id sid tid token lemma pos iajs-1919 1 1 microsoft microsoft PROPN iajs-1919 1 2 word word NOUN iajs-1919 1 3 167 167 NUM iajs-1919 1 4 - - SYM iajs-1919 1 5 177 177 NUM iajs-1919 1 6         SPACE iajs-1919 1 7 167 167 NUM iajs-1919 1 8     SPACE iajs-1919 1 9 ibn ibn PROPN iajs-1919 1 10 al al PROPN iajs-1919 1 11 - - PUNCT iajs-1919 1 12 haitham haitham PROPN iajs-1919 1 13 jour.for jour.for PROPN iajs-1919 1 14 pure&appl.sci pure&appl.sci PROPN iajs-1919 1 15 . . PUNCT iajs-1919 1 16 ihjpas ihjpas PROPN iajs-1919 1 17 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 1 18 vol vol NOUN iajs-1919 1 19 . . PUNCT iajs-1919 2 1 32 32 NUM iajs-1919 3 1 ( ( PUNCT iajs-1919 3 2 1 1 NUM iajs-1919 3 3 ) ) PUNCT iajs-1919 3 4 2019 2019 NUM iajs-1919 3 5 strongly strongly ADV iajs-1919 3 6 𝓚-nonsingular 𝓚-nonsingular ADJ iajs-1919 3 7 modules module NOUN iajs-1919 3 8 tha'ar tha'ar PUNCT iajs-1919 3 9 younis younis NOUN iajs-1919 3 10 ghawi ghawi ADJ iajs-1919 3 11 thar.younis@qu.edu.iq thar.younis@qu.edu.iq PROPN iajs-1919 3 12 department department PROPN iajs-1919 3 13 of of ADP iajs-1919 3 14 mathematics mathematics PROPN iajs-1919 3 15 , , PUNCT iajs-1919 3 16 college college NOUN iajs-1919 3 17 of of ADP iajs-1919 3 18 education education NOUN iajs-1919 3 19 , , PUNCT iajs-1919 3 20 al al PROPN iajs-1919 3 21 - - PUNCT iajs-1919 3 22 qadisiyah qadisiyah PROPN iajs-1919 3 23 university university PROPN iajs-1919 3 24 al al PROPN iajs-1919 3 25 - - PUNCT iajs-1919 3 26 qadisiyah qadisiyah PROPN iajs-1919 3 27 , , PUNCT iajs-1919 3 28 iraq iraq PROPN iajs-1919 3 29 . . PUNCT iajs-1919 4 1 article article NOUN iajs-1919 4 2 history history NOUN iajs-1919 4 3 : : PUNCT iajs-1919 4 4 received receive VERB iajs-1919 4 5 12 12 NUM iajs-1919 4 6 august august PROPN iajs-1919 4 7 2018 2018 NUM iajs-1919 4 8 , , PUNCT iajs-1919 4 9 accepted accept VERB iajs-1919 4 10 26 26 NUM iajs-1919 4 11 september september PROPN iajs-1919 4 12 2018 2018 NUM iajs-1919 4 13 , , PUNCT iajs-1919 4 14 publish publish VERB iajs-1919 4 15 january january PROPN iajs-1919 4 16 2019 2019 NUM iajs-1919 4 17 abstract abstract ADV iajs-1919 4 18 a a DET iajs-1919 4 19 submodule submodule NOUN iajs-1919 4 20 n n PROPN iajs-1919 4 21 of of ADP iajs-1919 4 22 a a DET iajs-1919 4 23 module module NOUN iajs-1919 4 24 m m NOUN iajs-1919 4 25 is be AUX iajs-1919 4 26 said say VERB iajs-1919 4 27 to to PART iajs-1919 4 28 be be AUX iajs-1919 4 29 s s NOUN iajs-1919 4 30 - - ADJ iajs-1919 4 31 essential essential ADJ iajs-1919 4 32 if if SCONJ iajs-1919 4 33 it it PRON iajs-1919 4 34 has have VERB iajs-1919 4 35 nonzero nonzero ADJ iajs-1919 4 36 intersection intersection NOUN iajs-1919 4 37 with with ADP iajs-1919 4 38 any any DET iajs-1919 4 39 nonzero nonzero ADJ iajs-1919 4 40 small small ADJ iajs-1919 4 41 submodule submodule NOUN iajs-1919 4 42 in in ADP iajs-1919 4 43 m. m. NOUN iajs-1919 4 44 in in ADP iajs-1919 4 45 this this DET iajs-1919 4 46 article article NOUN iajs-1919 4 47 , , PUNCT iajs-1919 4 48 we we PRON iajs-1919 4 49 introduce introduce VERB iajs-1919 4 50 and and CCONJ iajs-1919 4 51 study study VERB iajs-1919 4 52 a a DET iajs-1919 4 53 class class NOUN iajs-1919 4 54 of of ADP