id sid tid token lemma pos iajs-3181 1 1 ihjpas ihjpas PROPN iajs-3181 1 2 . . PUNCT iajs-3181 2 1 36 36 NUM iajs-3181 2 2 ( ( PUNCT iajs-3181 2 3 4 4 NUM iajs-3181 2 4 ) ) PUNCT iajs-3181 2 5 2023 2023 NUM iajs-3181 2 6 377 377 NUM iajs-3181 2 7 this this DET iajs-3181 2 8 work work NOUN iajs-3181 2 9 is be AUX iajs-3181 2 10 licensed license VERB iajs-3181 2 11 under under ADP iajs-3181 2 12 a a DET iajs-3181 2 13 creative creative ADJ iajs-3181 2 14 commons common NOUN iajs-3181 2 15 attribution attribution NOUN iajs-3181 2 16 4.0 4.0 NUM iajs-3181 2 17 international international ADJ iajs-3181 2 18 license license NOUN iajs-3181 2 19 abstract abstract NOUN iajs-3181 2 20 suppose suppose VERB iajs-3181 2 21 that that SCONJ iajs-3181 2 22 a a PRON iajs-3181 2 23 is be AUX iajs-3181 2 24 an an DET iajs-3181 2 25 abelain abelain ADJ iajs-3181 2 26 ring ring NOUN iajs-3181 2 27 with with ADP iajs-3181 2 28 identity identity NOUN iajs-3181 2 29 and and CCONJ iajs-3181 2 30 b b NOUN iajs-3181 2 31 is be AUX iajs-3181 2 32 a a DET iajs-3181 2 33 unitary unitary ADJ iajs-3181 2 34 ( ( PUNCT iajs-3181 2 35 left left ADJ iajs-3181 2 36 ) ) PUNCT iajs-3181 2 37 a a DET iajs-3181 2 38 - - PUNCT iajs-3181 2 39 module module NOUN iajs-3181 2 40 . . PUNCT iajs-3181 3 1 in in ADP iajs-3181 3 2 this this DET iajs-3181 3 3 paper paper NOUN iajs-3181 3 4 , , PUNCT iajs-3181 3 5 we we PRON iajs-3181 3 6 introduce introduce VERB iajs-3181 3 7 a a DET iajs-3181 3 8 type type NOUN iajs-3181 3 9 of of ADP iajs-3181 3 10 module module NOUN iajs-3181 3 11 , , PUNCT iajs-3181 3 12 namely namely ADV iajs-3181 3 13 quasi quasi NOUN iajs-3181 3 14 - - NOUN iajs-3181 3 15 semiprime semiprime NOUN iajs-3181 3 16 . . PUNCT iajs-3181 4 1 a a DET iajs-3181 4 2 - - PUNCT iajs-3181 4 3 module module NOUN iajs-3181 4 4 , , PUNCT iajs-3181 4 5 whenever whenever SCONJ iajs-3181 4 6 √[𝑁 √[𝑁 ADV iajs-3181 4 7 : : PUNCT iajs-3181 4 8 𝐡 𝐡 NOUN iajs-3181 4 9 ] ] PUNCT iajs-3181 4 10 is be AUX iajs-3181 4 11 a a DET iajs-3181 4 12 prime prime ADJ iajs-3181 4 13 ideal ideal NOUN iajs-3181 4 14 for for ADP iajs-3181 4 15 proper proper ADJ iajs-3181 4 16 submodule submodule NOUN iajs-3181 4 17 n n PROPN iajs-3181 4 18 of of ADP iajs-3181 4 19 b b PROPN iajs-3181 4 20 , , PUNCT iajs-3181 4 21 then then ADV iajs-3181 4 22 b b PROPN iajs-3181 4 23 is be AUX iajs-3181 4 24 called call VERB iajs-3181 4 25 quasi quasi PROPN iajs-3181 4 26 -semiprime -semiprime PROPN iajs-3181 4 27 module module NOUN iajs-3181 4 28 , , PUNCT iajs-3181 4 29 which which PRON iajs-3181 4 30 is be AUX iajs-3181 4 31 a a DET iajs-3181 4 32 generalization generalization NOUN iajs-3181 4 33 of of ADP iajs-3181 4 34 quasi quasi ADJ iajs-3181 4 35 - - ADJ iajs-3181 4 36 prime prime ADJ iajs-3181 4 37 a a DET iajs-3181 4 38 - - PUNCT iajs-3181 4 39 module module NOUN iajs-3181 4 40 , , PUNCT iajs-3181 4 41 whenever whenever SCONJ iajs-3181 4 42 annan annan PROPN iajs-3181 4 43 is be AUX iajs-3181 4 44 a a DET iajs-3181 4 45 prime prime ADJ iajs-3181 4 46 ideal ideal NOUN iajs-3181 4 47 for for ADP iajs-3181 4 48 proper proper ADJ iajs-3181 4 49 submodule submodule NOUN iajs-3181 4 50 n n PROPN iajs-3181 4 51 of of ADP iajs-3181 4 52 b b PROPN iajs-3181 4 53 , , PUNCT iajs-3181 4 54 then then ADV iajs-3181 4 55 b b NOUN iajs-3181 4 56 is be AUX iajs-3181 4 57 quasi quasi ADJ iajs-3181 4 58 - - ADJ iajs-3181 4 59 prime prime ADJ iajs-3181 4 60 module module NOUN iajs-3181 4 61 . . PUNCT iajs-3181 5 1 a a DET iajs-3181 5 2 comprehensive comprehensive ADJ iajs-3181 5 3 study study NOUN iajs-3181 5 4 of of ADP iajs-3181 5 5 these these DET iajs-3181 5 6 modules module NOUN iajs-3181 5 7 is be AUX iajs-3181 5 8 given give VERB iajs-3181 5 9 , , PUNCT iajs-3181 5 10 and and CCONJ iajs-3181 5 11 we we PRON iajs-3181 5 12 study study VERB iajs-3181 5 13 the the DET iajs-3181 5 14 relationship relationship NOUN iajs-3181 5 15 between between ADP iajs-3181 5 16 quasi quasi ADJ iajs-3181 5 17 - - ADJ iajs-3181 5 18 semiprime semiprime ADJ iajs-3181 5 19 modules module NOUN iajs-3181 5 20 and and CCONJ iajs-3181 5 21 quasi quasi NOUN iajs-3181 5 22 - - NOUN iajs-3181 5 23 prime prime ADJ iajs-3181 5 24 . . PUNCT iajs-3181 6 1 we we PRON iajs-3181 6 2 put put VERB iajs-3181 6 3 the the DET iajs-3181 6 4 condition condition NOUN iajs-3181 6 5 coprime coprime ADV iajs-3181 6 6 over over ADP iajs-3181 6 7 cosemiprime cosemiprime NOUN iajs-3181 6 8 ring ring NOUN iajs-3181 6 9 for for ADP iajs-3181 6 10 the the DET iajs-3181 6 11 two two NUM iajs-3181 6 12 cocept cocept NOUN iajs-3181 6 13 quasi quasi ADJ iajs-3181 6 14 - - ADJ iajs-3181 6 15 prime prime ADJ iajs-3181 6 16 modules module NOUN iajs-3181 6 17 and and CCONJ iajs-3181 6 18 quasi quasi ADJ iajs-3181 6 19 - - ADJ iajs-3181 6 20 semiprime semiprime ADJ iajs-3181 6 21 modules module NOUN iajs-3181 6 22 , , PUNCT iajs-3181 6 23 which which PRON iajs-3181 6 24 are be AUX iajs-3181 6 25 equivalent equivalent ADJ iajs-3181 6 26 . . PUNCT iajs-3181 7 1 the the DET iajs-3181 7 2 concepts concept NOUN iajs-3181 7 3 of of ADP iajs-3181 7 4 prime prime ADJ iajs-3181 7 5 modules module NOUN iajs-3181 7 6 and and CCONJ iajs-3181 7 7 quasi quasi ADJ iajs-3181 7 8 - - ADJ iajs-3181 7 9 semiprime semiprime ADJ iajs-3181 7 10 modules module NOUN iajs-3181 7 11 are be AUX iajs-3181 7 12 equivalent equivalent ADJ iajs-3181 7 13 . . PUNCT iajs-3181 8 1 the the DET iajs-3181 8 2 condition condition NOUN iajs-3181 8 3 of of ADP iajs-3181 8 4 anti anti ADJ iajs-3181 8 5 - - ADJ iajs-3181 8 6 hopfain hopfain ADJ iajs-3181 8 7 makes make VERB iajs-3181 8 8 quasi quasi ADJ iajs-3181 8 9 - - ADJ iajs-3181 8 10 prime prime ADJ iajs-3181 8 11 is be AUX iajs-3181 8 12 quasi quasi ADJ iajs-3181 8 13 - - ADJ iajs-3181 8 14 semiprime semiprime ADJ iajs-3181 8 15 a a DET iajs-3181 8 16 - - PUNCT iajs-3181 8 17 module module NOUN iajs-3181 8 18 . . PUNCT iajs-3181 9 1 whenever whenever SCONJ iajs-3181 9 2 b b NOUN iajs-3181 9 3 is be AUX iajs-3181 9 4 cyclic cyclic ADJ iajs-3181 9 5 , , PUNCT iajs-3181 9 6 coprime coprime ADJ iajs-3181 9 7 c c NOUN iajs-3181 9 8 - - PUNCT iajs-3181 9 9 module module NOUN iajs-3181 9 10 , , PUNCT iajs-3181 9 11 where where SCONJ iajs-3181 9 12 c c PROPN iajs-3181 9 13 is be AUX iajs-3181 9 14 the the DET iajs-3181 9 15 ring ring NOUN iajs-3181 9 16 , , PUNCT iajs-3181 9 17 each each DET iajs-3181 9 18 ideal ideal NOUN iajs-3181 9 19 is be AUX iajs-3181 9 20 semiprime semiprime NOUN iajs-3181 9 21 , , PUNCT iajs-3181 9 22 which which PRON iajs-3181 9 23 implies imply VERB iajs-3181 9 24 quasi quasi ADJ iajs-3181 9 25 - - ADJ iajs-3181 9 26 prime prime ADJ iajs-3181 9 27 , , PUNCT iajs-3181 9 28 quasi quasi NOUN iajs-3181 9 29 - - NOUN iajs-3181 9 30 simepime simepime NOUN iajs-3181 9 31 , , PUNCT iajs-3181 9 32 and and CCONJ iajs-3181 9 33 anncb anncb PROPN iajs-3181 9 34 are be AUX iajs-3181 9 35 prime prime ADJ iajs-3181 9 36 ideals ideal NOUN iajs-3181 9 37 . . PUNCT iajs-3181 10 1 if if SCONJ iajs-3181 10 2 f f PROPN iajs-3181 10 3 is be AUX iajs-3181 10 4 an an DET iajs-3181 10 5 epimorphism epimorphism NOUN iajs-3181 10 6 from from ADP iajs-3181 10 7 b1 b1 PROPN iajs-3181 10 8 β†’ β†’ SYM iajs-3181 10 9 b2 b2 NOUN iajs-3181 10 10 , , PUNCT iajs-3181 10 11 whenever whenever SCONJ iajs-3181 10 12 b1 b1 PROPN iajs-3181 10 13 is be AUX iajs-3181 10 14 a a DET iajs-3181 10 15 quasi quasi ADJ iajs-3181 10 16 - - ADJ iajs-3181 10 17 prime prime ADJ iajs-3181 10 18 module module NOUN iajs-3181 10 19 , , PUNCT iajs-3181 10 20 it it PRON iajs-3181 10 21 implies imply VERB iajs-3181 10 22 b2 b2 NOUN iajs-3181 10 23 is be AUX iajs-3181 10 24 a a DET iajs-3181 10 25 quasi quasi ADJ iajs-3181 10 26 - - ADJ iajs-3181 10 27 prime prime ADJ iajs-3181 10 28 a a DET iajs-3181 10 29 - - PUNCT iajs-3181 10 30 module module NOUN iajs-3181 10 31 , , PUNCT iajs-3181 10 32 and and CCONJ iajs-3181 10 33 the the DET iajs-3181 10 34 inverse inverse ADJ iajs-3181 10 35 image image NOUN iajs-3181 10 36 of of ADP iajs-3181 10 37 quasi quasi NOUN iajs-3181 10 38 - - ADJ iajs-3181 10 39 semiprime semiprime NOUN iajs-3181 10 40 is be AUX iajs-3181 10 41 a a DET iajs-3181 10 42 quasi quasi ADJ iajs-3181 10 43 - - ADJ iajs-3181 10 44 prime prime ADJ iajs-3181 10 45 a a DET iajs-3181 10 46 - - PUNCT iajs-3181 10 47 module module NOUN iajs-3181 10 48 . . PUNCT iajs-3181 11 1 keywords keyword NOUN iajs-3181 11 2 : : PUNCT iajs-3181 11 3 prime prime ADJ iajs-3181 11 4 module module NOUN iajs-3181 11 5 , , PUNCT iajs-3181 11 6 quasi quasi ADJ iajs-3181 11 7 - - ADJ iajs-3181 11 8 prime prime ADJ iajs-3181 11 9 r r NOUN iajs-3181 11 10 - - PUNCT iajs-3181 11 11 modules module NOUN iajs-3181 11 12 , , PUNCT iajs-3181 11 13 quasi quasi ADJ iajs-3181 11 14 - - ADJ iajs-3181 11 15 semiprime semiprime ADJ iajs-3181 11 16 r r NOUN iajs-3181 11 17 - - PUNCT iajs-3181 11 18 modules module NOUN iajs-3181 11 19 , , PUNCT iajs-3181 11 20 coprime coprime NOUN iajs-3181 11 21 rmodules rmodule NOUN iajs-3181 11 22 , , PUNCT iajs-3181 11 23 antihopfian antihopfian ADJ iajs-3181 11 24 r r NOUN iajs-3181 11 25 - - PUNCT iajs-3181 11 26 modules module NOUN iajs-3181 11 27 . . PUNCT iajs-3181 12 1 1 1 X iajs-3181 12 2 . . X iajs-3181 12 3 introduction introduction NOUN iajs-3181 12 4 suppose suppose VERB iajs-3181 12 5 that that SCONJ iajs-3181 12 6 w w NOUN iajs-3181 12 7 is be AUX iajs-3181 12 8 a a DET iajs-3181 12 9 left left ADJ iajs-3181 12 10 a a DET iajs-3181 12 11 - - PUNCT iajs-3181 12 12 module module NOUN iajs-3181 12 13 , , PUNCT iajs-3181 12 14 where where SCONJ iajs-3181 12 15 a a PRON iajs-3181 12 16 is be AUX iajs-3181 12 17 a a DET iajs-3181 12 18 ring ring NOUN iajs-3181 12 19 with with ADP iajs-3181 12 20 unity unity NOUN iajs-3181 12 21 . . PUNCT iajs-3181 13 1 an an DET iajs-3181 13 2 a a DET iajs-3181 13 3 - - PUNCT iajs-3181 13 4 module module NOUN iajs-3181 13 5 b b NOUN iajs-3181 13 6 is be AUX iajs-3181 13 7 said say VERB iajs-3181 13 8 to to PART iajs-3181 13 9 be be AUX iajs-3181 13 10 prime prime ADJ iajs-3181 13 11 whenever whenever SCONJ iajs-3181 13 12 annab annab NOUN iajs-3181 13 13 = = PRON iajs-3181 13 14 annan annan PROPN iajs-3181 13 15 for for ADP iajs-3181 13 16 each each DET iajs-3181 13 17 non non ADJ iajs-3181 13 18 - - ADJ iajs-3181 13 19 zero zero NUM iajs-3181 13 20 submodule submodule PROPN iajs-3181 13 21 nof nof PROPN iajs-3181 13 22 b b PROPN iajs-3181 13 23 , , PUNCT iajs-3181 13 24 where where SCONJ iajs-3181 13 25 annab={a annab={a PROPN iajs-3181 13 26 ∈ ∈ PROPN iajs-3181 13 27 a;bx=0 a;bx=0 NOUN iajs-3181 13 28 for for ADP iajs-3181 13 29 each each DET iajs-3181 13 30 b b PROPN iajs-3181 13 31 ∈ ∈ PROPN iajs-3181 13 32 b}[1,2 b}[1,2 NOUN iajs-3181 13 33 ] ] PUNCT iajs-3181 13 34 . . PUNCT iajs-3181 14 1 hasan hasan PROPN iajs-3181 14 2 in in ADP iajs-3181 14 3 [ [ X iajs-3181 14 4 3 3 NUM iajs-3181 14 5 ] ] PUNCT iajs-3181 14 6 introduced introduce VERB iajs-3181 14 7 the the DET iajs-3181 14 8 concept concept NOUN iajs-3181 14 9 of of ADP iajs-3181 14 10 qasi qasi NOUN iajs-3181 14 11 - - PUNCT iajs-3181 14 12 prime prime ADJ iajs-3181 14 13 a a NOUN iajs-3181 14 14 - - PUNCT iajs-3181 14 15 modules module NOUN iajs-3181 14 16 , , PUNCT iajs-3181 14 17 which which PRON iajs-3181 14 18 is be AUX iajs-3181 14 19 a a DET