , , PUNCT iajs-2429 415 4 university university NOUN iajs-2429 415 5 of of ADP iajs-2429 415 6 baghdad baghdad PROPN iajs-2429 415 7 , , PUNCT iajs-2429 415 8 baghdad baghdad PROPN iajs-2429 415 9 , , PUNCT iajs-2429 415 10 iraq iraq PROPN iajs-2429 415 11 , , PUNCT iajs-2429 415 12 2003 2003 NUM iajs-2429 415 13 . . PUNCT iajs-2429 416 1 9 9 X iajs-2429 416 2 . . PUNCT iajs-2429 416 3 dauns daun NOUN iajs-2429 416 4 , , PUNCT iajs-2429 416 5 j. j. PROPN iajs-2429 416 6 prime prime PROPN iajs-2429 416 7 submodules submodules PROPN iajs-2429 416 8 . . PUNCT iajs-2429 417 1 j. j. PROPN iajs-2429 417 2 reine reine PROPN iajs-2429 417 3 angew angew PROPN iajs-2429 417 4 . . PUNCT iajs-2429 418 1 math.1978 math.1978 NOUN iajs-2429 418 2 , , PUNCT iajs-2429 418 3 298 298 NUM iajs-2429 418 4 , , PUNCT iajs-2429 418 5 156–181 156–181 NUM iajs-2429 418 6 . . 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