iajs-1919 4 55 modules module NOUN iajs-1919 4 56 in in ADP iajs-1919 4 57 which which PRON iajs-1919 4 58 all all DET iajs-1919 4 59 its its PRON iajs-1919 4 60 nonzero nonzero NOUN iajs-1919 4 61 endomorphisms endomorphism NOUN iajs-1919 4 62 have have VERB iajs-1919 4 63 non non ADJ iajs-1919 4 64 - - ADJ iajs-1919 4 65 s s ADJ iajs-1919 4 66 - - ADJ iajs-1919 4 67 essential essential ADJ iajs-1919 4 68 kernels kernel NOUN iajs-1919 4 69 , , PUNCT iajs-1919 4 70 named name VERB iajs-1919 4 71 , , PUNCT iajs-1919 4 72 strongly strongly ADV iajs-1919 4 73 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 4 74 . . PUNCT iajs-1919 5 1 we we PRON iajs-1919 5 2 investigate investigate VERB iajs-1919 5 3 some some DET iajs-1919 5 4 properties property NOUN iajs-1919 5 5 of of ADP iajs-1919 5 6 strongly strongly ADV iajs-1919 5 7 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 5 8 modules module NOUN iajs-1919 5 9 . . PUNCT iajs-1919 6 1 direct direct ADJ iajs-1919 6 2 summand summand NOUN iajs-1919 6 3 , , PUNCT iajs-1919 6 4 direct direct ADJ iajs-1919 6 5 sums sum NOUN iajs-1919 6 6 and and CCONJ iajs-1919 6 7 some some DET iajs-1919 6 8 connections connection NOUN iajs-1919 6 9 of of ADP iajs-1919 6 10 such such ADJ iajs-1919 6 11 modules module NOUN iajs-1919 6 12 are be AUX iajs-1919 6 13 discussed discuss VERB iajs-1919 6 14 . . PUNCT iajs-1919 7 1 keywords keyword NOUN iajs-1919 7 2 : : PUNCT iajs-1919 7 3 modules module NOUN iajs-1919 7 4 ; ; PUNCT iajs-1919 7 5 s s X iajs-1919 7 6 - - ADJ iajs-1919 7 7 essential essential ADJ iajs-1919 7 8 submodules submodule NOUN iajs-1919 7 9 ; ; PUNCT iajs-1919 7 10 nonsingular nonsingular ADJ iajs-1919 7 11 modules module NOUN iajs-1919 7 12 ; ; PUNCT iajs-1919 7 13 strongly strongly ADV iajs-1919 7 14 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 7 15 modules module NOUN iajs-1919 7 16 . . PUNCT iajs-1919 8 1 1 1 X iajs-1919 8 2 . . X iajs-1919 8 3 introduction introduction NOUN iajs-1919 8 4 a a DET iajs-1919 8 5 proper proper ADJ iajs-1919 8 6 submodule submodule NOUN iajs-1919 8 7 n n PROPN iajs-1919 8 8 of of ADP iajs-1919 8 9 a a DET iajs-1919 8 10 module module NOUN iajs-1919 8 11 m m NOUN iajs-1919 8 12 is be AUX iajs-1919 8 13 said say VERB iajs-1919 8 14 to to PART iajs-1919 8 15 be be AUX iajs-1919 8 16 small small ADJ iajs-1919 8 17 if if SCONJ iajs-1919 8 18 for for ADP iajs-1919 8 19 any any DET iajs-1919 8 20 submodule submodule NOUN iajs-1919 8 21 k k PROPN iajs-1919 8 22 of of ADP iajs-1919 8 23 m m PROPN iajs-1919 8 24 with with ADP iajs-1919 8 25 𝑁 𝑁 PROPN iajs-1919 8 26 𝐾 𝐾 PROPN iajs-1919 8 27 𝑀 𝑀 PROPN iajs-1919 8 28 implies imply VERB iajs-1919 8 29 𝐾 𝐾 PROPN iajs-1919 8 30 𝑀[1 𝑀[1 PROPN iajs-1919 8 31 ] ] PUNCT iajs-1919 8 32 . . PUNCT iajs-1919 9 1 a a DET iajs-1919 9 2 nonzero nonzero NOUN iajs-1919 9 3 module module NOUN iajs-1919 9 4 m m NOUN iajs-1919 9 5 is be AUX iajs-1919 9 6 called call VERB iajs-1919 9 7 hollow hollow ADJ iajs-1919 9 8 if if SCONJ iajs-1919 9 9 all all PRON iajs-1919 9 10 its its PRON iajs-1919 9 11 proper proper ADJ iajs-1919 9 12 submodules submodule NOUN iajs-1919 9 13 are be AUX iajs-1919 9 14 small small ADJ iajs-1919 9 15 [ [ X iajs-1919 9 16 2 2 NUM