iajs-3181 14 20 generalization generalization NOUN iajs-3181 14 21 of of ADP iajs-3181 14 22 prime prime ADJ iajs-3181 14 23 a a NOUN iajs-3181 14 24 - - PUNCT iajs-3181 14 25 modules module NOUN iajs-3181 14 26 , , PUNCT iajs-3181 14 27 where where SCONJ iajs-3181 14 28 an an DET iajs-3181 14 29 a a DET iajs-3181 14 30 - - PUNCT iajs-3181 14 31 module module NOUN iajs-3181 14 32 w w NOUN iajs-3181 14 33 is be AUX iajs-3181 14 34 called call VERB iajs-3181 14 35 quasi quasi ADJ iajs-3181 14 36 - - ADJ iajs-3181 14 37 prime prime ADJ iajs-3181 14 38 modulles modulle NOUN iajs-3181 14 39 if if SCONJ iajs-3181 14 40 and and CCONJ iajs-3181 14 41 only only ADV iajs-3181 14 42 if if SCONJ iajs-3181 14 43 for for ADP iajs-3181 14 44 each each DET iajs-3181 14 45 non non ADJ iajs-3181 14 46 - - ADJ iajs-3181 14 47 zero zero NUM iajs-3181 14 48 submodule submodule NOUN iajs-3181 14 49 n n PROPN iajs-3181 14 50 of of ADP iajs-3181 14 51 w w PROPN iajs-3181 14 52 , , PUNCT iajs-3181 14 53 anna anna PROPN iajs-3181 14 54 n n PART iajs-3181 14 55 is be AUX iajs-3181 14 56 a a DET iajs-3181 14 57 prime prime ADJ iajs-3181 14 58 ideal ideal NOUN iajs-3181 14 59 . . PUNCT iajs-3181 15 1 annin annin ADJ iajs-3181 16 1 [ [ X iajs-3181 16 2 9 9 NUM iajs-3181 16 3 ] ] PUNCT iajs-3181 16 4 calls call VERB iajs-3181 16 5 an an DET iajs-3181 16 6 a a DET iajs-3181 16 7 - - PUNCT iajs-3181 16 8 module module NOUN iajs-3181 16 9 w w NOUN iajs-3181 16 10 a a DET iajs-3181 16 11 coprime coprime NOUN iajs-3181 16 12 ( ( PUNCT iajs-3181 16 13 dual dual ADJ iajs-3181 16 14 notion notion NOUN iajs-3181 16 15 of of ADP iajs-3181 16 16 prime prime ADJ iajs-3181 16 17 modules module NOUN iajs-3181 16 18 ) ) PUNCT iajs-3181 16 19 if if SCONJ iajs-3181 16 20 annaw annaw NOUN iajs-3181 16 21 = = VERB iajs-3181 16 22 annaw annaw NOUN iajs-3181 16 23 / / SYM iajs-3181 16 24 a a NOUN iajs-3181 16 25 for for ADP iajs-3181 16 26 every every DET iajs-3181 16 27 proper proper ADJ iajs-3181 16 28 submodule submodule NOUN iajs-3181 16 29 a a PRON iajs-3181 16 30 of of ADP iajs-3181 16 31 w. w. NOUN iajs-3181 16 32 in in ADP iajs-3181 16 33 this this DET iajs-3181 16 34 paper paper NOUN iajs-3181 16 35 , , PUNCT iajs-3181 16 36 we we PRON iajs-3181 16 37 study study VERB iajs-3181 16 38 a a DET iajs-3181 16 39 generalization generalization NOUN iajs-3181 16 40 of of ADP iajs-3181 16 41 the the DET iajs-3181 16 42 quasi quasi ADJ iajs-3181 16 43 - - ADJ iajs-3181 16 44 prime prime ADJ iajs-3181 16 45 module module NOUN iajs-3181 16 46 which which PRON iajs-3181 16 47 we we PRON iajs-3181 16 48 called call VERB iajs-3181 16 49 the the DET iajs-3181 16 50 quasi quasi NOUN iajs-3181 16 51 - - ADJ iajs-3181 16 52 semiprime semiprime ADJ iajs-3181 16 53 a a DET iajs-3181 16 54 - - PUNCT iajs-3181 16 55 module module NOUN iajs-3181 16 56 if if SCONJ iajs-3181 16 57 √annb √annb NOUN iajs-3181 16 58 / / SYM iajs-3181 16 59 n n NOUN iajs-3181 16 60 = = PUNCT iajs-3181 16 61 √[n √[n NOUN iajs-3181 16 62 : : PUNCT iajs-3181 16 63 b b X iajs-3181 16 64 ] ] PUNCT iajs-3181 16 65 is be AUX iajs-3181 16 66 a a DET iajs-3181 16 67 prime prime ADJ iajs-3181 16 68 ideal ideal NOUN iajs-3181 16 69 for for ADP iajs-3181 16 70 each each DET iajs-3181 16 71 submodule submodule NOUN iajs-3181 16 72 n n PROPN iajs-3181 16 73 of of ADP iajs-3181 16 74 b. b. PROPN iajs-3181 16 75 this this DET iajs-3181 16 76 paper paper NOUN iajs-3181 16 77 consists consist VERB iajs-3181 16 78 of of ADP iajs-3181 16 79 two two NUM iajs-3181 16 80 sections section NOUN iajs-3181 16 81 . . PUNCT iajs-3181 17 1 in in ADP iajs-3181 17 2 section section NOUN iajs-3181 17 3 one one NUM iajs-3181 17 4 ; ; PUNCT iajs-3181 17 5 we we PRON iajs-3181 17 6 study study VERB iajs-3181 17 7 the the DET iajs-3181 17 8 basic basic ADJ iajs-3181 17 9 properties property NOUN iajs-3181 17 10 of of ADP iajs-3181 17 11 a a DET iajs-3181 17 12 quasidoi.org/10.30526/36.4.3181 quasidoi.org/10.30526/36.4.3181 NOUN iajs-3181 17 13 article article NOUN iajs-3181 17 14 history history NOUN iajs-3181 17 15 : : PUNCT iajs-3181 17 16 received receive VERB iajs-3181 17 17 9 9 NUM iajs-3181 17 18 january january PROPN iajs-3181 17 19 2023 2023 NUM iajs-3181 17 20 , , PUNCT iajs-3181 17 21 accepted accept VERB iajs-3181 17 22 20 20 NUM iajs-3181 17 23 february february NOUN iajs-3181 17 24 2023 2023 NUM iajs-3181 17 25 , , PUNCT iajs-3181 17 26 published publish VERB iajs-3181 17 27 in in ADP iajs-3181 17 28 october october PROPN iajs-3181 17 29 2023 2023 NUM iajs-3181 17 30 ibn ibn PROPN iajs-3181 17 31 al al PROPN iajs-3181 17 32 - - PUNCT iajs-3181 17 33 haitham haitham PROPN iajs-3181 17 34 journal journal PROPN iajs-3181 17 35 for for ADP iajs-3181 17 36 pure pure ADJ iajs-3181 17 37 and and CCONJ iajs-3181 17 38 applied applied ADJ iajs-3181 17 39 sciences sciences PROPN iajs-3181 17 40 journal journal PROPN iajs-3181 17 41 homepage homepage NOUN iajs-3181 17 42 : : PUNCT iajs-3181 17 43 jih.uobaghdad.edu.iq jih.uobaghdad.edu.iq NOUN iajs-3181 17 44 quasi quasi ADJ iajs-3181 17 45 - - ADJ iajs-3181 17 46 semiprime semiprime ADJ iajs-3181 17 47 modules module NOUN iajs-3181 17 48 muntaha muntaha PROPN iajs-3181 17 49 abdulrazaq abdulrazaq PROPN iajs-3181 17 50 hasan hasan PROPN iajs-3181 17 51 department department PROPN iajs-3181 17 52 of of ADP iajs-3181 17 53 mathematics mathematics PROPN iajs-3181 17 54 , , PUNCT iajs-3181 17 55 college college NOUN iajs-3181 17 56 of of ADP iajs-3181 17 57 basic basic ADJ iajs-3181 17 58 education education NOUN iajs-3181 17 59 , , PUNCT iajs-3181 17 60 mustansiriyah mustansiriyah NOUN iajs-3181 17 61 university university NOUN iajs-3181 17 62 , , PUNCT iajs-3181 17 63 baghhdad baghhdad PROPN iajs-3181 17 64 - - PROPN iajs-3181 17 65 iraq iraq PROPN iajs-3181 17 66 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3181 17 67 mailto:muntha_1974rzaq.edbs@uomustansiriyah.edu.iq mailto:muntha_1974rzaq.edbs@uomustansiriyah.edu.iq PROPN iajs-3181 17 68 ihjpas ihjpas PROPN iajs-3181 17 69 . . PUNCT iajs-3181 18 1 36 36 NUM iajs-3181 18 2 ( ( PUNCT iajs-3181 18 3 4 4 NUM iajs-3181 18 4 ) ) PUNCT iajs-3181 18 5 2023 2023 NUM iajs-3181 18 6 378 378 NUM iajs-3181 18 7 semiprime semiprime NOUN iajs-3181 18 8 a a DET iajs-3181 18 9 - - PUNCT iajs-3181 18 10 module module NOUN iajs-3181 18 11 . . PUNCT iajs-3181 19 1 in in ADP iajs-3181 19 2 section section NOUN iajs-3181 19 3 two two NUM iajs-3181 19 4 , , PUNCT iajs-3181 19 5 we we PRON iajs-3181 19 6 study study VERB iajs-3181 19 7 the the DET iajs-3181 19 8 relation relation NOUN iajs-3181 19 9 between between ADP iajs-3181 19 10 quasi quasi ADJ iajs-3181 19 11 - - ADJ iajs-3181 19 12 semiprime semiprime ADJ iajs-3181 19 13 amodules amodule NOUN iajs-3181 19 14 and and CCONJ iajs-3181 19 15 prime prime ADJ iajs-3181 19 16 a a NOUN iajs-3181 19 17 - - PUNCT iajs-3181 19 18 modules module NOUN iajs-3181 19 19 . . PUNCT iajs-3181 20 1 2 2 X iajs-3181 20 2 . . X iajs-3181 20 3 materials material NOUN iajs-3181 20 4 and and CCONJ iajs-3181 20 5 methods method NOUN iajs-3181 20 6 definition definition NOUN iajs-3181 20 7 ( ( PUNCT iajs-3181 20 8 2.1 2.1 NUM iajs-3181 20 9 ) ) PUNCT iajs-3181 20 10 b b NOUN iajs-3181 20 11 is be AUX iajs-3181 20 12 said say VERB iajs-3181 20 13 to to PART iajs-3181 20 14 be be AUX iajs-3181 20 15 a a DET iajs-3181 20 16 quasi quasi NOUN iajs-3181 20 17 - - ADJ iajs-3181 20 18 semiprime semiprime ADJ iajs-3181 20 19 a a DET iajs-3181 20 20 - - PUNCT iajs-3181 20 21 module module NOUN iajs-3181 20 22 if if SCONJ iajs-3181 20 23 √[n √[n NOUN iajs-3181 20 24 : : PUNCT iajs-3181 20 25 w w X iajs-3181 20 26 ] ] X iajs-3181 20 27 is be AUX iajs-3181 20 28 a a DET iajs-3181 20 29 prime prime ADJ iajs-3181 20 30 ideal ideal NOUN iajs-3181 20 31 for for ADP iajs-3181 20 32 the the DET iajs-3181 20 33 proper proper ADJ iajs-3181 20 34 submodule submodule NOUN iajs-3181 20 35 n n PROPN iajs-3181 20 36 of of ADP iajs-3181 20 37 b. b. PROPN iajs-3181 20 38 examples example NOUN iajs-3181 20 39 and and CCONJ iajs-3181 20 40 remarks remark NOUN iajs-3181 20 41 ( ( PUNCT iajs-3181 20 42 2.2 2.2 NUM iajs-3181 20 43 ) ) PUNCT iajs-3181 20 44 1it 1it NOUN iajs-3181 20 45 is be AUX iajs-3181 20 46 clear clear ADJ iajs-3181 20 47 that that SCONJ iajs-3181 20 48 zn zn PROPN iajs-3181 20 49 is be AUX iajs-3181 20 50 a a DET iajs-3181 20 51 quasi quasi NOUN iajs-3181 20 52 - - ADJ iajs-3181 20 53 semiprime semiprime ADJ iajs-3181 20 54 a a DET iajs-3181 20 55 - - PUNCT iajs-3181 20 56 module module NOUN iajs-3181 20 57 if if SCONJ iajs-3181 20 58 and and CCONJ iajs-3181 20 59 only only ADV iajs-3181 20 60 if if SCONJ iajs-3181 20 61 n n PRON iajs-3181 20 62 is be AUX iajs-3181 20 63 a a DET iajs-3181 20 64 prime prime ADJ iajs-3181 20 65 number number NOUN iajs-3181 20 66 . . PUNCT iajs-3181 21 1 2if 2if NOUN iajs-3181 21 2 n n NOUN iajs-3181 21 3 can can AUX iajs-3181 21 4 be be AUX iajs-3181 21 5 written write VERB iajs-3181 21 6 as as ADP iajs-3181 21 7 a a DET iajs-3181 21 8 product product NOUN iajs-3181 21 9 of of ADP iajs-3181 21 10 two two NUM iajs-3181 21 11 prime prime ADJ iajs-3181 21 12 numbers number NOUN iajs-3181 21 13 , , PUNCT iajs-3181 21 14 then then ADV iajs-3181 21 15 zn zn PROPN iajs-3181 21 16 is be AUX iajs-3181 21 17 a a DET iajs-3181 21 18 quasi quasi NOUN iajs-3181 21 19 - - ADJ iajs-3181 21 20 semiprime semiprime ADJ iajs-3181 21 21 a a DET iajs-3181 21 22 - - PUNCT iajs-3181 21 23 module module NOUN iajs-3181 21 24 . . PUNCT iajs-3181 22 1 proof proof NOUN iajs-3181 22 2 : : PUNCT iajs-3181 22 3 let let VERB iajs-3181 22 4 n n PRON iajs-3181 22 5 = = X iajs-3181 22 6 p1p2 p1p2 PRON iajs-3181 22 7 ; ; PUNCT iajs-3181 22 8 p1 p1 NOUN iajs-3181 22 9 , , PUNCT iajs-3181 22 10 p2 p2 PROPN iajs-3181 22 11 be be VERB iajs-3181 22 12 two two NUM iajs-3181 22 13 prime prime ADJ iajs-3181 22 14 numbers number NOUN iajs-3181 22 15 , , PUNCT iajs-3181 22 16 so so ADV iajs-3181 22 17 n1= n1= PROPN iajs-3181 22 18 ( ( PUNCT iajs-3181 22 19 p1),n2=(p2 p1),n2=(p2 ADJ iajs-3181 22 20 ) ) PUNCT iajs-3181 22 21 , , PUNCT iajs-3181 22 22 then√[(𝑃1 then√[(𝑃1 VERB iajs-3181 22 23 : : PUNCT iajs-3181 23 1 𝑍𝑛 𝑍𝑛 ADJ iajs-3181 23 2 ] ] X iajs-3181 23 3 = = SYM iajs-3181 23 4 √(𝑝1 √(𝑝1 PROPN iajs-3181 23 5 ) ) PUNCT iajs-3181 23 6 = = PUNCT iajs-3181 23 7 ( ( PUNCT iajs-3181 23 8 p1 p1 PROPN iajs-3181 23 9 ) ) PUNCT iajs-3181 23 10 , , PUNCT iajs-3181 23 11 n2=(p2 n2=(p2 NOUN iajs-3181 23 12 ) ) PUNCT iajs-3181 23 13 , , PUNCT iajs-3181 23 14 then then ADV iajs-3181 23 15 √[(𝑝2):𝑍𝑛 √[(𝑝2):𝑍𝑛 ADV iajs-3181 23 16 ] ] X iajs-3181 23 17 = = PUNCT iajs-3181 23 18 √(𝑝2 √(𝑝2 PROPN iajs-3181 23 19 ) ) PUNCT iajs-3181 23 20 = = PRON iajs-3181 23 21 ( ( PUNCT iajs-3181 23 22 p2 p2 PROPN iajs-3181 23 23 ) ) PUNCT iajs-3181 23 24 is be AUX iajs-3181 23 25 a a DET iajs-3181 23 26 prime prime ADJ iajs-3181 23 27 ideal ideal NOUN iajs-3181 23 28 , , PUNCT iajs-3181 23 29 henc henc PROPN iajs-3181 23 30 zn zn PROPN iajs-3181 23 31 is be AUX iajs-3181 23 32 a a DET iajs-3181 23 33 quasi quasi ADJ iajs-3181 23 34 - - ADJ iajs-3181 23 35 semiprime semiprime ADJ iajs-3181 23 36 amodule amodule NOUN iajs-3181 23 37 . . PUNCT iajs-3181 24 1 3zp∞ 3zp∞ NUM iajs-3181 24 2 is be AUX iajs-3181 24 3 not not PART iajs-3181 24 4 a a DET iajs-3181 24 5 quasi quasi ADJ iajs-3181 24 6 - - ADJ iajs-3181 24 7 semiprime semiprime ADJ iajs-3181 24 8 module module NOUN iajs-3181 24 9 , , PUNCT iajs-3181 24 10 since since SCONJ iajs-3181 24 11 we we PRON iajs-3181 24 12 know know VERB iajs-3181 24 13 that that SCONJ iajs-3181 24 14 every every DET iajs-3181 24 15 submodule submodule NOUN iajs-3181 24 16 of of ADP iajs-3181 24 17 zp∞ zp∞ PROPN iajs-3181 24 18 is be AUX iajs-3181 24 19 of of ADP iajs-3181 24 20 the the DET iajs-3181 24 21 form form NOUN iajs-3181 24 22 ( ( PUNCT iajs-3181 24 23 1 1 NUM iajs-3181 24 24 / / SYM iajs-3181 24 25 pn pn NOUN iajs-3181 24 26 + + NOUN iajs-3181 24 27 z z NOUN iajs-3181 24 28 ) ) PUNCT iajs-3181 24 29 , , PUNCT iajs-3181 24 30 where where SCONJ iajs-3181 24 31 n n PRON iajs-3181 24 32 is be AUX iajs-3181 24 33 a a DET iajs-3181 24 34 non non ADJ iajs-3181 24 35 - - ADJ iajs-3181 24 36 negative negative ADJ iajs-3181 24 37 integer integer NOUN iajs-3181 24 38 , , PUNCT iajs-3181 24 39 so so ADV iajs-3181 24 40 √ √ PROPN iajs-3181 24 41 [ [ PUNCT iajs-3181 24 42 1 1 NUM iajs-3181 24 43 𝑝𝑛 𝑝𝑛 ADP iajs-3181 24 44 + + X iajs-3181 24 45 𝑍 𝑍 NOUN iajs-3181 24 46 : : PUNCT iajs-3181 24 47 π‘π‘βˆž π‘π‘βˆž NUM iajs-3181 24 48 ] ] X iajs-3181 24 49 = = SYM iajs-3181 24 50 √ √ NUM iajs-3181 24 51 [ [ PUNCT iajs-3181 24 52 1 1 NUM iajs-3181 24 53 𝑃𝑛 𝑃𝑛 PROPN iajs-3181 24 54 + + CCONJ iajs-3181 24 55 𝑍 𝑍 PROPN iajs-3181 24 56 ] ] X iajs-3181 24 57 = = SYM iajs-3181 24 58 ( ( PUNCT iajs-3181 24 59 pn pn PROPN iajs-3181 24 60 z z PROPN iajs-3181 24 61 ) ) PUNCT iajs-3181 24 62 is be AUX iajs-3181 24 63 not not PART iajs-3181 24 64 prime prime ADJ iajs-3181 24 65 ideal ideal NOUN iajs-3181 24 66 . . PUNCT iajs-3181 25 1 4suppose 4suppose NUM iajs-3181 25 2 b b NOUN iajs-3181 25 3 is be AUX iajs-3181 25 4 a a DET iajs-3181 25 5 simple simple ADJ iajs-3181 25 6 a a DET iajs-3181 25 7 - - PUNCT iajs-3181 25 8 module module NOUN iajs-3181 25 9 , , PUNCT iajs-3181 25 10 then then ADV iajs-3181 25 11 b b PROPN iajs-3181 25 12 is be AUX iajs-3181 25 13 a a DET iajs-3181 25 14 quasi quasi NOUN iajs-3181 25 15 - - ADJ iajs-3181 25 16 semiprime semiprime ADJ iajs-3181 25 17 a a DET iajs-3181 25 18 - - PUNCT iajs-3181 25 19 module module NOUN iajs-3181 25 20 . . PUNCT iajs-3181 26 1 proof proof NOUN iajs-3181 26 2 : : PUNCT iajs-3181 26 3 it it PRON iajs-3181 26 4 is be AUX iajs-3181 26 5 clear clear ADJ iajs-3181 26 6 . . PUNCT iajs-3181 27 1 proposition proposition NOUN iajs-3181 27 2 ( ( PUNCT iajs-3181 27 3 2.3 2.3 NUM iajs-3181 27 4 ) ) PUNCT iajs-3181 27 5 every every DET iajs-3181 27 6 proper proper ADJ iajs-3181 27 7 submodule submodule NOUN iajs-3181 27 8 n n PROPN iajs-3181 27 9 of of ADP iajs-3181 27 10 quasi quasi ADJ iajs-3181 27 11 - - ADJ iajs-3181 27 12 semiprime semiprime ADJ iajs-3181 27 13 module module NOUN iajs-3181 27 14 is be AUX iajs-3181 27 15 a a DET iajs-3181 27 16 quasi quasi ADJ iajs-3181 27 17 - - ADJ iajs-3181 27 18 semiprime semiprime ADJ iajs-3181 27 19 module module NOUN iajs-3181 27 20 . . PUNCT iajs-3181 28 1 proof proof NOUN iajs-3181 28 2 : : PUNCT iajs-3181 28 3 suppose suppose VERB iajs-3181 28 4 n n PRON iajs-3181 28 5 is be AUX iajs-3181 28 6 a a DET iajs-3181 28 7 proper proper ADJ iajs-3181 28 8 submodule submodule NOUN iajs-3181 28 9 of of ADP iajs-3181 28 10 quasi quasi NOUN iajs-3181 28 11 - - ADJ iajs-3181 28 12 semiprime semiprime ADJ iajs-3181 28 13 a a DET iajs-3181 28 14 - - PUNCT iajs-3181 28 15 module module NOUN iajs-3181 28 16 w. w. NOUN iajs-3181 28 17 let let VERB iajs-3181 28 18 k k PRON iajs-3181 28 19 be be AUX iajs-3181 28 20 a a DET iajs-3181 28 21 proper proper ADJ iajs-3181 28 22 submodule submodule NOUN iajs-3181 28 23 of of ADP iajs-3181 28 24 n n PART iajs-3181 28 25 to to PART iajs-3181 28 26 show show VERB iajs-3181 28 27 that that SCONJ iajs-3181 28 28 √[𝐾 √[𝐾 NOUN iajs-3181 28 29 : : PUNCT iajs-3181 28 30 𝑁 𝑁 PROPN iajs-3181 28 31 ] ] PUNCT iajs-3181 28 32 is be AUX iajs-3181 28 33 a a DET iajs-3181 28 34 prime prime ADJ iajs-3181 28 35 ideal ideal NOUN iajs-3181 28 36 if if SCONJ iajs-3181 28 37 ab∈ ab∈ PROPN iajs-3181 28 38 √[𝐾 √[𝐾 NOUN iajs-3181 28 39 : : PUNCT iajs-3181 28 40 𝑁 𝑁 PROPN iajs-3181 28 41 ] ] PUNCT iajs-3181 28 42 , , PUNCT iajs-3181 28 43 so so ADV iajs-3181 28 44 anbn∈ anbn∈ ADP iajs-3181 28 45 [ [ X iajs-3181 28 46 k k X iajs-3181 28 47 : : PUNCT iajs-3181 28 48 n n X iajs-3181 28 49 ] ] PUNCT iajs-3181 28 50 , , PUNCT iajs-3181 28 51 so so SCONJ iajs-3181 28 52 that that SCONJ iajs-3181 28 53 anbnn anbnn NOUN iajs-3181 28 54 βŠ† βŠ† NUM iajs-3181 28 55 k k PROPN iajs-3181 28 56 βŠ† βŠ† NUM iajs-3181 28 57 w w NOUN iajs-3181 28 58 that that PRON iajs-3181 28 59 is be AUX iajs-3181 28 60 anbn anbn PROPN iajs-3181 28 61 ∈ ∈ PROPN iajs-3181 28 62 [ [ X iajs-3181 28 63 n n NUM iajs-3181 28 64 : : PUNCT iajs-3181 28 65 w w NOUN iajs-3181 28 66 ] ] X iajs-3181 28 67 , , PUNCT iajs-3181 28 68 but but CCONJ iajs-3181 28 69 w w NOUN iajs-3181 28 70 is be AUX iajs-3181 28 71 a a DET iajs-3181 28 72 quasi quasi NOUN iajs-3181 28 73 - - ADJ iajs-3181 28 74 semiprime semiprime ADJ iajs-3181 28 75 a a PRON iajs-3181 28 76 - - PUNCT iajs-3181 28 77 modul modul NOUN iajs-3181 28 78 implies imply VERB iajs-3181 28 79 either either CCONJ iajs-3181 28 80 an an DET iajs-3181 28 81 ∈ ∈ PROPN iajs-3181 29 1 [ [ X iajs-3181 29 2 n n CCONJ iajs-3181 29 3 : : PUNCT iajs-3181 29 4 w w NOUN iajs-3181 29 5 ] ] PUNCT iajs-3181 29 6 or or CCONJ iajs-3181 29 7 bn bn ADP iajs-3181 29 8 ∈ ∈ PROPN iajs-3181 30 1 [ [ X iajs-3181 30 2 n n CCONJ iajs-3181 30 3 : : PUNCT iajs-3181 30 4 w w NOUN iajs-3181 30 5 ] ] X iajs-3181 30 6 , , PUNCT iajs-3181 30 7 thus thus ADV iajs-3181 30 8 either either CCONJ iajs-3181 30 9 an∈ an∈ VERB iajs-3181 31 1 [ [ X iajs-3181 31 2 k k X iajs-3181 31 3 : : PUNCT iajs-3181 31 4 n n CCONJ iajs-3181 31 5 ] ] PUNCT iajs-3181 31 6 or or CCONJ iajs-3181 31 7 bn bn ADP iajs-3181 31 8 ∈ ∈ PROPN iajs-3181 32 1 [ [ X iajs-3181 32 2 k k X iajs-3181 32 3 : : PUNCT iajs-3181 32 4 n n CCONJ iajs-3181 32 5 ] ] PUNCT iajs-3181 32 6 which which PRON iajs-3181 32 7 means mean VERB iajs-3181 32 8 either either CCONJ iajs-3181 32 9 a∈ a∈ PROPN iajs-3181 32 10 √[𝐾 √[𝐾 PROPN iajs-3181 32 11 : : PUNCT iajs-3181 32 12 𝑁 𝑁 PROPN iajs-3181 32 13 ] ] PUNCT iajs-3181 32 14 or or CCONJ iajs-3181 32 15 b∈ b∈ PROPN iajs-3181 32 16 √[𝐾 √[𝐾 PROPN iajs-3181 32 17 : : PUNCT iajs-3181 33 1 𝑁 𝑁 PROPN iajs-3181 33 2 ] ] SYM iajs-3181 33 3 , , PUNCT iajs-3181 33 4 so so ADV iajs-3181 33 5 √[𝐾 √[𝐾 NOUN iajs-3181 33 6 : : PUNCT iajs-3181 33 7 𝑁 𝑁 PROPN iajs-3181 33 8 ] ] PUNCT iajs-3181 33 9 is be AUX iajs-3181 33 10 a a DET iajs-3181 33 11 prime prime ADJ iajs-3181 33 12 ideal ideal NOUN iajs-3181 33 13 . . PUNCT iajs-3181 34 1 recall recall VERB iajs-3181 34 2 that that SCONJ iajs-3181 34 3 whenever whenever SCONJ iajs-3181 34 4 b b PROPN iajs-3181 34 5 β‰… β‰… PROPN iajs-3181 34 6 b b PROPN iajs-3181 34 7 / / SYM iajs-3181 34 8 n n PROPN iajs-3181 34 9 for for ADP iajs-3181 34 10 all all DET iajs-3181 34 11 proper proper ADJ iajs-3181 34 12 submodule submodule PROPN iajs-3181 34 13 nof nof PROPN iajs-3181 34 14 modules modules PROPN iajs-3181 34 15 b b PROPN iajs-3181 34 16 , , PUNCT iajs-3181 34 17 then then ADV iajs-3181 34 18 we we PRON iajs-3181 34 19 said say VERB iajs-3181 34 20 that that SCONJ iajs-3181 34 21 anonsimple anonsimple PROPN iajs-3181 34 22 a a DET iajs-3181 34 23 - - PUNCT iajs-3181 34 24 module module NOUN iajs-3181 34 25 b b NOUN iajs-3181 34 26 anti anti ADJ iajs-3181 34 27 - - ADJ iajs-3181 34 28 hopfian hopfian ADJ iajs-3181 34 29 module module NOUN iajs-3181 34 30 [ [ X iajs-3181 34 31 4 4 NUM iajs-3181 34 32 , , PUNCT iajs-3181 34 33 5 5 NUM iajs-3181 34 34 ] ] PUNCT iajs-3181 34 35 . . PUNCT iajs-3181 35 1 proposition proposition NOUN iajs-3181 35 2 ( ( PUNCT iajs-3181 35 3 2.4 2.4 NUM iajs-3181 35 4 ) ) PUNCT iajs-3181 35 5 suppose suppose VERB iajs-3181 35 6 that that SCONJ iajs-3181 35 7 b b PROPN iajs-3181 35 8 is be AUX iajs-3181 35 9 an an DET iajs-3181 35 10 anti anti ADJ iajs-3181 35 11 - - ADJ iajs-3181 35 12 hopfian hopfian ADJ iajs-3181 35 13 quasi quasi ADJ iajs-3181 35 14 - - ADJ iajs-3181 35 15 prime prime ADJ iajs-3181 35 16 a a DET iajs-3181 35 17 - - PUNCT iajs-3181 35 18 module module NOUN iajs-3181 35 19 , , PUNCT iajs-3181 35 20 then then ADV iajs-3181 35 21 b b PROPN iajs-3181 35 22 is be AUX iajs-3181 35 23 quasi quasi ADJ iajs-3181 35 24 - - NOUN iajs-3181 35 25 semiprime semiprime NOUN iajs-3181 35 26 . . PUNCT iajs-3181 36 1 proof proof NOUN iajs-3181 36 2 : : PUNCT iajs-3181 36 3 since since SCONJ iajs-3181 36 4 b b PROPN iajs-3181 36 5 is be AUX iajs-3181 36 6 an an DET iajs-3181 36 7 anti anti ADJ iajs-3181 36 8 - - ADJ iajs-3181 36 9 hopfian hopfian ADJ iajs-3181 36 10 module module NOUN iajs-3181 36 11 , , PUNCT iajs-3181 36 12 then then ADV iajs-3181 36 13 b b PROPN iajs-3181 36 14 β‰… β‰… PROPN iajs-3181 36 15 b b PROPN iajs-3181 36 16 / / SYM iajs-3181 36 17 n n PROPN iajs-3181 36 18 for for ADP iajs-3181 36 19 n n VERB iajs-3181 36 20 be be AUX iajs-3181 36 21 a a DET iajs-3181 36 22 proper proper ADJ iajs-3181 36 23 submodule submodule NOUN iajs-3181 36 24 of of ADP iajs-3181 36 25 b b PROPN iajs-3181 36 26 , , PUNCT iajs-3181 36 27 so so SCONJ iajs-3181 36 28 there there PRON iajs-3181 36 29 exists exist VERB iajs-3181 36 30 an an DET iajs-3181 36 31 isomorphism isomorphism NOUN iajs-3181 36 32 function function NOUN iajs-3181 37 1 f f X iajs-3181 37 2 : : PUNCT iajs-3181 37 3 bβ†’ bβ†’ PROPN iajs-3181 37 4 b b X iajs-3181 37 5 / / SYM iajs-3181 37 6 n n CCONJ iajs-3181 37 7 ; ; PUNCT iajs-3181 37 8 f(b)=b+n f(b)=b+n VERB iajs-3181 37 9 for for ADP iajs-3181 37 10 each each DET iajs-3181 37 11 b b PROPN iajs-3181 37 12 ∈ ∈ PROPN iajs-3181 37 13 b b PROPN iajs-3181 37 14 , , PUNCT iajs-3181 37 15 so so SCONJ iajs-3181 37 16 it it PRON iajs-3181 37 17 is be AUX iajs-3181 37 18 easy easy ADJ iajs-3181 37 19 to to PART iajs-3181 37 20 check check VERB iajs-3181 37 21 that that PRON iajs-3181 37 22 ihjpas ihjpa NOUN iajs-3181 37 23 . . PUNCT iajs-3181 38 1 36 36 NUM iajs-3181 38 2 ( ( PUNCT iajs-3181 38 3 4 4 NUM iajs-3181 38 4 ) ) PUNCT iajs-3181 38 5 2023 2023 NUM iajs-3181 38 6 379 379 NUM iajs-3181 38 7 annab annab NOUN iajs-3181 38 8 = = NOUN iajs-3181 38 9 annab annab NOUN iajs-3181 38 10 / / SYM iajs-3181 38 11 n n CCONJ iajs-3181 38 12 , , PUNCT iajs-3181 38 13 then then ADV iajs-3181 38 14 by by ADP iajs-3181 38 15 [ [ PUNCT iajs-3181 38 16 6 6 NUM iajs-3181 38 17 ] ] PUNCT iajs-3181 38 18 , , PUNCT iajs-3181 38 19 every every DET iajs-3181 38 20 anti anti ADJ iajs-3181 38 21 - - ADJ iajs-3181 38 22 hopfian hopfian ADJ iajs-3181 38 23 a a PRON iajs-3181 38 24 - - PUNCT iajs-3181 38 25 module module NOUN iajs-3181 38 26 is be AUX iajs-3181 38 27 a a DET iajs-3181 38 28 coprime coprime ADJ iajs-3181 38 29 e e NOUN iajs-3181 38 30 - - NOUN iajs-3181 38 31 module module NOUN iajs-3181 38 32 , , PUNCT iajs-3181 38 33 where where SCONJ iajs-3181 38 34 e e NOUN iajs-3181 38 35 = = NOUN iajs-3181 38 36 end(w end(w VERB iajs-3181 38 37 ) ) PUNCT iajs-3181 38 38 and and CCONJ iajs-3181 38 39 by by ADP iajs-3181 38 40 [ [ X iajs-3181 38 41 5 5 NUM iajs-3181 38 42 ] ] PUNCT iajs-3181 38 43 every every DET iajs-3181 38 44 