iajs-1919 9 17 ] ] PUNCT iajs-1919 9 18 . . PUNCT iajs-1919 10 1 the the DET iajs-1919 10 2 dual dual ADJ iajs-1919 10 3 concept concept NOUN iajs-1919 10 4 of of ADP iajs-1919 10 5 small small ADJ iajs-1919 10 6 submodule submodule NOUN iajs-1919 10 7 is be AUX iajs-1919 10 8 an an DET iajs-1919 10 9 essential essential ADJ iajs-1919 10 10 submodule submodule NOUN iajs-1919 10 11 , , PUNCT iajs-1919 10 12 where where SCONJ iajs-1919 10 13 a a DET iajs-1919 10 14 nonzero nonzero PROPN iajs-1919 10 15 submodule submodule PROPN iajs-1919 10 16 n n PROPN iajs-1919 10 17 of of ADP iajs-1919 10 18 a a DET iajs-1919 10 19 module module NOUN iajs-1919 10 20 m m VERB iajs-1919 10 21 is be AUX iajs-1919 10 22 called call VERB iajs-1919 10 23 essential essential ADJ iajs-1919 10 24 if if SCONJ iajs-1919 10 25 for for ADP iajs-1919 10 26 any any DET iajs-1919 10 27 submodule submodule NOUN iajs-1919 10 28 k k PROPN iajs-1919 10 29 of of ADP iajs-1919 10 30 m m PROPN iajs-1919 10 31 with with ADP iajs-1919 10 32 𝑁 𝑁 PROPN iajs-1919 10 33 ∩ ∩ ADJ iajs-1919 10 34 𝐾 𝐾 PROPN iajs-1919 10 35 0 0 NUM iajs-1919 10 36 implies imply VERB iajs-1919 10 37 𝐾 𝐾 PROPN iajs-1919 10 38 0 0 PROPN iajs-1919 10 39 . . PUNCT iajs-1919 11 1 a a DET iajs-1919 11 2 nonzero nonzero ADJ iajs-1919 11 3 r r NOUN iajs-1919 11 4 - - PUNCT iajs-1919 11 5 module module NOUN iajs-1919 11 6 m m NOUN iajs-1919 11 7 is be AUX iajs-1919 11 8 said say VERB iajs-1919 11 9 to to PART iajs-1919 11 10 be be AUX iajs-1919 11 11 uniform uniform ADJ iajs-1919 11 12 if if SCONJ iajs-1919 11 13 all all DET iajs-1919 11 14 its its PRON iajs-1919 11 15 nonzero nonzero NOUN iajs-1919 11 16 submodules submodule NOUN iajs-1919 11 17 are be AUX iajs-1919 11 18 essential essential ADJ iajs-1919 11 19 [ [ X iajs-1919 11 20 3 3 NUM iajs-1919 11 21 ] ] PUNCT iajs-1919 11 22 . . PUNCT iajs-1919 12 1 as as ADP iajs-1919 12 2 mixing mix VERB iajs-1919 12 3 of of ADP iajs-1919 12 4 concepts concept NOUN iajs-1919 12 5 small small ADJ iajs-1919 12 6 and and CCONJ iajs-1919 12 7 essential essential ADJ iajs-1919 12 8 submodules submodule NOUN iajs-1919 12 9 , , PUNCT iajs-1919 12 10 we we PRON iajs-1919 12 11 introduced introduce VERB iajs-1919 12 12 the the DET iajs-1919 12 13 following follow VERB iajs-1919 12 14 class class NOUN iajs-1919 12 15 of of ADP iajs-1919 12 16 submodules submodule NOUN iajs-1919 12 17 . . PUNCT iajs-1919 13 1 a a DET iajs-1919 13 2 submodule submodule NOUN iajs-1919 13 3 n n PROPN iajs-1919 13 4 of of ADP iajs-1919 13 5 m m PROPN iajs-1919 13 6 is be AUX iajs-1919 13 7 said say VERB iajs-1919 13 8 to to PART iajs-1919 13 9 be be AUX iajs-1919 13 10 s s NOUN iajs-1919 13 11 - - ADJ iajs-1919 13 12 essential essential ADJ iajs-1919 13 13 if if SCONJ iajs-1919 13 14 for for ADP iajs-1919 13 15 any any DET iajs-1919 13 16 small small ADJ iajs-1919 13 17 k k PROPN iajs-1919 13 18 in in ADP iajs-1919 13 19 m m PROPN iajs-1919 13 20 with with ADP iajs-1919 13 21 𝑁 𝑁 PROPN iajs-1919 13 22 ∩ ∩ ADJ iajs-1919 13 23 𝐾 𝐾 PROPN iajs-1919 13 24 0 0 NUM iajs-1919 13 25 implies imply VERB iajs-1919 13 26 𝐾 𝐾 PROPN iajs-1919 13 27 0 0 PUNCT iajs-1919 14 1 [ [ X iajs-1919 14 2 4 4 NUM iajs-1919 14 3 ] ] PUNCT iajs-1919 14 4 . . PUNCT iajs-1919 15 1 it it PRON iajs-1919 15 2 is be AUX iajs-1919 15 3 clear clear ADJ iajs-1919 15 4 essential essential ADJ iajs-1919 15 5 submdules submdule NOUN iajs-1919 15 6 implies imply VERB iajs-1919 15 7 s s NOUN iajs-1919 15 8 - - ADJ iajs-1919 15 9 essential essential ADJ iajs-1919 15 10 . . PUNCT iajs-1919 16 1 roman roman PROPN iajs-1919 16 2 c.s c.s PROPN iajs-1919 16 3 . . PROPN iajs-1919 17 1 in in ADP iajs-1919 17 2 [ [ X iajs-1919 17 3 5 5 NUM iajs-1919 17 4 ] ] PUNCT iajs-1919 17 5 , , PUNCT iajs-1919 17 6 recall recall VERB iajs-1919 17 7 that that SCONJ iajs-1919 17 8 an an DET iajs-1919 17 9 r r NOUN iajs-1919 17 10 - - PUNCT iajs-1919 17 11 module module NOUN iajs-1919 17 12 m m NOUN iajs-1919 17 13 is be AUX iajs-1919 17 14 called call VERB iajs-1919 17 15 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 17 16 if if SCONJ iajs-1919 17 17 for for ADP iajs-1919 17 18 any any DET iajs-1919 17 19 endomorphism endomorphism NOUN iajs-1919 17 20 𝜑 𝜑 PROPN iajs-1919 17 21 of of ADP iajs-1919 17 22 m m PRON iajs-1919 17 23 which which PRON iajs-1919 17 24 has have VERB iajs-1919 17 25 essential essential ADJ iajs-1919 17 26 kernel kernel NOUN iajs-1919 17 27 , , PUNCT iajs-1919 17 28 𝜑 𝜑 PROPN iajs-1919 17 29 0 0 NUM iajs-1919 17 30 . . PUNCT iajs-1919 18 1 𝒦-a 𝒦-a PROPN iajs-1919 18 2 nonsingular nonsingular PROPN iajs-1919 18 3 module module NOUN iajs-1919 18 4 is be AUX iajs-1919 18 5 studied study VERB iajs-1919 18 6 in in ADP iajs-1919 18 7 detail detail NOUN iajs-1919 18 8 by by ADP iajs-1919 18 9 [ [ X iajs-1919 18 10 6 6 NUM iajs-1919 18 11 ] ] PUNCT iajs-1919 18 12 . . PUNCT iajs-1919 19 1 in in ADP iajs-1919 19 2 this this DET iajs-1919 19 3 research research NOUN iajs-1919 19 4 , , PUNCT iajs-1919 19 5 we we PRON iajs-1919 19 6 introduced introduce VERB iajs-1919 19 7 concept concept NOUN iajs-1919 19 8 of of ADP iajs-1919 19 9 strongly strongly ADV iajs-1919 19 10 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 19 11 modules module NOUN iajs-1919 19 12 which which PRON iajs-1919 19 13 is be AUX iajs-1919 19 14 stronger strong ADJ iajs-1919 19 15 than than ADP iajs-1919 19 16 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 19 17 modules module NOUN iajs-1919 19 18 . . PUNCT iajs-1919 20 1 an an DET iajs-1919 20 2 r r NOUN iajs-1919 20 3 - - PUNCT iajs-1919 20 4 module module NOUN iajs-1919 20 5 m m NOUN iajs-1919 20 6 is be AUX iajs-1919 20 7 said say VERB iajs-1919 20 8 to to PART iajs-1919 20 9 be be AUX iajs-1919 20 10 strongly strongly ADV iajs-1919 20 11 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 20 12 if if SCONJ iajs-1919 20 13 for for ADP iajs-1919 20 14 each each DET iajs-1919 20 15 endomorphism endomorphism NOUN iajs-1919 20 16 of of ADP iajs-1919 20 17 m m PRON iajs-1919 20 18 which which PRON iajs-1919 20 19 has have VERB iajs-1919 20 20 s s NOUN iajs-1919 20 21 - - ADJ iajs-1919 20 22 essential essential ADJ iajs-1919 20 23 kernel kernel NOUN iajs-1919 20 24 , , PUNCT iajs-1919 20 25 is be AUX iajs-1919 20 26 zero zero NUM iajs-1919 20 27 . . PUNCT iajs-1919 21 1 in in ADP iajs-1919 21 2 section section NOUN iajs-1919 21 3 2 2 NUM iajs-1919 21 4 , , PUNCT iajs-1919 21 5 we we PRON iajs-1919 