f f PROPN iajs-3181 38 45 ∈ ∈ PROPN iajs-3181 38 46 e e PROPN iajs-3181 38 47 , , PUNCT iajs-3181 38 48 either either CCONJ iajs-3181 38 49 f=0 f=0 PROPN iajs-3181 38 50 or or CCONJ iajs-3181 38 51 f f PROPN iajs-3181 38 52 is be AUX iajs-3181 38 53 subjective subjective ADJ iajs-3181 38 54 , , PUNCT iajs-3181 38 55 thus thus ADV iajs-3181 38 56 f(b)=0 f(b)=0 NUM iajs-3181 38 57 or or CCONJ iajs-3181 38 58 f(b)=b f(b)=b NOUN iajs-3181 38 59 for for ADP iajs-3181 38 60 every every DET iajs-3181 38 61 w w PROPN iajs-3181 38 62 ∈ ∈ PROPN iajs-3181 38 63 w w NOUN iajs-3181 38 64 if if SCONJ iajs-3181 38 65 f(w)=0 f(w)=0 NOUN iajs-3181 38 66 implies imply VERB iajs-3181 38 67 w w NOUN iajs-3181 38 68 = = NOUN iajs-3181 38 69 n n PRON iajs-3181 38 70 which which PRON iajs-3181 38 71 is be AUX iajs-3181 38 72 a a DET iajs-3181 38 73 contradiction contradiction NOUN iajs-3181 38 74 , , PUNCT iajs-3181 38 75 so so ADV iajs-3181 38 76 f(w)=w f(w)=w PROPN iajs-3181 38 77 , , PUNCT iajs-3181 38 78 which which PRON iajs-3181 38 79 means mean VERB iajs-3181 38 80 annab=[n annab=[n ADJ iajs-3181 38 81 : : PUNCT iajs-3181 38 82 b b X iajs-3181 38 83 ] ] X iajs-3181 38 84 , , PUNCT iajs-3181 38 85 implies imply VERB iajs-3181 38 86 βˆšπ‘Žπ‘›π‘›π΄π΅ βˆšπ‘Žπ‘›π‘›π΄π΅ PROPN iajs-3181 38 87 = = SYM iajs-3181 38 88 βˆšπ‘Žπ‘›π‘›π‘ βˆšπ‘Žπ‘›π‘›π‘ NOUN iajs-3181 38 89 𝐡 𝐡 NOUN iajs-3181 38 90 if if SCONJ iajs-3181 38 91 anbn anbn ADP iajs-3181 38 92 ∈[n ∈[n PROPN iajs-3181 38 93 : : PUNCT iajs-3181 38 94 b b NOUN iajs-3181 38 95 ] ] X iajs-3181 38 96 , , PUNCT iajs-3181 38 97 then then ADV iajs-3181 38 98 anbn anbn PROPN iajs-3181 38 99 ∈ ∈ PROPN iajs-3181 38 100 annab annab NOUN iajs-3181 39 1 but but CCONJ iajs-3181 39 2 b b NOUN iajs-3181 39 3 is be AUX iajs-3181 39 4 a a DET iajs-3181 39 5 quasi quasi ADJ iajs-3181 39 6 - - ADJ iajs-3181 39 7 prime prime ADJ iajs-3181 39 8 a a DET iajs-3181 39 9 - - PUNCT iajs-3181 39 10 module module NOUN iajs-3181 39 11 , , PUNCT iajs-3181 39 12 so so ADV iajs-3181 39 13 by by ADP iajs-3181 39 14 [ [ X iajs-3181 39 15 3 3 NUM iajs-3181 39 16 ] ] PUNCT iajs-3181 39 17 implies imply VERB iajs-3181 39 18 annaw annaw NOUN iajs-3181 39 19 is be AUX iajs-3181 39 20 a a DET iajs-3181 39 21 prime prime ADJ iajs-3181 39 22 ideal ideal NOUN iajs-3181 39 23 , , PUNCT iajs-3181 39 24 so so CCONJ iajs-3181 39 25 either either CCONJ iajs-3181 39 26 an∈ an∈ ADP iajs-3181 39 27 annaw annaw NOUN iajs-3181 39 28 or or CCONJ iajs-3181 39 29 bn bn PROPN iajs-3181 39 30 ∈ ∈ PROPN iajs-3181 39 31 annab annab NOUN iajs-3181 39 32 . . PUNCT iajs-3181 40 1 thus thus ADV iajs-3181 40 2 , , PUNCT iajs-3181 40 3 either either CCONJ iajs-3181 40 4 a a DET iajs-3181 40 5 ∈ ∈ NOUN iajs-3181 40 6 √[𝑁 √[𝑁 PRON iajs-3181 40 7 : : PUNCT iajs-3181 40 8 𝐡 𝐡 NOUN iajs-3181 40 9 ] ] PUNCT iajs-3181 40 10 or or CCONJ iajs-3181 40 11 b∈ b∈ NOUN iajs-3181 40 12 √[𝑁 √[𝑁 NUM iajs-3181 40 13 : : PUNCT iajs-3181 40 14 𝐡 𝐡 NOUN iajs-3181 40 15 ] ] PUNCT iajs-3181 40 16 , , PUNCT iajs-3181 40 17 which which PRON iajs-3181 40 18 means mean VERB iajs-3181 40 19 b b NOUN iajs-3181 40 20 is be AUX iajs-3181 40 21 a a DET iajs-3181 40 22 quasi quasi NOUN iajs-3181 40 23 - - ADJ iajs-3181 40 24 semiprime semiprime ADJ iajs-3181 40 25 a a DET iajs-3181 40 26 - - PUNCT iajs-3181 40 27 module module NOUN iajs-3181 40 28 . . PUNCT iajs-3181 41 1 the the DET iajs-3181 41 2 condition condition NOUN iajs-3181 41 3 anti anti ADJ iajs-3181 41 4 - - ADJ iajs-3181 41 5 hopfian hopfian ADJ iajs-3181 41 6 we we PRON iajs-3181 41 7 can can AUX iajs-3181 41 8 not not PART iajs-3181 41 9 drop drop VERB iajs-3181 41 10 for for ADP iajs-3181 41 11 example example NOUN iajs-3181 41 12 : : PUNCT iajs-3181 41 13 z6 z6 PROPN iajs-3181 41 14 is be AUX iajs-3181 41 15 quasi quasi ADJ iajs-3181 41 16 - - ADJ iajs-3181 41 17 semiprime semiprime ADJ iajs-3181 41 18 a a DET iajs-3181 41 19 - - PUNCT iajs-3181 41 20 module module NOUN iajs-3181 41 21 by by ADP iajs-3181 41 22 ( ( PUNCT iajs-3181 41 23 2.2 2.2 NUM iajs-3181 41 24 ) ) PUNCT iajs-3181 41 25 , , PUNCT iajs-3181 41 26 while while SCONJ iajs-3181 41 27 it it PRON iajs-3181 41 28 is be AUX iajs-3181 41 29 not not PART iajs-3181 41 30 quasi quasi ADJ iajs-3181 41 31 - - NOUN iajs-3181 41 32 prime prime ADJ iajs-3181 41 33 by by ADP iajs-3181 41 34 [ [ X iajs-3181 41 35 3 3 NUM iajs-3181 41 36 ] ] PUNCT iajs-3181 41 37 , , PUNCT iajs-3181 41 38 and and CCONJ iajs-3181 41 39 z6 z6 PROPN iajs-3181 41 40 is be AUX iajs-3181 41 41 not not PART iajs-3181 41 42 anti anti ADJ iajs-3181 41 43 - - ADJ iajs-3181 41 44 hopfain hopfain ADJ iajs-3181 41 45 by by ADP iajs-3181 41 46 [ [ X iajs-3181 41 47 6 6 NUM iajs-3181 41 48 ] ] PUNCT iajs-3181 41 49 . . PUNCT iajs-3181 42 1 prorosition prorosition NOUN iajs-3181 42 2 ( ( PUNCT iajs-3181 42 3 2.5 2.5 NUM iajs-3181 42 4 ) ) PUNCT iajs-3181 42 5 suppose suppose VERB iajs-3181 42 6 b b NOUN iajs-3181 42 7 is be AUX iajs-3181 42 8 a a DET iajs-3181 42 9 coprime coprime NOUN iajs-3181 42 10 a a DET iajs-3181 42 11 - - PUNCT iajs-3181 42 12 module module NOUN iajs-3181 42 13 of of ADP iajs-3181 42 14 quasi quasi NOUN iajs-3181 42 15 - - ADJ iajs-3181 42 16 prime prime ADJ iajs-3181 42 17 , , PUNCT iajs-3181 42 18 then then ADV iajs-3181 42 19 b b NOUN iajs-3181 42 20 is be AUX iajs-3181 42 21 quasi quasi ADJ iajs-3181 42 22 - - ADJ iajs-3181 42 23 semiprime semiprime ADJ iajs-3181 42 24 a a DET iajs-3181 42 25 - - PUNCT iajs-3181 42 26 module module NOUN iajs-3181 42 27 . . PUNCT iajs-3181 43 1 proof proof NOUN iajs-3181 43 2 let let VERB iajs-3181 43 3 b b X iajs-3181 43 4 be be AUX iajs-3181 43 5 a a DET iajs-3181 43 6 quasi quasi ADJ iajs-3181 43 7 - - ADJ iajs-3181 43 8 prime prime ADJ iajs-3181 43 9 a a DET iajs-3181 43 10 - - PUNCT iajs-3181 43 11 module module NOUN iajs-3181 43 12 , , PUNCT iajs-3181 43 13 then then ADV iajs-3181 43 14 by by ADP iajs-3181 43 15 [ [ X iajs-3181 43 16 3 3 NUM iajs-3181 43 17 ] ] PUNCT iajs-3181 43 18 , , PUNCT iajs-3181 43 19 annab annab PROPN iajs-3181 43 20 is be AUX iajs-3181 43 21 a a DET iajs-3181 43 22 prime prime ADJ iajs-3181 43 23 ideal ideal NOUN iajs-3181 43 24 , , PUNCT iajs-3181 43 25 but but CCONJ iajs-3181 43 26 w w NOUN iajs-3181 43 27 is be AUX iajs-3181 43 28 a a DET iajs-3181 43 29 coprime coprime ADJ iajs-3181 43 30 amodule amodule NOUN iajs-3181 43 31 , , PUNCT iajs-3181 43 32 so so ADV iajs-3181 43 33 annab annab PROPN iajs-3181 43 34 / / SYM iajs-3181 43 35 n n PROPN iajs-3181 43 36 is be AUX iajs-3181 43 37 a a DET iajs-3181 43 38 prime prime ADJ iajs-3181 43 39 ideal ideal NOUN iajs-3181 43 40 for for ADP iajs-3181 43 41 each each DET iajs-3181 43 42 non non ADJ iajs-3181 43 43 - - ADJ iajs-3181 43 44 zero zero NUM iajs-3181 43 45 submodule submodule NOUN iajs-3181 43 46 n n PROPN iajs-3181 43 47 of of ADP iajs-3181 43 48 b b PROPN iajs-3181 43 49 , , PUNCT iajs-3181 43 50 which which PRON iajs-3181 43 51 means√[𝑁 means√[𝑁 X iajs-3181 43 52 : : PUNCT iajs-3181 43 53 𝐡 𝐡 NOUN iajs-3181 43 54 ] ] PUNCT iajs-3181 43 55 is be AUX iajs-3181 43 56 a a DET iajs-3181 43 57 prime prime ADJ iajs-3181 43 58 ideal ideal NOUN iajs-3181 43 59 . . PUNCT iajs-3181 44 1 thus thus ADV iajs-3181 44 2 , , PUNCT iajs-3181 44 3 w w PROPN iajs-3181 44 4 is be AUX iajs-3181 44 5 a a DET iajs-3181 44 6 quasi quasi NOUN iajs-3181 44 7 - - ADJ iajs-3181 44 8 semiprime semiprime ADJ iajs-3181 44 9 a a DET iajs-3181 44 10 - - PUNCT iajs-3181 44 11 module module NOUN iajs-3181 44 12 . . PUNCT iajs-3181 45 1 recall recall VERB iajs-3181 45 2 that that SCONJ iajs-3181 45 3 an an DET iajs-3181 45 4 ideal ideal NOUN iajs-3181 45 5 k k PROPN iajs-3181 45 6 of of ADP iajs-3181 45 7 the the DET iajs-3181 45 8 ring ring NOUN iajs-3181 45 9 a a PRON iajs-3181 45 10 is be AUX iajs-3181 45 11 called call VERB iajs-3181 45 12 nil nil ADV iajs-3181 45 13 radical radical ADJ iajs-3181 45 14 and and CCONJ iajs-3181 45 15 denoted denote VERB iajs-3181 45 16 by√𝐾 by√𝐾 NOUN iajs-3181 45 17 , , PUNCT iajs-3181 45 18 and and CCONJ iajs-3181 45 19 is be AUX iajs-3181 45 20 defined define VERB iajs-3181 45 21 by by ADP iajs-3181 45 22 : : PUNCT iajs-3181 45 23 √𝐾 √𝐾 PROPN iajs-3181 46 1 = = SYM iajs-3181 46 2 { { PUNCT iajs-3181 46 3 a∈ a∈ PROPN iajs-3181 46 4 a a NOUN iajs-3181 46 5 ; ; PUNCT iajs-3181 47 1 an∈ an∈ ADP iajs-3181 47 2 k k X iajs-3181 47 3 , , PUNCT iajs-3181 47 4 for for ADP iajs-3181 47 5 some some DET iajs-3181 47 6 n n DET iajs-3181 47 7 z+}[8 z+}[8 NOUN iajs-3181 47 8 ] ] PUNCT iajs-3181 47 9 . . PUNCT iajs-3181 48 1 not not PART iajs-3181 48 2 ( ( PUNCT iajs-3181 48 3 2.6 2.6 NUM iajs-3181 48 4 ) ) PUNCT iajs-3181 48 5 suppose suppose VERB iajs-3181 48 6 c c NOUN iajs-3181 48 7 is be AUX iajs-3181 48 8 a a DET iajs-3181 48 9 ring ring NOUN iajs-3181 48 10 where where SCONJ iajs-3181 48 11 every every DET iajs-3181 48 12 ideal ideal NOUN iajs-3181 48 13 is be AUX iajs-3181 48 14 nil nil ADJ iajs-3181 48 15 radical radical ADJ iajs-3181 48 16 , , PUNCT iajs-3181 48 17 which which PRON iajs-3181 48 18 we we PRON iajs-3181 48 19 call call VERB iajs-3181 48 20 cosemiprime cosemiprime NOUN iajs-3181 48 21 ring ring NOUN iajs-3181 48 22 . . PUNCT iajs-3181 49 1 theorem theorem ADJ iajs-3181 49 2 m m PROPN iajs-3181 49 3 ( ( PUNCT iajs-3181 49 4 2.7 2.7 NUM iajs-3181 49 5 ) ) PUNCT iajs-3181 49 6 suppose suppose VERB iajs-3181 49 7 that that SCONJ iajs-3181 49 8 b b PROPN iajs-3181 49 9 is be AUX iajs-3181 49 10 a a DET iajs-3181 49 11 coprime coprime ADJ iajs-3181 49 12 c c NOUN iajs-3181 49 13 - - NOUN iajs-3181 49 14 module module NOUN iajs-3181 49 15 . . PUNCT iajs-3181 50 1 the the DET iajs-3181 50 2 following follow VERB iajs-3181 50 3 statements statement NOUN iajs-3181 50 4 are be AUX iajs-3181 50 5 equivalent equivalent ADJ iajs-3181 50 6 : : PUNCT iajs-3181 50 7 1 1 NUM iajs-3181 50 8 ) ) PUNCT iajs-3181 50 9 b b NOUN iajs-3181 50 10 is be AUX iajs-3181 50 11 a a DET iajs-3181 50 12 quasi quasi ADJ iajs-3181 50 13 - - ADJ iajs-3181 50 14 prime prime ADJ iajs-3181 50 15 module module NOUN iajs-3181 50 16 . . PUNCT iajs-3181 51 1 2 2 NUM iajs-3181 51 2 ) ) PUNCT iajs-3181 51 3 b b NOUN iajs-3181 51 4 is be AUX iajs-3181 51 5 a a DET iajs-3181 51 6 quasi quasi ADJ iajs-3181 51 7 - - ADJ iajs-3181 51 8 semiprime semiprime ADJ iajs-3181 51 9 module module NOUN iajs-3181 51 10 . . PUNCT iajs-3181 52 1 proof proof NOUN iajs-3181 52 2 : : PUNCT iajs-3181 52 3 1 1 X iajs-3181 52 4 ) ) PUNCT iajs-3181 52 5 β†’ β†’ X iajs-3181 52 6 ( ( PUNCT iajs-3181 52 7 2 2 NUM iajs-3181 52 8 ) ) PUNCT iajs-3181 52 9 ( ( PUNCT iajs-3181 52 10 by by ADP iajs-3181 52 11 theorem theorem NOUN iajs-3181 52 12 ( ( PUNCT iajs-3181 52 13 2 2 NUM iajs-3181 52 14 - - SYM iajs-3181 52 15 5 5 NUM iajs-3181 52 16 ) ) PUNCT iajs-3181 52 17 ) ) PUNCT iajs-3181 52 18 ( ( PUNCT iajs-3181 52 19 2 2 X iajs-3181 52 20 ) ) PUNCT iajs-3181 52 21 β†’ β†’ SYM iajs-3181 52 22 ( ( PUNCT iajs-3181 52 23 1 1 X iajs-3181 52 24 ) ) PUNCT iajs-3181 52 25 for for ADP iajs-3181 52 26 each each DET iajs-3181 52 27 a a NOUN iajs-3181 52 28 , , PUNCT iajs-3181 52 29 b b X iajs-3181 52 30 ∈ ∈ PROPN iajs-3181 52 31 c c NOUN iajs-3181 52 32 if if SCONJ iajs-3181 52 33 ab ab PROPN iajs-3181 52 34 annc annc VERB iajs-3181 52 35 n n CCONJ iajs-3181 52 36 , , PUNCT iajs-3181 52 37 then then ADV iajs-3181 52 38 abn=0 abn=0 PROPN iajs-3181 52 39 implies imply VERB iajs-3181 52 40 ab ab PROPN iajs-3181 52 41 ∈ ∈ PROPN iajs-3181 53 1 [ [ X iajs-3181 53 2 ( ( PUNCT iajs-3181 53 3 0):n],which 0):n],which NOUN iajs-3181 53 4 means mean VERB iajs-3181 53 5 ab∈ ab∈ PROPN iajs-3181 53 6 √[(0 √[(0 PROPN iajs-3181 53 7 ): ): PUNCT iajs-3181 53 8 𝑁 𝑁 PROPN iajs-3181 53 9 ] ] X iajs-3181 53 10 , , PUNCT iajs-3181 53 11 but but CCONJ iajs-3181 53 12 w w NOUN iajs-3181 53 13 is be AUX iajs-3181 53 14 quasi quasi ADJ iajs-3181 53 15 - - ADJ iajs-3181 53 16 semiprime semiprime ADJ iajs-3181 53 17 module module NOUN iajs-3181 53 18 so so ADV iajs-3181 53 19 either either CCONJ iajs-3181 53 20 a∈ a∈ PROPN iajs-3181 53 21 √[(0 √[(0 PROPN iajs-3181 53 22 ): ): PUNCT iajs-3181 54 1 𝑁 𝑁 PROPN iajs-3181 54 2 ] ] PUNCT iajs-3181 54 3 or or CCONJ iajs-3181 54 4 b∈ b∈ PROPN iajs-3181 54 5 √[(0 √[(0 PROPN iajs-3181 54 6 ): ): PUNCT iajs-3181 54 7 𝑁 𝑁 PROPN iajs-3181 54 8 ] ] PUNCT iajs-3181 54 9 implies imply VERB iajs-3181 54 10 either either CCONJ iajs-3181 54 11 a∈ a∈ PROPN iajs-3181 54 12 anncn anncn NOUN iajs-3181 54 13 or or CCONJ iajs-3181 54 14 b∈ b∈ PROPN iajs-3181 54 15 anncn anncn NOUN iajs-3181 54 16 . . PUNCT iajs-3181 55 1 thus thus ADV iajs-3181 55 2 , , PUNCT iajs-3181 55 3 b b PROPN iajs-3181 55 4 is be AUX iajs-3181 55 5 a a DET iajs-3181 55 6 quasi quasi ADJ iajs-3181 55 7 - - ADJ iajs-3181 55 8 prime prime ADJ iajs-3181 55 9 c c NOUN iajs-3181 55 10 - - PUNCT iajs-3181 55 11 module module NOUN iajs-3181 55 12 . . PUNCT iajs-3181 56 1 theorem theorem NOUN iajs-3181 56 2 ( ( PUNCT iajs-3181 56 3 2.8 2.8 NUM iajs-3181 56 4 ) ) PUNCT iajs-3181 56 5 let let VERB iajs-3181 56 6 b b X iajs-3181 56 7 be be AUX iajs-3181 56 8 a a DET iajs-3181 56 9 cyclic cyclic ADJ iajs-3181 56 10 coprime coprime NOUN iajs-3181 56 11 c c NOUN iajs-3181 56 12 - - PUNCT iajs-3181 56 13 module module NOUN iajs-3181 56 14 , , PUNCT iajs-3181 56 15 then then ADV iajs-3181 56 16 the the DET iajs-3181 56 17 following following ADJ iajs-3181 56 18 statements statement NOUN iajs-3181 56 19 are be AUX iajs-3181 56 20 equivalent equivalent ADJ iajs-3181 56 21 : : PUNCT iajs-3181 56 22 1b 1b NUM iajs-3181 56 23 is be AUX iajs-3181 56 24 a a DET iajs-3181 56 25 quasi quasi ADJ iajs-3181 56 26 - - ADJ iajs-3181 56 27 prime prime ADJ iajs-3181 56 28 c c NOUN iajs-3181 56 29 - - PUNCT iajs-3181 56 30 module module NOUN iajs-3181 56 31 . . PUNCT iajs-3181 57 1 2b 2b NUM iajs-3181 57 2 is be AUX iajs-3181 57 3 a a DET iajs-3181 57 4 quasi quasi NOUN iajs-3181 57 5 - - ADJ iajs-3181 57 6 semiprime semiprime ADJ iajs-3181 57 7 c c NOUN iajs-3181 57 8 - - PUNCT iajs-3181 57 9 module module NOUN iajs-3181 57 10 . . PUNCT iajs-3181 58 1 3anncb 3anncb NUM iajs-3181 58 2 is be AUX iajs-3181 58 3 a a DET iajs-3181 58 4 prime prime ADJ iajs-3181 58 5 ideal ideal NOUN iajs-3181 58 6 . . PUNCT iajs-3181 59 1 ihjpas ihjpas PROPN iajs-3181 59 2 . . PUNCT iajs-3181 60 1 36 36 NUM iajs-3181 60 2 ( ( PUNCT iajs-3181 60 3 4 4 NUM iajs-3181 60 4 ) ) PUNCT iajs-3181 60 5 2023 2023 NUM iajs-3181 60 6 380 380 NUM iajs-3181 60 7 proof proof NOUN iajs-3181 60 8 : : PUNCT iajs-3181 60 9 1β†’ 1β†’ NUM iajs-3181 60 10 2 2 NUM iajs-3181 60 11 ( ( PUNCT iajs-3181 60 12 by by ADP iajs-3181 60 13 theorem theorem NOUN iajs-3181 60 14 ( ( PUNCT iajs-3181 60 15 2.5 2.5 NUM iajs-3181 60 16 ) ) PUNCT iajs-3181 60 17 ) ) PUNCT iajs-3181 60 18 2 2 NUM iajs-3181 60 19 β†’ β†’ SYM iajs-3181 60 20 3 3 NUM iajs-3181 60 21 if if SCONJ iajs-3181 60 22 ab ab PROPN iajs-3181 60 23 ∈ ∈ PROPN iajs-3181 60 24 anncb anncb PROPN iajs-3181 60 25 , , PUNCT iajs-3181 60 26 then then ADV iajs-3181 60 27 ab∈ ab∈ NOUN iajs-3181 60 28 βˆšπ‘Žπ‘›π‘›π΅. βˆšπ‘Žπ‘›π‘›π΅. NOUN iajs-3181 60 29 thus thus ADV iajs-3181 60 30 , , PUNCT iajs-3181 60 31 ab∈ ab∈ NOUN iajs-3181 60 32 √[𝑁 √[𝑁 NUM iajs-3181 60 33 : : PUNCT iajs-3181 60 34 𝐡 𝐡 NOUN iajs-3181 60 35 ] ] PUNCT iajs-3181 60 36 for for ADP iajs-3181 60 37 every every DET iajs-3181 60 38 submodule submodule NOUN iajs-3181 60 39 n n PROPN iajs-3181 60 40 of of ADP iajs-3181 60 41 b b NUM iajs-3181 60 42 which which PRON iajs-3181 60 43 means mean VERB iajs-3181 60 44 anbn anbn PROPN iajs-3181 60 45 ∈ ∈ PROPN iajs-3181 60 46 [ [ X iajs-3181 60 47 n n NUM iajs-3181 60 48 : : PUNCT iajs-3181 60 49 b b NOUN iajs-3181 60 50 ] ] X iajs-3181 60 51 , , PUNCT iajs-3181 60 52 but but CCONJ iajs-3181 60 53 b b X iajs-3181 60 54 is be AUX iajs-3181 60 55 a a DET iajs-3181 60 56 quasi quasi NOUN iajs-3181 60 57 - - ADJ iajs-3181 60 58 semiprime semiprime ADJ iajs-3181 60 59 c c NOUN iajs-3181 60 60 - - PUNCT iajs-3181 60 61 module module NOUN iajs-3181 60 62 , , PUNCT iajs-3181 60 63 so so SCONJ iajs-3181 60 64 either either CCONJ iajs-3181 60 65 an∈ an∈ VERB iajs-3181 61 1 [ [ X iajs-3181 61 2 n n CCONJ iajs-3181 61 3 : : PUNCT iajs-3181 61 4 b b NOUN iajs-3181 61 5 ] ] PUNCT iajs-3181 61 6 or or CCONJ iajs-3181 61 7 bn bn ADP iajs-3181 61 8 ∈ ∈ PROPN iajs-3181 62 1 [ [ X iajs-3181 62 2 n n X iajs-3181 62 3 : : PUNCT iajs-3181 62 4 b b NOUN iajs-3181 62 5 ] ] X iajs-3181 62 6 , , PUNCT iajs-3181 62 7 but but CCONJ iajs-3181 62 8 b b NOUN iajs-3181 62 9 is be AUX iajs-3181 62 10 coprime coprime ADJ iajs-3181 62 11 by by ADP iajs-3181 62 12 [ [ PUNCT iajs-3181 62 13 9 9 NUM iajs-3181 62 14 ] ] PUNCT iajs-3181 62 15 implies imply VERB iajs-3181 62 16 either either CCONJ iajs-3181 62 17 anb=0 anb=0 NOUN iajs-3181 62 18 or or CCONJ iajs-3181 62 19 bnb=0 bnb=0 NOUN iajs-3181 62 20 , , PUNCT iajs-3181 62 21 which which PRON iajs-3181 62 22 means mean VERB iajs-3181 62 23 either either CCONJ iajs-3181 62 24 a∈ a∈ PROPN iajs-3181 62 25 βˆšπ‘Žπ‘›π‘›π΅ βˆšπ‘Žπ‘›π‘›π΅ PROPN iajs-3181 62 26 or or CCONJ iajs-3181 62 27 b∈ b∈ PROPN iajs-3181 62 28 βˆšπ‘Žπ‘›π‘›π΅.thus βˆšπ‘Žπ‘›π‘›π΅.thus PROPN iajs-3181 62 29 , , PUNCT iajs-3181 62 30 either either CCONJ iajs-3181 62 31 a∈ a∈ PROPN iajs-3181 62 32 anncw anncw PROPN iajs-3181 62 33 or or CCONJ iajs-3181 62 34 b∈ b∈ PROPN iajs-3181 62 35 anncb anncb PROPN iajs-3181 62 36 . . PROPN iajs-3181 63 1 3 3 NUM iajs-3181 63 2 β†’ β†’ SYM iajs-3181 63 3 1 1 NUM iajs-3181 63 4 by by ADP iajs-3181 63 5 [ [ X iajs-3181 63 6 3 3 NUM iajs-3181 63 7 ] ] PUNCT iajs-3181 63 8 implies imply VERB iajs-3181 63 9 the the DET iajs-3181 63 10 result result NOUN iajs-3181 63 11 . . PUNCT iajs-3181 64 1 proposition proposition NOUN iajs-3181 64 2 ( ( PUNCT iajs-3181 64 3 2.9 2.9 NUM iajs-3181 64 4 ) ) PUNCT iajs-3181 64 5 suppose suppose VERB iajs-3181 64 6 that that SCONJ iajs-3181 64 7 b b PROPN iajs-3181 64 8 is be AUX iajs-3181 64 9 an an DET iajs-3181 64 10 a a DET iajs-3181 64 11 - - PUNCT iajs-3181 64 12 module module NOUN iajs-3181 64 13 and and CCONJ iajs-3181 64 14 j j PROPN iajs-3181 64 15 is be AUX iajs-3181 64 16 an an DET iajs-3181 64 17 ideal ideal NOUN iajs-3181 64 18 of of ADP iajs-3181 64 19 a a DET iajs-3181 64 20 which which PRON iajs-3181 64 21 that that PRON iajs-3181 64 22 is be AUX iajs-3181 64 23 contained contain VERB iajs-3181 64 24 in in ADP iajs-3181 64 25 annab annab PROPN iajs-3181 64 26 / / SYM iajs-3181 64 27 n n PROPN iajs-3181 64 28 where where SCONJ iajs-3181 64 29 n n X iajs-3181 64 30 is be AUX iajs-3181 64 31 a a DET iajs-3181 64 32 submodule submodule NOUN iajs-3181 64 33 of of ADP iajs-3181 64 34 b. b. PROPN iajs-3181 64 35 then then ADV iajs-3181 64 36 , , PUNCT iajs-3181 64 37 b b PROPN iajs-3181 64 38 is be AUX iajs-3181 64 39 a a DET iajs-3181 64 40 quasi quasi NOUN iajs-3181 64 41 - - ADJ iajs-3181 64 42 semiprime semiprime ADJ iajs-3181 64 43 a a DET iajs-3181 64 44 - - PUNCT iajs-3181 64 45 module module NOUN iajs-3181 64 46 ⟷ ⟷ NUM iajs-3181 64 47 b b NOUN iajs-3181 64 48 is be AUX iajs-3181 64 49 a a DET iajs-3181 64 50 quasi quasi NOUN iajs-3181 64 51 - - ADJ iajs-3181 64 52 semiprime semiprime ADJ iajs-3181 64 53 a a DET iajs-3181 64 54 / / SYM iajs-3181 64 55 jmodule jmodule NOUN iajs-3181 64 56 . . PUNCT iajs-3181 65 1 proof proof NOUN iajs-3181 65 2 to to PART iajs-3181 65 3 show show VERB iajs-3181 65 4 b b PROPN iajs-3181 65 5 is be AUX iajs-3181 65 6 quasi quasi ADJ iajs-3181 65 7 - - ADJ iajs-3181 65 8 semiprime semiprime ADJ iajs-3181 65 9 a\j a\j NOUN iajs-3181 65 10 - - PUNCT iajs-3181 65 11 module module NOUN iajs-3181 65 12 if if SCONJ iajs-3181 65 13 ( ( PUNCT iajs-3181 65 14 a1+j)(a2+j)∈ a1+j)(a2+j)∈ PROPN iajs-3181 65 15 √[𝑁:𝐴/𝐽 √[𝑁:𝐴/𝐽 PROPN iajs-3181 65 16 𝐡 𝐡 PROPN iajs-3181 65 17 ] ] PUNCT iajs-3181 65 18 , , PUNCT iajs-3181 65 19 where where SCONJ iajs-3181 65 20 a1+j a1+j NOUN iajs-3181 65 21 , , PUNCT iajs-3181 65 22 a2+j∈ a2+j∈ PROPN iajs-3181 65 23 b b PROPN iajs-3181 65 24 / / SYM iajs-3181 65 25 j j PROPN iajs-3181 65 26 , , PUNCT iajs-3181 65 27 then then ADV iajs-3181 65 28 ( ( PUNCT iajs-3181 65 29 a1a2+j)n∈ a1a2+j)n∈ X iajs-3181 65 30 [ [ X iajs-3181 65 31 n n CCONJ iajs-3181 65 32 : : PUNCT iajs-3181 65 33 a a PRON iajs-3181 65 34 / / SYM iajs-3181 65 35 ib ib NOUN iajs-3181 65 36 ] ] X iajs-3181 65 37 . . PUNCT iajs-3181 66 1 thus thus ADV iajs-3181 66 2 , , PUNCT iajs-3181 66 3 ( ( PUNCT iajs-3181 66 4 a1 a1 VERB iajs-3181 66 5 na2 na2 ADV iajs-3181 66 6 n+j)x=0 n+j)x=0 PROPN iajs-3181 66 7 for for ADP iajs-3181 66 8 all all DET iajs-3181 66 9 x∈ x∈ PROPN iajs-3181 66 10 anna anna PROPN iajs-3181 66 11 / / SYM iajs-3181 66 12 jb jb PROPN iajs-3181 66 13 / / SYM iajs-3181 66 14 n. n. PROPN iajs-3181 66 15 hence hence ADV iajs-3181 66 16 , , PUNCT iajs-3181 66 17 a1 a1 VERB iajs-3181 66 18 na2 na2 NOUN iajs-3181 66 19 nx=0 nx=0 PROPN iajs-3181 66 20 for for ADP iajs-3181 66 21 all all DET iajs-3181 66 22 x∈ x∈ PROPN iajs-3181 66 23 annab annab NOUN iajs-3181 66 24 / / SYM iajs-3181 66 25 n n NUM iajs-3181 66 26 which which PRON iajs-3181 66 27 means mean VERB iajs-3181 66 28 a1 a1 NOUN iajs-3181 66 29 na2 na2 ADV iajs-3181 66 30 n∈ n∈ NOUN iajs-3181 67 1 [ [ X iajs-3181 67 2 n n X iajs-3181 67 3 : : PUNCT iajs-3181 67 4 b b NOUN iajs-3181 67 5 ] ] X iajs-3181 67 6 , , PUNCT iajs-3181 67 7 so so CCONJ iajs-3181 67 8 a1a2 a1a2 ADP iajs-3181 67 9 ∈ ∈ PROPN iajs-3181 67 10 √[𝑁:𝐴 √[𝑁:𝐴 PROPN iajs-3181 67 11 𝐡 𝐡 PROPN iajs-3181 67 12 ] ] X iajs-3181 67 13 , , PUNCT iajs-3181 67 14 but but CCONJ iajs-3181 67 15 b b X iajs-3181 67 16 is be AUX iajs-3181 67 17 quasi quasi ADJ iajs-3181 67 18 - - ADJ iajs-3181 67 19 semiprime semiprime ADJ iajs-3181 67 20 a a DET iajs-3181 67 21 - - PUNCT iajs-3181 67 22 module module NOUN iajs-3181 67 23 , , PUNCT iajs-3181 67 24 which which PRON iajs-3181 67 25 implies imply VERB iajs-3181 67 26 either either CCONJ iajs-3181 67 27 a1 a1 PROPN iajs-3181 67 28 ∈ ∈ PROPN iajs-3181 67 29 √[𝑁:𝐴 √[𝑁:𝐴 PROPN iajs-3181 67 30 𝐡 𝐡 PROPN iajs-3181 67 31 ] ] PUNCT iajs-3181 67 32 or or CCONJ iajs-3181 67 33 a2∈ a2∈ PROPN iajs-3181 67 34 √[𝑁:𝐴 √[𝑁:𝐴 PROPN iajs-3181 67 35 𝐡 𝐡 PROPN iajs-3181 67 36 ] ] PUNCT iajs-3181 67 37 . . PUNCT iajs-3181 68 1 thus thus ADV iajs-3181 68 2 , , PUNCT iajs-3181 68 3 either either CCONJ iajs-3181 68 4 a1 a1 VERB iajs-3181 68 5 n n PRON iajs-3181 68 6 ∈ ∈ PROPN iajs-3181 69 1 [ [ X iajs-3181 69 2 n:𝐴b n:𝐴b X iajs-3181 69 3 ] ] X iajs-3181 69 4 or or CCONJ iajs-3181 69 5 a2 a2 PROPN iajs-3181 69 6 n n PRON iajs-3181 69 7 ∈[n:𝐴b ∈[n:𝐴b NUM iajs-3181 69 8 ] ] PUNCT iajs-3181 69 9 . . PUNCT iajs-3181 70 1 however however