21 6 give give VERB iajs-1919 21 7 some some DET iajs-1919 21 8 characterizations characterization NOUN iajs-1919 21 9 and and CCONJ iajs-1919 21 10 properties property NOUN iajs-1919 21 11 of of ADP iajs-1919 21 12 this this DET iajs-1919 21 13 concept concept NOUN iajs-1919 21 14 . . PUNCT iajs-1919 22 1 in in ADP iajs-1919 22 2 section section NOUN iajs-1919 22 3 3 3 NUM iajs-1919 22 4 , , PUNCT iajs-1919 22 5 we we PRON iajs-1919 22 6 proved prove VERB iajs-1919 22 7 a a DET iajs-1919 22 8 strongly strongly ADV iajs-1919 22 9 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 22 10 module module NOUN iajs-1919 22 11 is be AUX iajs-1919 22 12 inherited inherit VERB iajs-1919 22 13 by by ADP iajs-1919 22 14 direct direct ADJ iajs-1919 22 15 summands summand NOUN iajs-1919 22 16 . . PUNCT iajs-1919 23 1 also also ADV iajs-1919 23 2 , , PUNCT iajs-1919 23 3 we we PRON iajs-1919 23 4 give give VERB iajs-1919 23 5 a a DET iajs-1919 23 6 condition condition NOUN iajs-1919 23 7 for for ADP iajs-1919 23 8 finite finite ADJ iajs-1919 23 9 direct direct ADJ iajs-1919 23 10 sums sum NOUN iajs-1919 23 11 of of ADP iajs-1919 23 12 strongly strongly ADV iajs-1919 23 13 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 23 14 modules module NOUN iajs-1919 23 15 to to PART iajs-1919 23 16 be be AUX iajs-1919 23 17 strongly strongly ADV iajs-1919 23 18 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 23 19 . . PUNCT iajs-1919 24 1 several several ADJ iajs-1919 24 2 connections connection NOUN iajs-1919 24 3 between between ADP iajs-1919 24 4 strongly strongly ADV iajs-1919 24 5 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 24 6 and and CCONJ iajs-1919 24 7 other other ADJ iajs-1919 24 8 classes class NOUN iajs-1919 24 9 , , PUNCT iajs-1919 24 10 also also ADV iajs-1919 24 11 some some DET iajs-1919 24 12 examples example NOUN iajs-1919 24 13 are be AUX iajs-1919 24 14 proved prove VERB iajs-1919 24 15 in in ADP iajs-1919 24 16 section section NOUN iajs-1919 24 17 4 4 NUM iajs-1919 24 18 . . PUNCT iajs-1919 25 1 throughout throughout ADP iajs-1919 25 2 this this DET iajs-1919 25 3 work work NOUN iajs-1919 25 4 , , PUNCT iajs-1919 25 5 all all DET iajs-1919 25 6 rings ring NOUN iajs-1919 25 7 are be AUX iajs-1919 25 8 associative associative ADJ iajs-1919 25 9 with with ADP iajs-1919 25 10 identity identity NOUN iajs-1919 25 11 and and CCONJ iajs-1919 25 12 all all DET iajs-1919 25 13 modules module NOUN iajs-1919 25 14 are be AUX iajs-1919 25 15 unitary unitary ADJ iajs-1919 25 16 right right ADJ iajs-1919 25 17 r r NOUN iajs-1919 25 18 - - PUNCT iajs-1919 25 19 modules module NOUN iajs-1919 25 20 . . PUNCT iajs-1919 26 1 for for ADP iajs-1919 26 2 a a DET iajs-1919 26 3 right right ADJ iajs-1919 26 4 r r NOUN iajs-1919 26 5 - - PUNCT iajs-1919 26 6 module module NOUN iajs-1919 26 7 m m NOUN iajs-1919 26 8 , , PUNCT iajs-1919 26 9 the the DET iajs-1919 26 10 notations𝑁 notations𝑁 PROPN iajs-1919 26 11 ⊆ ⊆ NUM iajs-1919 26 12 𝑀 𝑀 PROPN iajs-1919 26 13 , , PUNCT iajs-1919 26 14 𝑁 𝑁 PROPN iajs-1919 26 15 𝑀 𝑀 PROPN iajs-1919 26 16 , , PUNCT iajs-1919 26 17 𝑁 𝑁 PROPN iajs-1919 26 18 ≪ ≪ ADJ iajs-1919 26 19 𝑀 𝑀 PROPN iajs-1919 26 20 , , PUNCT iajs-1919 26 21 𝑁 𝑁 