ADV iajs-3181 70 2 , , PUNCT iajs-3181 70 3 a1 a1 NOUN iajs-3181 70 4 n+i n+i NUM iajs-3181 70 5 ∈ ∈ PROPN iajs-3181 70 6 annab annab NOUN iajs-3181 70 7 / / SYM iajs-3181 70 8 n n PROPN iajs-3181 70 9 or or CCONJ iajs-3181 70 10 a2 a2 PROPN iajs-3181 70 11 n+j n+j PROPN iajs-3181 70 12 ∈ ∈ PROPN iajs-3181 70 13 ann ann PROPN iajs-3181 70 14 b b PROPN iajs-3181 70 15 / / SYM iajs-3181 70 16 n. n. PROPN iajs-3181 70 17 thus thus ADV iajs-3181 70 18 , , PUNCT iajs-3181 70 19 either either CCONJ iajs-3181 70 20 ( ( PUNCT iajs-3181 70 21 a1+i)∈ a1+i)∈ PROPN iajs-3181 70 22 √[𝑁:𝐴/𝐽 √[𝑁:𝐴/𝐽 PROPN iajs-3181 70 23 𝐡 𝐡 PROPN iajs-3181 70 24 ] ] PUNCT iajs-3181 70 25 or or CCONJ iajs-3181 70 26 ( ( PUNCT iajs-3181 70 27 a2+i)∈ a2+i)∈ PROPN iajs-3181 70 28 √[𝑁:𝐴/𝐽 √[𝑁:𝐴/𝐽 PROPN iajs-3181 70 29 𝐡 𝐡 PROPN iajs-3181 70 30 ] ] PUNCT iajs-3181 70 31 , , PUNCT iajs-3181 70 32 which which PRON iajs-3181 70 33 means mean VERB iajs-3181 70 34 b b NOUN iajs-3181 70 35 is be AUX iajs-3181 70 36 a a DET iajs-3181 70 37 quasi quasi ADJ iajs-3181 70 38 - - ADJ iajs-3181 70 39 semiprime semiprime ADJ iajs-3181 70 40 b b NOUN iajs-3181 70 41 / / SYM iajs-3181 70 42 i i PROPN iajs-3181 70 43 module module NOUN iajs-3181 70 44 . . PUNCT iajs-3181 71 1 conversely conversely ADV iajs-3181 71 2 , , PUNCT iajs-3181 71 3 if if SCONJ iajs-3181 71 4 b b NOUN iajs-3181 71 5 is be AUX iajs-3181 71 6 quasi quasi ADJ iajs-3181 71 7 - - ADJ iajs-3181 71 8 semiprime semiprime ADJ iajs-3181 71 9 a a DET iajs-3181 71 10 / / SYM iajs-3181 71 11 j j NOUN iajs-3181 71 12 - - NOUN iajs-3181 71 13 module module NOUN iajs-3181 71 14 , , PUNCT iajs-3181 71 15 let let VERB iajs-3181 71 16 n n PRON iajs-3181 71 17 be be AUX iajs-3181 71 18 a a DET iajs-3181 71 19 nonzero nonzero NOUN iajs-3181 71 20 a a DET iajs-3181 71 21 - - PUNCT iajs-3181 71 22 submodule submodule NOUN iajs-3181 71 23 of of ADP iajs-3181 71 24 b b PROPN iajs-3181 71 25 , , PUNCT iajs-3181 71 26 let let VERB iajs-3181 71 27 a1,a2∈ a1,a2∈ NUM iajs-3181 71 28 √[𝑁:𝐴 √[𝑁:𝐴 PROPN iajs-3181 71 29 𝐡 𝐡 PROPN iajs-3181 71 30 ] ] PUNCT iajs-3181 71 31 , , PUNCT iajs-3181 71 32 then then ADV iajs-3181 71 33 a1 a1 NOUN iajs-3181 71 34 n n PROPN iajs-3181 71 35 a2 a2 PROPN iajs-3181 71 36 nx=0 nx=0 PROPN iajs-3181 71 37 for for ADP iajs-3181 71 38 all all DET iajs-3181 71 39 x x SYM iajs-3181 71 40 ∈annab ∈annab NOUN iajs-3181 71 41 / / SYM iajs-3181 71 42 n.hence n.hence NOUN iajs-3181 71 43 ( ( PUNCT iajs-3181 71 44 a1 a1 PROPN iajs-3181 71 45 n+j)(a2 n+j)(a2 PROPN iajs-3181 71 46 n+j)x=0 n+j)x=0 VERB iajs-3181 71 47 for for ADP iajs-3181 71 48 all all DET iajs-3181 71 49 x∈ x∈ PROPN iajs-3181 71 50 anna anna PROPN iajs-3181 71 51 / / SYM iajs-3181 71 52 jw jw PROPN iajs-3181 71 53 / / SYM iajs-3181 71 54 n n PROPN iajs-3181 71 55 , , PUNCT iajs-3181 71 56 so(a1+j)(a2+j)∈ so(a1+j)(a2+j)∈ PROPN iajs-3181 71 57 √[𝑁:𝐴/𝐽 √[𝑁:𝐴/𝐽 PROPN iajs-3181 71 58 𝐡 𝐡 PROPN iajs-3181 71 59 ] ] PUNCT iajs-3181 71 60 , , PUNCT iajs-3181 71 61 whille whille PROPN iajs-3181 71 62 is be AUX iajs-3181 71 63 a a DET iajs-3181 71 64 prime prime ADJ iajs-3181 71 65 ideal ideal NOUN iajs-3181 71 66 , , PUNCT iajs-3181 71 67 so so SCONJ iajs-3181 71 68 either either ADV iajs-3181 71 69 ( ( PUNCT iajs-3181 71 70 a1+i)∈ a1+i)∈ PROPN iajs-3181 71 71 √[𝑁:𝐴/𝐽 √[𝑁:𝐴/𝐽 PROPN iajs-3181 71 72 𝐡 𝐡 PROPN iajs-3181 71 73 ] ] PUNCT iajs-3181 71 74 or or CCONJ iajs-3181 71 75 ( ( PUNCT iajs-3181 71 76 a2+i)∈ a2+i)∈ PROPN iajs-3181 71 77 √[𝑁:𝐴/𝐽 √[𝑁:𝐴/𝐽 PROPN iajs-3181 71 78 𝐡 𝐡 PROPN iajs-3181 71 79 ] ] PUNCT iajs-3181 71 80 .then .then VERB iajs-3181 71 81 we we PRON iajs-3181 71 82 get get VERB iajs-3181 71 83 either either CCONJ iajs-3181 71 84 a1 a1 VERB iajs-3181 71 85 n n CCONJ iajs-3181 71 86 x=0 x=0 PROPN iajs-3181 71 87 or or CCONJ iajs-3181 71 88 a2 a2 PROPN iajs-3181 72 1 nx=0 nx=0 PROPN iajs-3181 72 2 f0r f0r PROPN iajs-3181 72 3 each each DET iajs-3181 72 4 x x PROPN iajs-3181 72 5 ∈ ∈ PROPN iajs-3181 72 6 annab annab NOUN iajs-3181 72 7 / / SYM iajs-3181 72 8 n n CCONJ iajs-3181 72 9 , , PUNCT iajs-3181 72 10 so so SCONJ iajs-3181 72 11 either either CCONJ iajs-3181 72 12 a1∈ a1∈ PROPN iajs-3181 72 13 √[𝑁:𝐴 √[𝑁:𝐴 PROPN iajs-3181 72 14 𝐡 𝐡 PROPN iajs-3181 72 15 ] ] PUNCT iajs-3181 72 16 or or CCONJ iajs-3181 72 17 a2 a2 PROPN iajs-3181 72 18 ∈ ∈ PROPN iajs-3181 72 19 √[𝑁:𝐴 √[𝑁:𝐴 PROPN iajs-3181 72 20 𝐡 𝐡 PROPN iajs-3181 72 21 ] ] PUNCT iajs-3181 72 22 . . PUNCT iajs-3181 73 1 theorem theorem NOUN iajs-3181 73 2 ( ( PUNCT iajs-3181 73 3 2.10 2.10 NUM iajs-3181 73 4 ) ) PUNCT iajs-3181 73 5 suppose suppose VERB iajs-3181 73 6 b1 b1 NOUN iajs-3181 73 7 , , PUNCT iajs-3181 73 8 and and CCONJ iajs-3181 73 9 b2 b2 NOUN iajs-3181 73 10 are be AUX iajs-3181 73 11 two two NUM iajs-3181 73 12 a a DET iajs-3181 73 13 - - PUNCT iajs-3181 73 14 modules module NOUN iajs-3181 73 15 , , PUNCT iajs-3181 73 16 if if SCONJ iajs-3181 73 17 f f X iajs-3181 73 18 : : PUNCT iajs-3181 73 19 b1 b1 PROPN iajs-3181 73 20 β†’ β†’ SYM iajs-3181 73 21 b2 b2 NOUN iajs-3181 73 22 , , PUNCT iajs-3181 73 23 is be AUX iajs-3181 73 24 an an DET iajs-3181 73 25 epimorphism epimorphism NOUN iajs-3181 73 26 function function NOUN iajs-3181 73 27 , , PUNCT iajs-3181 73 28 then then ADV iajs-3181 73 29 if if SCONJ iajs-3181 73 30 b1 b1 NOUN iajs-3181 73 31 is be AUX iajs-3181 73 32 a a DET iajs-3181 73 33 quasi quasi ADJ iajs-3181 73 34 - - ADJ iajs-3181 73 35 semiprime semiprime ADJ iajs-3181 73 36 module module NOUN iajs-3181 73 37 , , PUNCT iajs-3181 73 38 then then ADV iajs-3181 73 39 b2 b2 NOUN iajs-3181 73 40 is be AUX iajs-3181 73 41 a a DET iajs-3181 73 42 quasi quasi NOUN iajs-3181 73 43 - - ADJ iajs-3181 73 44 semiprime semiprime ADJ iajs-3181 73 45 a a DET iajs-3181 73 46 - - PUNCT iajs-3181 73 47 module module NOUN iajs-3181 73 48 . . PUNCT iajs-3181 74 1 proof proof NOUN iajs-3181 74 2 : : PUNCT iajs-3181 74 3 since since SCONJ iajs-3181 74 4 b1 b1 NOUN iajs-3181 74 5 is be AUX iajs-3181 74 6 a a DET iajs-3181 74 7 quasi quasi NOUN iajs-3181 74 8 - - ADJ iajs-3181 74 9 semiprime semiprime ADJ iajs-3181 74 10 a a DET iajs-3181 74 11 - - PUNCT iajs-3181 74 12 module module NOUN iajs-3181 74 13 , , PUNCT iajs-3181 74 14 so so CCONJ iajs-3181 74 15 if if SCONJ iajs-3181 74 16 anbnb1βŠ† anbnb1βŠ† PROPN iajs-3181 74 17 n1 n1 NOUN iajs-3181 74 18 for for ADP iajs-3181 74 19 each each DET iajs-3181 74 20 a a PRON iajs-3181 74 21 , , PUNCT iajs-3181 74 22 b∈ b∈ PROPN iajs-3181 74 23 a a NOUN iajs-3181 74 24 , , PUNCT iajs-3181 74 25 then then ADV iajs-3181 74 26 either either CCONJ iajs-3181 74 27 anb1βŠ† anb1βŠ† ADP iajs-3181 74 28 n n PRON iajs-3181 74 29 or or CCONJ iajs-3181 74 30 bnb1βŠ† bnb1βŠ† ADP iajs-3181 74 31 n. n. NOUN iajs-3181 74 32 thus thus ADV iajs-3181 74 33 , , PUNCT iajs-3181 74 34 f(anbnb1)βŠ† f(anbnb1)βŠ† NOUN iajs-3181 74 35 f(n1 f(n1 NOUN iajs-3181 74 36 ) ) PUNCT iajs-3181 74 37 since since SCONJ iajs-3181 74 38 f f PROPN iajs-3181 74 39 is be AUX iajs-3181 74 40 a a DET iajs-3181 74 41 homomorphism homomorphism NOUN iajs-3181 74 42 implies imply VERB iajs-3181 74 43 f(an).f(bn)∈ f(an).f(bn)∈ PROPN iajs-3181 74 44 [ [ X iajs-3181 74 45 f(n):f(w1 f(n):f(w1 NOUN iajs-3181 74 46 ) ) PUNCT iajs-3181 74 47 ] ] PUNCT iajs-3181 74 48 . . PUNCT iajs-3181 75 1 suppose suppose VERB iajs-3181 75 2 f(a)=x f(a)=x NOUN iajs-3181 75 3 , , PUNCT iajs-3181 75 4 f(b)=y f(b)=y NOUN iajs-3181 75 5 . . PROPN iajs-3181 75 6 thus thus ADV iajs-3181 75 7 , , PUNCT iajs-3181 75 8 either either DET iajs-3181 75 9 f(anb1 f(anb1 NOUN iajs-3181 75 10 ) ) PUNCT iajs-3181 75 11 βŠ†f(n1 βŠ†f(n1 NOUN iajs-3181 75 12 ) ) PUNCT iajs-3181 75 13 or or CCONJ iajs-3181 75 14 f(bnb1)βŠ† f(bnb1)βŠ† PROPN iajs-3181 75 15 f(n f(n PROPN iajs-3181 75 16 ) ) PUNCT iajs-3181 75 17 , , PUNCT iajs-3181 75 18 so so ADV iajs-3181 75 19 either either DET iajs-3181 75 20 f(an)f(w1 f(an)f(w1 NOUN iajs-3181 75 21 ) ) PUNCT iajs-3181 75 22 βŠ†f(n βŠ†f(n PROPN iajs-3181 75 23 ) ) PUNCT iajs-3181 75 24 or or CCONJ iajs-3181 75 25 f(bn)f(b1)βŠ† f(bn)f(b1)βŠ† PROPN iajs-3181 75 26 f(n f(n PROPN iajs-3181 75 27 ) ) PUNCT iajs-3181 75 28 implies imply VERB iajs-3181 75 29 either either CCONJ iajs-3181 75 30 xnf(b1)βŠ† xnf(b1)βŠ† PROPN iajs-3181 75 31 f(n f(n PROPN iajs-3181 75 32 ) ) PUNCT iajs-3181 75 33 or or CCONJ iajs-3181 75 34 ynf(b1)βŠ† ynf(b1)βŠ† NUM iajs-3181 75 35 f(n1 f(n1 NOUN iajs-3181 75 36 ) ) PUNCT iajs-3181 75 37 , , PUNCT iajs-3181 75 38 but but CCONJ iajs-3181 75 39 f f PROPN iajs-3181 75 40 is be AUX iajs-3181 75 41 onto onto ADP iajs-3181 75 42 , , PUNCT iajs-3181 75 43 so so ADV iajs-3181 75 44 f(b1)=b2,f(n)=n2 f(b1)=b2,f(n)=n2 PROPN iajs-3181 75 45 , , PUNCT iajs-3181 75 46 which which PRON iajs-3181 75 47 means mean VERB iajs-3181 75 48 either either CCONJ iajs-3181 75 49 x∈ x∈ PROPN iajs-3181 75 50 √[𝑁2∢𝐡2 √[𝑁2∢𝐡2 NOUN iajs-3181 75 51 or or CCONJ iajs-3181 75 52 y∈ y∈ NOUN iajs-3181 75 53 βˆšπ‘2 βˆšπ‘2 NOUN iajs-3181 75 54 : : PUNCT iajs-3181 75 55 𝐡2 𝐡2 NOUN iajs-3181 75 56 , , PUNCT iajs-3181 75 57 whenever whenever SCONJ iajs-3181 75 58 xy∈ xy∈ PROPN iajs-3181 75 59 βˆšπ‘2 βˆšπ‘2 VERB iajs-3181 75 60 : : PUNCT iajs-3181 75 61 𝐡2 𝐡2 NOUN iajs-3181 75 62 . . PUNCT iajs-3181 76 1 thus thus ADV iajs-3181 76 2 , , PUNCT iajs-3181 76 3 βˆšπ‘2 βˆšπ‘2 VERB iajs-3181 76 4 : : PUNCT iajs-3181 76 5 𝐡2 𝐡2 NOUN iajs-3181 76 6 is be AUX iajs-3181 76 7 a a DET iajs-3181 76 8 prime prime ADJ iajs-3181 76 9 ideal ideal NOUN iajs-3181 76 10 , , PUNCT iajs-3181 76 11 which which PRON iajs-3181 76 12 means mean VERB iajs-3181 76 13 b2 b2 NOUN iajs-3181 76 14 is be AUX iajs-3181 76 15 a a DET iajs-3181 76 16 quasi quasi ADJ iajs-3181 76 17 - - ADJ iajs-3181 76 18 semiprime semiprime ADJ iajs-3181 76 19 module module NOUN iajs-3181 76 20 . . PUNCT iajs-3181 77 1 ihjpas ihjpas PROPN iajs-3181 77 2 . . PUNCT iajs-3181 78 1 36 36 NUM iajs-3181 78 2 ( ( PUNCT iajs-3181 78 3 4 4 NUM iajs-3181 78 4 ) ) PUNCT iajs-3181 78 5 2023 2023 NUM iajs-3181 78 6 381 381 NUM iajs-3181 78 7 corollary corollary ADJ iajs-3181 78 8 ( ( PUNCT iajs-3181 78 9 2.11 2.11 NUM iajs-3181 78 10 ) ) PUNCT iajs-3181 78 11 the the DET iajs-3181 78 12 inverse inverse ADJ iajs-3181 78 13 image image NOUN iajs-3181 78 14 of of ADP iajs-3181 78 15 the the DET iajs-3181 78 16 quasi quasi ADJ iajs-3181 78 17 - - ADJ iajs-3181 78 18 semiprime semiprime ADJ iajs-3181 78 19 module module NOUN iajs-3181 78 20 is be AUX iajs-3181 78 21 a a DET iajs-3181 78 22 quasi quasi ADJ iajs-3181 78 23 – – PUNCT iajs-3181 78 24 semiprime semiprime NOUN iajs-3181 78 25 module module NOUN iajs-3181 78 26 . . PUNCT iajs-3181 79 1 theorem theorem NOUN iajs-3181 79 2 ( ( PUNCT iajs-3181 79 3 2.12 2.12 NUM iajs-3181 79 4 ) ) PUNCT iajs-3181 79 5 let let VERB iajs-3181 79 6 b1 b1 NOUN iajs-3181 79 7 and and CCONJ iajs-3181 79 8 b2 b2 NOUN iajs-3181 79 9 be be AUX iajs-3181 79 10 two two NUM iajs-3181 79 11 quasi quasi ADJ iajs-3181 79 12 - - ADJ iajs-3181 79 13 semiprime semiprime ADJ iajs-3181 79 14 a a DET iajs-3181 79 15 - - PUNCT iajs-3181 79 16 modules module NOUN iajs-3181 79 17 such such ADJ iajs-3181 79 18 that that PRON iajs-3181 79 19 for for ADP iajs-3181 79 20 each each DET iajs-3181 79 21 proper proper ADJ iajs-3181 79 22 submodule submodule PROPN iajs-3181 79 23 k k PROPN iajs-3181 79 24 , , PUNCT iajs-3181 79 25 t t PROPN iajs-3181 79 26 of of ADP iajs-3181 79 27 b1,b2 b1,b2 PROPN iajs-3181 79 28 , , PUNCT iajs-3181 79 29 respectively respectively ADV iajs-3181 79 30 , , PUNCT iajs-3181 79 31 if if SCONJ iajs-3181 79 32 [ [ X iajs-3181 79 33 k⨁ k⨁ NOUN iajs-3181 79 34 t t PROPN iajs-3181 79 35 : : PUNCT iajs-3181 79 36 w]=[k w]=[k NUM iajs-3181 79 37 : : PUNCT iajs-3181 79 38 w]∩ w]∩ X iajs-3181 79 39 [ [ X iajs-3181 79 40 t t X iajs-3181 79 41 : : PUNCT iajs-3181 79 42 w w NOUN iajs-3181 79 43 ] ] X iajs-3181 79 44 , , PUNCT iajs-3181 79 45 then then ADV iajs-3181 79 46 b b X iajs-3181 79 47 = = ADJ iajs-3181 79 48 b1⨁ b1⨁ PROPN iajs-3181 79 49 b2 b2 NOUN iajs-3181 79 50 is be AUX iajs-3181 79 51 a a DET iajs-3181 79 52 quasi quasi ADJ iajs-3181 79 53 - - ADJ iajs-3181 79 54 semiprime semiprime ADJ iajs-3181 79 55 amodule amodule NOUN iajs-3181 79 56 , , PUNCT iajs-3181 79 57 where√[𝐾 where√[𝐾 NOUN iajs-3181 79 58 : : PUNCT iajs-3181 79 59 𝐡 𝐡 NOUN iajs-3181 79 60 ] ] PUNCT iajs-3181 79 61 βŠ† βŠ† NUM iajs-3181 79 62 √[𝑇 √[𝑇 NOUN iajs-3181 79 63 : : PUNCT iajs-3181 79 64 𝐡 𝐡 NOUN iajs-3181 79 65 ] ] PUNCT iajs-3181 79 66 or or CCONJ iajs-3181 79 67 √[𝑇 √[𝑇 NUM iajs-3181 79 68 : : PUNCT iajs-3181 79 69 𝐡 𝐡 NOUN iajs-3181 79 70 ] ] PUNCT iajs-3181 79 71 βŠ† βŠ† NUM iajs-3181 79 72 √[𝐾 √[𝐾 NOUN iajs-3181 79 73 : : PUNCT iajs-3181 79 74 𝐡 𝐡 NOUN iajs-3181 79 75 ] ] PUNCT iajs-3181 79 76 . . PUNCT iajs-3181 80 1 proof proof NOUN iajs-3181 80 2 we we PRON iajs-3181 80 3 must must AUX iajs-3181 80 4 prove prove VERB iajs-3181 80 5 √[𝐾⨁𝑇 √[𝐾⨁𝑇 PROPN iajs-3181 80 6 : : PUNCT iajs-3181 80 7 𝐡 𝐡 NOUN iajs-3181 80 8 ] ] PUNCT iajs-3181 80 9 is be AUX iajs-3181 80 10 a a DET iajs-3181 80 11 prime prime ADJ iajs-3181 80 12 ideal ideal NOUN iajs-3181 80 13 for for ADP iajs-3181 80 14 the the DET iajs-3181 80 15 proper proper ADJ iajs-3181 80 16 submodules submodule NOUN iajs-3181 80 17 k k PROPN iajs-3181 80 18 , , PUNCT iajs-3181 80 19 t t PROPN iajs-3181 80 20 of of ADP iajs-3181 80 21 b1 b1 NOUN iajs-3181 80 22 and and CCONJ iajs-3181 80 23 b2 b2 NOUN iajs-3181 80 24 in in ADP iajs-3181 80 25 the the DET iajs-3181 80 26 order order NOUN iajs-3181 80 27 . . PUNCT iajs-3181 81 1 since since SCONJ iajs-3181 81 2 √[𝐾 √[𝐾 NOUN iajs-3181 81 3 ⨁𝑇 ⨁𝑇 NOUN iajs-3181 81 4 : : PUNCT iajs-3181 81 5 𝐡 𝐡 NOUN iajs-3181 81 6 ] ] PUNCT iajs-3181 81 7 = = SYM iajs-3181 81 8 √[𝐾 √[𝐾 NOUN iajs-3181 81 9 : : PUNCT iajs-3181 81 10 𝐡 𝐡 NOUN iajs-3181 81 11 ] ] PUNCT iajs-3181 81 12 ∩ ∩ ADJ iajs-3181 81 13 √[𝑇 √[𝑇 NOUN iajs-3181 81 14 : : PUNCT iajs-3181 81 15 𝐡 𝐡 NOUN iajs-3181 81 16 ] ] PUNCT iajs-3181 81 17 where where SCONJ iajs-3181 81 18 either√[𝐾 either√[𝐾 ADJ iajs-3181 81 19 : : PUNCT iajs-3181 81 20 𝐡 𝐡 NOUN iajs-3181 81 21 ] ] PUNCT iajs-3181 81 22 βŠ† βŠ† NUM iajs-3181 81 23 √[𝑇 √[𝑇 NOUN iajs-3181 81 24 : : PUNCT iajs-3181 81 25 𝐡 𝐡 NOUN iajs-3181 81 26 ] ] PUNCT iajs-3181 81 27 . . PUNCT iajs-3181 82 1 or√[𝑇 or√[𝑇 ADJ iajs-3181 82 2 : : PUNCT iajs-3181 82 3 𝐡 𝐡 NOUN iajs-3181 82 4 ] ] PUNCT iajs-3181 82 5 βŠ† βŠ† NUM iajs-3181 82 6 √[𝐾 √[𝐾 NOUN iajs-3181 82 7 : : PUNCT iajs-3181 82 8 𝐡 𝐡 NOUN iajs-3181 82 9 ] ] PUNCT iajs-3181 82 10 . . PUNCT iajs-3181 83 1 thus thus ADV iajs-3181 83 2 , , PUNCT iajs-3181 83 3 either either CCONJ iajs-3181 83 4 √[𝐾⨁𝑇 √[𝐾⨁𝑇 PROPN iajs-3181 83 5 : : PUNCT iajs-3181 83 6 𝐡 𝐡 NOUN iajs-3181 83 7 ] ] PUNCT iajs-3181 83 8 = = SYM iajs-3181 83 9 √[𝐾 √[𝐾 NOUN iajs-3181 83 10 : : PUNCT iajs-3181 83 11 𝐡 𝐡 NOUN iajs-3181 83 12 ] ] PUNCT iajs-3181 83 13 or or CCONJ iajs-3181 83 14 √[𝐾⨁𝑇 √[𝐾⨁𝑇 PROPN iajs-3181 83 15 : : PUNCT iajs-3181 83 16 𝐡 𝐡 NOUN iajs-3181 83 17 ] ] PUNCT iajs-3181 83 18 = = NOUN iajs-3181 83 19 √[𝑇 √[𝑇 NOUN iajs-3181 83 20 : : PUNCT iajs-3181 83 21 𝐡 𝐡 NOUN iajs-3181 83 22 ] ] PUNCT iajs-3181 83 23 , , PUNCT iajs-3181 83 24 but but CCONJ iajs-3181 83 25 w1 w1 NOUN iajs-3181 83 26 , , PUNCT iajs-3181 83 27 and and CCONJ iajs-3181 83 28 w2 w2 NOUN iajs-3181 83 29 are be AUX iajs-3181 83 30 quasi quasi ADJ iajs-3181 83 31 - - ADJ iajs-3181 83 32 semi semi ADJ iajs-3181 83 33 - - ADJ iajs-3181 83 34 prime prime ADJ iajs-3181 83 35 modules module NOUN iajs-3181 83 36 . . PUNCT iajs-3181 84 1 therefore,√[𝐾 therefore,√[𝐾 NOUN iajs-3181 84 2 : : PUNCT iajs-3181 84 3 π‘Š π‘Š NOUN iajs-3181 84 4 ] ] PUNCT iajs-3181 84 5 , , PUNCT iajs-3181 84 6 and and CCONJ iajs-3181 84 7 √[𝑇 √[𝑇 NOUN iajs-3181 84 8 : : PUNCT iajs-3181 84 9 π‘Š π‘Š NOUN iajs-3181 84 10 ] ] PUNCT iajs-3181 84 11 are be AUX iajs-3181 84 12 prime prime ADJ iajs-3181 84 13 ideals ideal NOUN iajs-3181 84 14 in in ADP iajs-3181 84 15 a. a. NOUN iajs-3181 84 16 implies imply VERB iajs-3181 84 17 √[𝐾⨁𝑇 √[𝐾⨁𝑇 PROPN iajs-3181 84 18 : : PUNCT iajs-3181 84 19 𝐡 𝐡 NOUN iajs-3181 84 20 ] ] PUNCT iajs-3181 84 21 is be AUX iajs-3181 84 22 a a DET iajs-3181 84 23 prime prime ADJ iajs-3181 84 24 ideal ideal NOUN iajs-3181 84 25 in in ADP iajs-3181 84 26 a. a. NOUN iajs-3181 84 27 thus thus ADV iajs-3181 84 28 , , PUNCT iajs-3181 84 29 b1⨁ b1⨁ PROPN iajs-3181 84 30 b2 b2 NOUN iajs-3181 84 31 is be AUX iajs-3181 84 32 quasi quasi ADJ iajs-3181 84 33 - - ADJ iajs-3181 84 34 semiprime semiprime ADJ iajs-3181 84 35 a a DET iajs-3181 84 36 - - PUNCT iajs-3181 84 37 modules module NOUN iajs-3181 84 38 . . PUNCT iajs-3181 85 1 the the DET iajs-3181 85 2 condition condition NOUN iajs-3181 85 3 √[𝐾 √[𝐾 NOUN iajs-3181 85 4 : : PUNCT iajs-3181 85 5 𝐡 𝐡 PROPN iajs-3181 85 6 ] ] PUNCT iajs-3181 85 7 βŠ† βŠ† NUM iajs-3181 85 8 βˆšπ‘‡ βˆšπ‘‡ NOUN iajs-3181 85 9 : : PUNCT iajs-3181 85 10 𝐡 𝐡 NOUN iajs-3181 85 11 ] ] PUNCT iajs-3181 85 12 or or CCONJ iajs-3181 85 13 √[𝑇 √[𝑇 NUM iajs-3181 85 14 : : PUNCT iajs-3181 85 15 𝐡 𝐡 NOUN iajs-3181 85 16 ] ] PUNCT iajs-3181 85 17 βŠ† βŠ† NUM iajs-3181 85 18 √[𝐾 √[𝐾 NOUN iajs-3181 85 19 : : PUNCT iajs-3181 85 20 𝐡 𝐡 NOUN iajs-3181 85 21 ] ] X iajs-3181 85 22 we we PRON iajs-3181 85 23 can can AUX iajs-3181 85 24 not not PART iajs-3181 85 25 be be AUX iajs-3181 85 26 dropped drop VERB iajs-3181 85 27 , , PUNCT iajs-3181 85 28 for for ADP iajs-3181 85 29 example example NOUN iajs-3181 85 30 , , PUNCT iajs-3181 85 31 let let VERB iajs-3181 85 32 b1 b1 NOUN iajs-3181 85 33 = = NOUN iajs-3181 85 34 z6 z6 PROPN iajs-3181 85 35 , , PUNCT iajs-3181 85 36 and and CCONJ iajs-3181 85 37 b2 b2 NOUN iajs-3181 85 38 = = NOUN iajs-3181 85 39 z3 z3 NOUN iajs-3181 85 40 are be AUX iajs-3181 85 41 two two NUM iajs-3181 85 42 quasi quasi ADJ iajs-3181 85 43 - - ADJ iajs-3181 85 44 semiprime semiprime ADJ iajs-3181 85 45 a a NOUN iajs-3181 85 46 - - PUNCT iajs-3181 85 47 modules module NOUN iajs-3181 85 48 by by ADP iajs-3181 85 49 ( ( PUNCT iajs-3181 85 50 examples example NOUN iajs-3181 85 51 and and CCONJ iajs-3181 85 52 remark remark NOUN iajs-3181 85 53 ( ( PUNCT iajs-3181 85 54 2.2 2.2 NUM iajs-3181 85 55 ) ) PUNCT iajs-3181 85 56 , , PUNCT iajs-3181 86 1 √[(2 √[(2 PROPN iajs-3181 86 2 ): ): PUNCT iajs-3181 86 3 𝑍18 𝑍18 PROPN iajs-3181 86 4 ] ] X iajs-3181 86 5 ⊈ ⊈ PROPN iajs-3181 86 6 √[(3 √[(3 PROPN iajs-3181 86 7 ): ): PUNCT iajs-3181 87 1 𝑍18 𝑍18 PROPN iajs-3181 87 2 and and CCONJ iajs-3181 87 3 √[(3 √[(3 PROPN iajs-3181 87 4 ): ): PUNCT iajs-3181 87 5 𝑍18 𝑍18 PROPN iajs-3181 87 6 ⊈ ⊈ PROPN iajs-3181 87 7 √[(2 √[(2 PROPN iajs-3181 87 8 ): ): PUNCT iajs-3181 87 9 𝑍18 𝑍18 PROPN iajs-3181 87 10 since√(3 since√(3 PROPN iajs-3181 87 11 ) ) PUNCT iajs-3181 87 12 ⊈ ⊈ PROPN iajs-3181 87 13 √(2 √(2 NOUN iajs-3181 87 14 ) ) PUNCT iajs-3181 87 15 and and CCONJ iajs-3181 87 16 √(2 √(2 PROPN iajs-3181 87 17 ) ) PUNCT iajs-3181 87 18 ⊈ ⊈ PROPN iajs-3181 87 19 √(3 √(3 PROPN iajs-3181 87 20 ) ) PUNCT iajs-3181 87 21 so√[𝑍2⨁ so√[𝑍2⨁ PART iajs-3181 87 22 𝑍3 𝑍3 NOUN iajs-3181 87 23 : : PUNCT iajs-3181 88 1 𝑍18 𝑍18 PROPN iajs-3181 88 2 β‰  β‰  PROPN iajs-3181 88 3 βˆšπ‘2 βˆšπ‘2 PROPN iajs-3181 88 4 : : PUNCT iajs-3181 88 5 𝑍18 𝑍18 PROPN iajs-3181 88 6 ∩ ∩ ADJ iajs-3181 88 7 βˆšπ‘3 βˆšπ‘3 NOUN iajs-3181 88 8 : : PUNCT iajs-3181 88 9 𝑍18 𝑍18 PROPN iajs-3181 88 10 = = SYM iajs-3181 88 11 √9𝑍 √9𝑍 X iajs-3181 88 12 ∩ ∩ NOUN iajs-3181 88 13 √6𝑍 √6𝑍 X iajs-3181 88 14 =( =( NOUN iajs-3181 88 15 3 3 NUM iajs-3181 88 16 ) ) PUNCT iajs-3181 88 17 ∩(6)=(6 ∩(6)=(6 NOUN iajs-3181 88 18 ) ) PUNCT iajs-3181 88 19 is be AUX iajs-3181 88 20 not not PART iajs-3181 88 21 a a DET iajs-3181 88 22 prime prime ADJ iajs-3181 88 23 ideal ideal NOUN iajs-3181 88 24 , , PUNCT iajs-3181 88 25 implying imply VERB iajs-3181 88 26 w w NOUN iajs-3181 88 27 = = NOUN iajs-3181 88 28 w1 w1 NOUN iajs-3181 88 29 ⨁w2 ⨁w2 NOUN iajs-3181 88 30 is be AUX iajs-3181 88 31 not not PART iajs-3181 88 32 a a DET iajs-3181 88 33 quasi quasi ADJ iajs-3181 88 34 - - ADJ iajs-3181 88 35 semiprime semiprime ADJ iajs-3181 88 36 module module NOUN iajs-3181 88 37 . . PUNCT iajs-3181 89 1 3 3 X iajs-3181 89 2 . . X iajs-3181 89 3 quasi quasi ADJ iajs-3181 89 4 - - ADJ iajs-3181 89 5 semi semi ADJ iajs-3181 89 6 - - ADJ iajs-3181 89 7 prime prime ADJ iajs-3181 89 8 a a DET iajs-3181 89 9 - - PUNCT iajs-3181 89 10 module module NOUN iajs-3181 89 11 and and CCONJ iajs-3181 89 12 prime prime ADJ iajs-3181 89 13 module module NOUN iajs-3181 89 14 now now ADV iajs-3181 89 15 , , PUNCT iajs-3181 89 16 we we PRON iajs-3181 89 17 turn turn VERB iajs-3181 89 18 our our PRON iajs-3181 89 19 attention attention NOUN iajs-3181 89 20 to to ADP iajs-3181 89 21 the the DET iajs-3181 89 22 relationship relationship NOUN iajs-3181 89 23 between between ADP iajs-3181 89 24 quasi quasi ADJ iajs-3181 89 25 - - ADJ iajs-3181 89 26 semiprime semiprime ADJ iajs-3181 89 27 modules module NOUN iajs-3181 89 28 and and CCONJ iajs-3181 89 29 prime prime ADJ iajs-3181 89 30 modules module NOUN iajs-3181 89 31 . . PUNCT iajs-3181 90 1 proposition proposition NOUN iajs-3181 90 2 ( ( PUNCT iajs-3181 90 3 3.1 3.1 NUM iajs-3181 90 4 ) ) PUNCT iajs-3181 90 5 suppose suppose VERB iajs-3181 90 6 b b NOUN iajs-3181 90 7 is be AUX iajs-3181 90 8 a a DET iajs-3181 90 9 coprime coprime NOUN iajs-3181 90 10 a a DET iajs-3181 90 11 - - PUNCT iajs-3181 90 12 module module NOUN iajs-3181 90 13 , , PUNCT iajs-3181 90 14 then then ADV iajs-3181 90 15 every every DET iajs-3181 90 16 prime prime ADJ iajs-3181 90 17 a a PRON iajs-3181 90 18 - - PUNCT iajs-3181 90 19 module module NOUN iajs-3181 90 20 is be AUX iajs-3181 90 21 a a DET iajs-3181 90 22 quasi quasi NOUN iajs-3181 90 23 - - ADJ iajs-3181 90 24 semiprime semiprime ADJ iajs-3181 90 25 a a DET iajs-3181 90 26 - - PUNCT iajs-3181 90 27 module module NOUN iajs-3181 90 28 . . PUNCT iajs-3181 91 1 proof proof NOUN iajs-3181 91 2 it it PRON iajs-3181 91 3 follows follow VERB iajs-3181 91 4 directly directly ADV iajs-3181 91 5 by by ADP iajs-3181 91 6 from from ADP iajs-3181 91 7 [ [ X iajs-3181 91 8 3 3 NUM iajs-3181 91 9 ] ] PUNCT iajs-3181 91 10 and and CCONJ iajs-3181 91 11 propositions proposition NOUN iajs-3181 91 12 ( ( PUNCT iajs-3181 91 13 2.5 2.5 NUM iajs-3181 91 14 ) ) PUNCT iajs-3181 91 15 . . PUNCT iajs-3181 92 1 the the DET iajs-3181 92 2 next next ADJ iajs-3181 92 3 example example NOUN iajs-3181 92 4 shows show VERB iajs-3181 92 5 that that SCONJ iajs-3181 92 6 the the DET iajs-3181 92 7 converse converse NOUN iajs-3181 92 8 of of ADP iajs-3181 92 9 proposition proposition NOUN iajs-3181 92 10 ( ( PUNCT iajs-3181 92 11 3.1 3.1 NUM iajs-3181 92 12 ) ) PUNCT iajs-3181 92 13 is be AUX