PROPN iajs-1919 26 22 ⊴ ⊴ NUM iajs-1919 26 23 𝑀 𝑀 PROPN iajs-1919 26 24 , , PUNCT iajs-1919 26 25 𝑁 𝑁 PROPN iajs-1919 26 26 ⊴ ⊴ ADP iajs-1919 26 27 𝑀 𝑀 PROPN iajs-1919 26 28 or or CCONJ iajs-1919 26 29 𝑁 𝑁 PROPN iajs-1919 26 30 ⨁ ⨁ PROPN iajs-1919 26 31 𝑀 𝑀 PROPN iajs-1919 26 32 denotes denote VERB iajs-1919 26 33 that that SCONJ iajs-1919 26 34 n n VERB iajs-1919 26 35 is be AUX iajs-1919 26 36 a a DET iajs-1919 26 37 subset subset NOUN iajs-1919 26 38 , , PUNCT iajs-1919 26 39 a a DET iajs-1919 26 40 submodule submodule NOUN iajs-1919 26 41 , , PUNCT iajs-1919 26 42 a a DET iajs-1919 26 43 small small ADJ iajs-1919 26 44 submodule submodule NOUN iajs-1919 26 45 , , PUNCT iajs-1919 26 46 an an DET iajs-1919 26 47 essential essential ADJ iajs-1919 26 48 submodule submodule NOUN iajs-1919 26 49 , , PUNCT iajs-1919 26 50 a a DET iajs-1919 26 51 s s ADJ iajs-1919 26 52 - - ADJ iajs-1919 26 53 essential essential ADJ iajs-1919 26 54 submodule submodule NOUN iajs-1919 26 55 , , PUNCT iajs-1919 26 56 or or CCONJ iajs-1919 26 57 direct direct ADJ iajs-1919 26 58 summand summand NOUN iajs-1919 26 59 of of ADP iajs-1919 26 60 m m PROPN iajs-1919 26 61 , , PUNCT iajs-1919 26 62         SPACE iajs-1919 26 63 168 168 NUM iajs-1919 26 64     SPACE iajs-1919 26 65 ibn ibn PROPN iajs-1919 26 66 al al PROPN iajs-1919 26 67 - - PUNCT iajs-1919 26 68 haitham haitham PROPN iajs-1919 26 69 jour.for jour.for PROPN iajs-1919 26 70 pure&appl.sci pure&appl.sci PROPN iajs-1919 26 71 . . PUNCT iajs-1919 27 1 ihjpas ihjpas PROPN iajs-1919 27 2 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 27 3 vol vol NOUN iajs-1919 27 4 . . PUNCT iajs-1919 27 5 32 32 NUM iajs-1919 27 6 ( ( PUNCT iajs-1919 27 7 1 1 NUM iajs-1919 27 8 ) ) PUNCT iajs-1919 27 9 2019 2019 NUM iajs-1919 27 10 respectively respectively ADV iajs-1919 27 11 . . PUNCT iajs-1919 28 1 also also ADV iajs-1919 28 2 , , PUNCT iajs-1919 28 3 for for ADP iajs-1919 28 4 𝑁 𝑁 PROPN iajs-1919 28 5 𝑀 𝑀 PROPN iajs-1919 28 6 , , PUNCT iajs-1919 28 7 we we PRON iajs-1919 28 8 denote denote VERB iajs-1919 28 9 the the DET iajs-1919 28 10 endomorphism endomorphism NOUN iajs-1919 28 11 ring ring NOUN iajs-1919 28 12 of of ADP iajs-1919 28 13 m m PRON iajs-1919 28 14 by by ADP iajs-1919 28 15 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 28 16 𝑀 𝑀 PROPN iajs-1919 28 17 , , PUNCT iajs-1919 28 18 𝑟 𝑟 X iajs-1919 28 19 𝑁 𝑁 ADP iajs-1919 28 20 𝑟 𝑟 PRON iajs-1919 28 21 ∈ ∈ NOUN iajs-1919 28 22 𝑅| 𝑅| PROPN iajs-1919 28 23 𝑁𝑟 𝑁𝑟 PROPN iajs-1919 28 24 0 0 NUM iajs-1919 28 25 and and CCONJ iajs-1919 28 26 𝑁 𝑁 PROPN iajs-1919 28 27 : : PUNCT iajs-1919 28 28 𝑀 𝑀 PROPN iajs-1919 28 29 𝑟 𝑟 NOUN iajs-1919 28 30 ∈ ∈ PROPN iajs-1919 28 31 𝑅| 𝑅| PROPN iajs-1919 28 32 𝑀𝑟 𝑀𝑟 PROPN iajs-1919 28 33 ⊆ ⊆ NUM iajs-1919 28 34 𝑁 𝑁 PROPN iajs-1919 28 35 . . PUNCT iajs-1919 29 1 starting start VERB iajs-1919 29 2 , , PUNCT iajs-1919 29 3 we we PRON iajs-1919 29 4 will will AUX iajs-1919 29 5 state state VERB iajs-1919 29 6 some some DET iajs-1919 29 7 properties property NOUN iajs-1919 29 8 of of ADP iajs-1919 29 9 s s NOUN iajs-1919 29 10 - - ADJ iajs-1919 29 11 essential essential ADJ iajs-1919 29 12 submodules submodule NOUN iajs-1919 29 13 in in ADP iajs-1919 29 14 [ [ X iajs-1919 29 15 4 4 NUM iajs-1919 29 16 , , PUNCT iajs-1919 29 17 prop prop NOUN iajs-1919 29 18 . . PUNCT iajs-1919 30 1 2.7 2.7 NUM iajs-1919 30 2 ] ] PUNCT iajs-1919 30 3 which which PRON iajs-1919 30 4 needed need VERB iajs-1919 30 5 in in ADP iajs-1919 30 6 this this DET iajs-1919 30 7 work work NOUN iajs-1919 30 8 . . PUNCT iajs-1919 31 1 proposition proposition NOUN iajs-1919 31 2 1 1 NUM iajs-1919 31 3 : : PUNCT iajs-1919 31 4 let let VERB iajs-1919 31 5 m m PRON iajs-1919 31 6 be be AUX iajs-1919 31 7 a a DET iajs-1919 31 8 module module NOUN iajs-1919 31 9 . . PUNCT iajs-1919 32 1 then then ADV iajs-1919 32 2 ; ; PUNCT iajs-1919 32 3 ( ( PUNCT iajs-1919 32 4 1 1 X iajs-1919 32 5 ) ) PUNCT iajs-1919 32 6 assume assume VERB iajs-1919 32 7 𝑁 𝑁 PROPN iajs-1919 32 8 , , PUNCT iajs-1919 32 9 𝐾 𝐾 PROPN iajs-1919 32 10 , , PUNCT iajs-1919 32 11 𝐿 𝐿 PROPN iajs-1919 32 12 are be AUX iajs-1919 32 13 submodules submodule NOUN iajs-1919 32 14 of of ADP iajs-1919 32 15 m m PROPN iajs-1919 32 16 with with ADP iajs-1919 32 17 𝐾 𝐾 PROPN iajs-1919 32 18 𝑁. 𝑁. PROPN iajs-1919 32 19 𝑖 𝑖 NOUN iajs-1919 32 20 if if SCONJ iajs-1919 32 21 𝐾 𝐾 PROPN iajs-1919 32 22 ⊴ ⊴ PROPN iajs-1919 32 23 𝑀 𝑀 PROPN iajs-1919 32 24 , , PUNCT iajs-1919 32 25 then then ADV iajs-1919 32 26 𝐾 𝐾 PROPN iajs-1919 32 27 ⊴ ⊴ ADP iajs-1919 32 28 𝑁 𝑁 PROPN iajs-1919 32 29 and and CCONJ iajs-1919 32 30 𝑁 𝑁 PROPN iajs-1919 32 31 ⊴ ⊴ ADP iajs-1919 32 32 𝑀. 𝑀. PROPN iajs-1919 32 33 𝑖𝑖 𝑖𝑖 NOUN iajs-1919 33 1 𝑁 𝑁 PROPN iajs-1919 33 2 ⊴ ⊴ ADP iajs-1919 33 3 𝑀 𝑀 PROPN iajs-1919 33 4 and and CCONJ iajs-1919 33 5 𝐿 𝐿 PROPN iajs-1919 33 6 ⊴ ⊴ ADP iajs-1919 33 7 𝑀 𝑀 PROPN iajs-1919 34 1 if if SCONJ iajs-1919 34 2 and and CCONJ iajs-1919 34 3 only only ADV iajs-1919 34 4 if if SCONJ iajs-1919 34 5 𝑁 𝑁 PROPN iajs-1919 34 6 ∩ ∩ ADJ iajs-1919 34 7 𝐿 𝐿 PROPN iajs-1919 34 8 ⊴ ⊴ PROPN iajs-1919 34 9 𝑀. 𝑀. PROPN iajs-1919 34 10 ( ( PUNCT iajs-1919 34 11 2 2 NUM iajs-1919 34 12 ) ) PUNCT iajs-1919 34 13 if if SCONJ iajs-1919 34 14 𝜑 𝜑 X iajs-1919 34 15 : : PUNCT iajs-1919 34 16 𝑀 𝑀 PROPN iajs-1919 34 17 → → SYM iajs-1919 34 18 𝑀 𝑀 PROPN iajs-1919 34 19 is be AUX iajs-1919 34 20 a a DET iajs-1919 34 21 homomorphism homomorphism NOUN iajs-1919 34 22 with with ADP iajs-1919 34 23 𝐾 𝐾 PROPN iajs-1919 34 24 ⊴ ⊴ PROPN iajs-1919 34 25 𝑀 𝑀 PROPN iajs-1919 34 26 , , PUNCT iajs-1919 34 27 then then ADV iajs-1919 34 28 𝜑 𝜑 PROPN iajs-1919 34 29 𝐾 𝐾 PROPN iajs-1919 34 30 ⊴ ⊴ ADP iajs-1919 34 31 𝑀. 𝑀. PROPN iajs-1919 34 32 ( ( PUNCT iajs-1919 34 33 3 3 NUM iajs-1919 34 34 ) ) PUNCT iajs-1919 34 35 if if SCONJ iajs-1919 34 36 𝐾 𝐾 PROPN iajs-1919 34 37 ⊆ ⊆ NUM iajs-1919 34 38 𝑀 𝑀 PROPN iajs-1919 34 39 ⊆ ⊆ NUM iajs-1919 34 40 𝑀 𝑀 PROPN iajs-1919 34 41 , , PUNCT iajs-1919 34 42 𝐾 𝐾 PROPN iajs-1919 34 43 ⊆ ⊆ NUM iajs-1919 34 44 𝑀 𝑀 PROPN iajs-1919 34 45 ⊆ ⊆ NUM iajs-1919 34 46 𝑀 𝑀 PROPN iajs-1919 34 47 and and CCONJ iajs-1919 34 48 𝑀 𝑀 PROPN iajs-1919 34 49 𝑀 𝑀 PROPN iajs-1919 34 50 ⨁𝑀 ⨁𝑀 PROPN iajs-1919 34 51 . . PUNCT iajs-1919 35 1 then then ADV iajs-1919 35 2 𝐾 𝐾 PROPN iajs-1919 35 3 ⨁𝐾 ⨁𝐾 X iajs-1919 35 4 ⊴ ⊴ ADP iajs-1919 35 5 𝑀 𝑀 PROPN iajs-1919 35 6 ⨁𝑀 ⨁𝑀 NOUN iajs-1919 35 7 if if SCONJ iajs-1919 35 8 and and CCONJ iajs-1919 