iajs-3181 92 14 not not PART iajs-3181 92 15 valid valid ADJ iajs-3181 92 16 in in ADP iajs-3181 92 17 general general ADJ iajs-3181 92 18 . . PUNCT iajs-3181 93 1 let let VERB iajs-3181 93 2 z6 z6 PROPN iajs-3181 93 3 as as ADP iajs-3181 93 4 a a DET iajs-3181 93 5 z z NOUN iajs-3181 93 6 – – PUNCT iajs-3181 93 7 module module NOUN iajs-3181 93 8 is be AUX iajs-3181 93 9 quasi quasi ADJ iajs-3181 93 10 - - ADJ iajs-3181 93 11 semiprime semiprime ADJ iajs-3181 93 12 module module NOUN iajs-3181 93 13 by by ADP iajs-3181 93 14 examples example NOUN iajs-3181 93 15 and and CCONJ iajs-3181 93 16 remarks remark NOUN iajs-3181 93 17 ( ( PUNCT iajs-3181 93 18 2.2 2.2 NUM iajs-3181 93 19 ) ) PUNCT iajs-3181 93 20 , , PUNCT iajs-3181 93 21 while while SCONJ iajs-3181 93 22 it it PRON iajs-3181 93 23 is be AUX iajs-3181 93 24 not not PART iajs-3181 93 25 a a DET iajs-3181 93 26 prime prime ADJ iajs-3181 93 27 module module NOUN iajs-3181 93 28 [ [ X iajs-3181 93 29 1 1 NUM iajs-3181 93 30 ] ] PUNCT iajs-3181 93 31 . . PUNCT iajs-3181 94 1 ihjpas ihjpas PROPN iajs-3181 94 2 . . PUNCT iajs-3181 95 1 36 36 NUM iajs-3181 95 2 ( ( PUNCT iajs-3181 95 3 4 4 NUM iajs-3181 95 4 ) ) PUNCT iajs-3181 95 5 2023 2023 NUM iajs-3181 95 6 382 382 NUM iajs-3181 95 7 theorem theorem NOUN iajs-3181 95 8 ( ( PUNCT iajs-3181 95 9 3.2 3.2 NUM iajs-3181 95 10 ) ) PUNCT iajs-3181 95 11 suppose suppose VERB iajs-3181 95 12 b b NOUN iajs-3181 95 13 is be AUX iajs-3181 95 14 a a DET iajs-3181 95 15 coprime coprime ADJ iajs-3181 95 16 c c NOUN iajs-3181 95 17 - - PUNCT iajs-3181 95 18 module module NOUN iajs-3181 95 19 , , PUNCT iajs-3181 95 20 then then ADV iajs-3181 95 21 the the DET iajs-3181 95 22 following following ADJ iajs-3181 95 23 statements statement NOUN iajs-3181 95 24 are be AUX iajs-3181 95 25 equivalent equivalent ADJ iajs-3181 95 26 : : PUNCT iajs-3181 95 27 1 1 NUM iajs-3181 95 28 - - SYM iajs-3181 95 29 b b NOUN iajs-3181 95 30 is be AUX iajs-3181 95 31 a a DET iajs-3181 95 32 prime prime ADJ iajs-3181 95 33 c c NOUN iajs-3181 95 34 - - PUNCT iajs-3181 95 35 module module NOUN iajs-3181 95 36 . . PUNCT iajs-3181 96 1 2 2 NUM iajs-3181 96 2 - - SYM iajs-3181 96 3 b b NOUN iajs-3181 96 4 is be AUX iajs-3181 96 5 a a DET iajs-3181 96 6 quasi quasi NOUN iajs-3181 96 7 - - ADJ iajs-3181 96 8 semiprime semiprime ADJ iajs-3181 96 9 c c NOUN iajs-3181 96 10 - - PUNCT iajs-3181 96 11 module module NOUN iajs-3181 96 12 . . PUNCT iajs-3181 97 1 proof proof NOUN iajs-3181 97 2 : : PUNCT iajs-3181 97 3 1β†’ 1β†’ NUM iajs-3181 97 4 2 2 NUM iajs-3181 97 5 by by ADP iajs-3181 97 6 ( ( PUNCT iajs-3181 97 7 proposition proposition NOUN iajs-3181 97 8 ( ( PUNCT iajs-3181 97 9 3 3 NUM iajs-3181 97 10 - - SYM iajs-3181 97 11 1 1 NUM iajs-3181 97 12 ) ) PUNCT iajs-3181 97 13 ) ) PUNCT iajs-3181 97 14 , , PUNCT iajs-3181 97 15 2 2 NUM iajs-3181 97 16 β†’ β†’ SYM iajs-3181 97 17 1 1 NUM iajs-3181 97 18 because because SCONJ iajs-3181 97 19 b b PROPN iajs-3181 97 20 is be AUX iajs-3181 97 21 a a DET iajs-3181 97 22 quasi quasi NOUN iajs-3181 97 23 - - ADJ iajs-3181 97 24 semiprime semiprime ADJ iajs-3181 97 25 c c NOUN iajs-3181 97 26 - - PUNCT iajs-3181 97 27 module module NOUN iajs-3181 97 28 , , PUNCT iajs-3181 97 29 so so ADV iajs-3181 97 30 √[𝑁 √[𝑁 ADJ iajs-3181 97 31 : : PUNCT iajs-3181 97 32 𝐡 𝐡 NOUN iajs-3181 97 33 ] ] PUNCT iajs-3181 97 34 is be AUX iajs-3181 97 35 a a DET iajs-3181 97 36 prime prime ADJ iajs-3181 97 37 ideal ideal NOUN iajs-3181 97 38 for for ADP iajs-3181 97 39 each each DET iajs-3181 97 40 n n PRON iajs-3181 97 41 submodule submodule NOUN iajs-3181 97 42 of of ADP iajs-3181 97 43 m m PROPN iajs-3181 97 44 , , PUNCT iajs-3181 97 45 so so SCONJ iajs-3181 97 46 [ [ X iajs-3181 97 47 n n CCONJ iajs-3181 97 48 : : PUNCT iajs-3181 97 49 b b X iajs-3181 97 50 ] ] PUNCT iajs-3181 97 51 is be AUX iajs-3181 97 52 a a DET iajs-3181 97 53 prime prime ADJ iajs-3181 97 54 ideal ideal NOUN iajs-3181 97 55 , , PUNCT iajs-3181 97 56 but but CCONJ iajs-3181 97 57 b b X iajs-3181 97 58 is be AUX iajs-3181 97 59 a a DET iajs-3181 97 60 coprime coprime ADJ iajs-3181 97 61 c c NOUN iajs-3181 97 62 - - PUNCT iajs-3181 97 63 module module NOUN iajs-3181 97 64 , , PUNCT iajs-3181 97 65 so so CCONJ iajs-3181 97 66 [ [ X iajs-3181 97 67 6 6 NUM iajs-3181 97 68 ] ] PUNCT iajs-3181 97 69 implies imply VERB iajs-3181 97 70 anncb anncb PROPN iajs-3181 97 71 is be AUX iajs-3181 97 72 a a DET iajs-3181 97 73 prime prime ADJ iajs-3181 97 74 ideal ideal NOUN iajs-3181 97 75 , , PUNCT iajs-3181 97 76 which which PRON iajs-3181 97 77 means mean VERB iajs-3181 97 78 if if SCONJ iajs-3181 97 79 rb=0 rb=0 PROPN iajs-3181 97 80 for for ADP iajs-3181 97 81 b∈ b∈ PROPN iajs-3181 97 82 b b PROPN iajs-3181 97 83 and and CCONJ iajs-3181 97 84 c∈ c∈ NOUN iajs-3181 97 85 c. c. NOUN iajs-3181 97 86 suppose suppose VERB iajs-3181 97 87 that that SCONJ iajs-3181 97 88 bβ‰  bβ‰  ADP iajs-3181 97 89 0 0 PUNCT iajs-3181 97 90 and and CCONJ iajs-3181 97 91 cbβ‰  cbβ‰  PROPN iajs-3181 97 92 o o NOUN iajs-3181 97 93 , , PUNCT iajs-3181 97 94 so so ADV iajs-3181 97 95 cb cb PROPN iajs-3181 97 96 = = PROPN iajs-3181 97 97 nβ‰  nβ‰  NOUN iajs-3181 97 98 0 0 NUM iajs-3181 97 99 , , PUNCT iajs-3181 97 100 thus thus ADV iajs-3181 97 101 there there PRON iajs-3181 97 102 exists exist VERB iajs-3181 97 103 that that SCONJ iajs-3181 97 104 b∈ b∈ PROPN iajs-3181 97 105 b b PROPN iajs-3181 97 106 and and CCONJ iajs-3181 97 107 n∈ n∈ NOUN iajs-3181 97 108 n n PRON iajs-3181 97 109 such such ADJ iajs-3181 97 110 that that SCONJ iajs-3181 97 111 cb cb PROPN iajs-3181 97 112 = = PROPN iajs-3181 97 113 n n NUM iajs-3181 97 114 , , PUNCT iajs-3181 97 115 this this PRON iajs-3181 97 116 means mean VERB iajs-3181 97 117 n=0 n=0 NUM iajs-3181 97 118 , , PUNCT iajs-3181 97 119 which which PRON iajs-3181 97 120 is be AUX iajs-3181 97 121 a a DET iajs-3181 97 122 contradiction contradiction NOUN iajs-3181 97 123 . . PUNCT iajs-3181 98 1 so so ADV iajs-3181 98 2 b b PROPN iajs-3181 98 3 is be AUX iajs-3181 98 4 a a DET iajs-3181 98 5 prime prime NOUN iajs-3181 98 6 . . PUNCT iajs-3181 99 1 proposition proposition NOUN iajs-3181 99 2 ( ( PUNCT iajs-3181 99 3 3.3 3.3 NUM iajs-3181 99 4 ) ) PUNCT iajs-3181 99 5 let let VERB iajs-3181 99 6 b b X iajs-3181 99 7 be be AUX iajs-3181 99 8 a a DET iajs-3181 99 9 coprime coprime ADJ iajs-3181 99 10 c c NOUN iajs-3181 99 11 - - PUNCT iajs-3181 99 12 module module NOUN iajs-3181 99 13 , , PUNCT iajs-3181 99 14 then then ADV iajs-3181 99 15 the the DET iajs-3181 99 16 following following ADJ iajs-3181 99 17 statements statement NOUN iajs-3181 99 18 are be AUX iajs-3181 99 19 equivalent equivalent ADJ iajs-3181 99 20 : : PUNCT iajs-3181 99 21 1bis 1bis NUM iajs-3181 99 22 a a DET iajs-3181 99 23 quasi quasi ADJ iajs-3181 99 24 - - ADJ iajs-3181 99 25 prime prime ADJ iajs-3181 99 26 module module NOUN iajs-3181 99 27 . . PUNCT iajs-3181 100 1 2b 2b NUM iajs-3181 100 2 is be AUX iajs-3181 100 3 a a DET iajs-3181 100 4 quasi quasi ADJ iajs-3181 100 5 - - ADJ iajs-3181 100 6 semiprime semiprime ADJ iajs-3181 100 7 modul modul PROPN iajs-3181 100 8 . . PUNCT iajs-3181 101 1 3b 3b PROPN iajs-3181 101 2 is be AUX iajs-3181 101 3 a a DET iajs-3181 101 4 prime prime ADJ iajs-3181 101 5 module module NOUN iajs-3181 101 6 . . PUNCT iajs-3181 102 1 proof proof NOUN iajs-3181 102 2 1 1 NUM iajs-3181 102 3 β†’ β†’ SYM iajs-3181 102 4 2 2 NUM iajs-3181 102 5 by by ADP iajs-3181 102 6 theorem(2.7 theorem(2.7 NOUN iajs-3181 102 7 ) ) PUNCT iajs-3181 102 8 . . PUNCT iajs-3181 103 1 2 2 NUM iajs-3181 103 2 β†’ β†’ SYM iajs-3181 103 3 3 3 NUM iajs-3181 103 4 by by ADP iajs-3181 103 5 theorem theorem NOUN iajs-3181 103 6 ( ( PUNCT iajs-3181 103 7 3.2 3.2 NUM iajs-3181 103 8 ) ) PUNCT iajs-3181 103 9 . . PUNCT iajs-3181 104 1 3β†’ 3β†’ NUM iajs-3181 104 2 1 1 NUM iajs-3181 104 3 by by ADP iajs-3181 104 4 [ [ X iajs-3181 104 5 3 3 NUM iajs-3181 104 6 ] ] PUNCT iajs-3181 104 7 . . PUNCT iajs-3181 105 1 corollary corollary ADJ iajs-3181 105 2 ( ( PUNCT iajs-3181 105 3 3.4 3.4 NUM iajs-3181 105 4 ) ) PUNCT iajs-3181 105 5 if if SCONJ iajs-3181 105 6 b b NOUN iajs-3181 105 7 is be AUX iajs-3181 105 8 a a DET iajs-3181 105 9 coprime coprime ADJ iajs-3181 105 10 c c NOUN iajs-3181 105 11 - - PUNCT iajs-3181 105 12 module module NOUN iajs-3181 105 13 , , PUNCT iajs-3181 105 14 then then ADV iajs-3181 105 15 b b PROPN iajs-3181 105 16 is be AUX iajs-3181 105 17 a a DET iajs-3181 105 18 quasi quasi NOUN iajs-3181 105 19 - - ADJ iajs-3181 105 20 semiprime semiprime ADJ iajs-3181 105 21 c c NOUN iajs-3181 105 22 - - PUNCT iajs-3181 105 23 module module NOUN iajs-3181 105 24 ⟷ ⟷ NUM iajs-3181 105 25 ( ( PUNCT iajs-3181 105 26 0 0 NUM iajs-3181 105 27 ) ) PUNCT iajs-3181 105 28 is be AUX iajs-3181 105 29 a a DET iajs-3181 105 30 prime prime ADJ iajs-3181 105 31 csubmodule csubmodule NOUN iajs-3181 105 32 . . PUNCT iajs-3181 106 1 proof proof NOUN iajs-3181 106 2 it it PRON iajs-3181 106 3 is be AUX iajs-3181 106 4 clear clear ADJ iajs-3181 106 5 . . PUNCT iajs-3181 107 1 conclusion conclusion NOUN iajs-3181 107 2 from from ADP iajs-3181 107 3 this this DET iajs-3181 107 4 research research NOUN iajs-3181 107 5 , , PUNCT iajs-3181 107 6 we we PRON iajs-3181 107 7 introduced introduce VERB iajs-3181 107 8 a a DET iajs-3181 107 9 new new ADJ iajs-3181 107 10 definition definition NOUN iajs-3181 107 11 of of ADP iajs-3181 107 12 quasi quasi ADJ iajs-3181 107 13 - - ADJ iajs-3181 107 14 semiprime semiprime ADJ iajs-3181 107 15 modules module NOUN iajs-3181 107 16 and and CCONJ iajs-3181 107 17 studied study VERB iajs-3181 107 18 the the DET iajs-3181 107 19 relationship relationship NOUN iajs-3181 107 20 between between ADP iajs-3181 107 21 quasi quasi ADJ iajs-3181 107 22 - - ADJ iajs-3181 107 23 semiprime semiprime ADJ iajs-3181 107 24 modules module NOUN iajs-3181 107 25 and and CCONJ iajs-3181 107 26 other other ADJ iajs-3181 107 27 modules module NOUN iajs-3181 107 28 , , PUNCT iajs-3181 107 29 such such ADJ iajs-3181 107 30 as as ADP iajs-3181 107 31 quasi quasi ADJ iajs-3181 107 32 - - ADJ iajs-3181 107 33 prime prime ADJ iajs-3181 107 34 modules module NOUN iajs-3181 107 35 and and CCONJ iajs-3181 107 36 prime prime ADJ iajs-3181 107 37 modules module NOUN iajs-3181 107 38 . . 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PROPN iajs-3181 110 6 anote anote VERB iajs-3181 110 7 on on ADP iajs-3181 110 8 prime prime ADJ iajs-3181 110 9 modules module NOUN iajs-3181 110 10 and and CCONJ iajs-3181 110 11 pure pure ADJ iajs-3181 110 12 submodules submodule NOUN iajs-3181 110 13 , , PUNCT iajs-3181 110 14 j. j. PROPN iajs-3181 110 15 sclence sclence PROPN iajs-3181 110 16 1996 1996 NUM iajs-3181 110 17 , , PUNCT iajs-3181 110 18 37 37 NUM iajs-3181 110 19 , , PUNCT iajs-3181 110 20 2 2 NUM iajs-3181 110 21 , , PUNCT iajs-3181 110 22 1431 1431 NUM iajs-3181 110 23 - - SYM iajs-3181 110 24 1441 1441 NUM iajs-3181 110 25 . . PUNCT iajs-3181 111 1 2 2 X iajs-3181 111 2 . . X iajs-3181 111 3 desale desale NOUN iajs-3181 111 4 , , PUNCT iajs-3181 111 5 g. g. PROPN iajs-3181 111 6 ; ; PUNCT iajs-3181 111 7 , , PUNCT iajs-3181 111 8 nicholson nicholson PROPN iajs-3181 111 9 , , PUNCT iajs-3181 111 10 w. w. PROPN iajs-3181 111 11 k. k. 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