35 9 only only ADV iajs-1919 35 10 if if SCONJ iajs-1919 35 11 𝐾 𝐾 PROPN iajs-1919 35 12 ⊴ ⊴ ADP iajs-1919 35 13 𝑀 𝑀 PROPN iajs-1919 35 14 for for ADP iajs-1919 35 15 𝑖 𝑖 PROPN iajs-1919 35 16 1,2 1,2 NUM iajs-1919 35 17 . . PUNCT iajs-1919 36 1 2 2 NUM iajs-1919 36 2 . . X iajs-1919 36 3 strongly strongly ADV iajs-1919 36 4 𝓚-nonsigular 𝓚-nonsigular ADJ iajs-1919 36 5 modules module NOUN iajs-1919 36 6 in in ADP iajs-1919 36 7 this this DET iajs-1919 36 8 section section NOUN iajs-1919 36 9 , , PUNCT iajs-1919 36 10 we we PRON iajs-1919 36 11 introduce introduce VERB iajs-1919 36 12 the the DET iajs-1919 36 13 class class NOUN iajs-1919 36 14 of of ADP iajs-1919 36 15 strongly strongly ADV iajs-1919 36 16 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 36 17 modules module NOUN iajs-1919 36 18 as as ADP iajs-1919 36 19 a a DET iajs-1919 36 20 stronger strong ADJ iajs-1919 36 21 class class NOUN iajs-1919 36 22 of of ADP iajs-1919 36 23 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 36 24 modules module NOUN iajs-1919 36 25 . . PUNCT iajs-1919 37 1 several several ADJ iajs-1919 37 2 various various ADJ iajs-1919 37 3 properties property NOUN iajs-1919 37 4 are be AUX iajs-1919 37 5 proved prove VERB iajs-1919 37 6 . . PUNCT iajs-1919 38 1 definition definition NOUN iajs-1919 38 2 2 2 NUM iajs-1919 38 3 . . PUNCT iajs-1919 39 1 an an DET iajs-1919 39 2 r r NOUN iajs-1919 39 3 - - PUNCT iajs-1919 39 4 module module NOUN iajs-1919 39 5 m m NOUN iajs-1919 39 6 is be AUX iajs-1919 39 7 said say VERB iajs-1919 39 8 to to PART iajs-1919 39 9 be be AUX iajs-1919 39 10 strongly strongly ADV iajs-1919 39 11 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 39 12 if if SCONJ iajs-1919 39 13 for for ADP iajs-1919 39 14 all all DET iajs-1919 39 15 𝜑 𝜑 PRON iajs-1919 39 16 ∈ ∈ PROPN iajs-1919 39 17 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 39 18 𝑀 𝑀 PROPN iajs-1919 39 19 with with ADP iajs-1919 39 20 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 39 21 is be AUX iajs-1919 39 22 s s NOUN iajs-1919 39 23 - - ADJ iajs-1919 39 24 essential essential ADJ iajs-1919 39 25 in in ADP iajs-1919 39 26 m m PROPN iajs-1919 39 27 , , PUNCT iajs-1919 39 28 implies imply VERB iajs-1919 39 29 𝜑 𝜑 PROPN iajs-1919 39 30 0 0 X iajs-1919 39 31 . . PUNCT iajs-1919 40 1 also also ADV iajs-1919 40 2 , , PUNCT iajs-1919 40 3 a a DET iajs-1919 40 4 ring ring NOUN iajs-1919 40 5 r r NOUN iajs-1919 40 6 is be AUX iajs-1919 40 7 strongly strongly ADV iajs-1919 40 8 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 40 9 if if SCONJ iajs-1919 40 10 it it PRON iajs-1919 40 11 is be AUX iajs-1919 40 12 a a DET iajs-1919 40 13 strongly strongly ADV iajs-1919 40 14 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 40 15 r r NOUN iajs-1919 40 16 - - PUNCT iajs-1919 40 17 module module NOUN iajs-1919 40 18 . . PUNCT iajs-1919 41 1 for for ADP iajs-1919 41 2 𝑁 𝑁 PROPN iajs-1919 41 3 𝑀 𝑀 PROPN iajs-1919 41 4 , , PUNCT iajs-1919 41 5 if if SCONJ iajs-1919 41 6 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 41 7 , , PUNCT iajs-1919 41 8 𝑀 𝑀 PROPN iajs-1919 41 9 0 0 NUM iajs-1919 41 10 then then ADV iajs-1919 41 11 n n PROPN iajs-1919 41 12 is be AUX iajs-1919 41 13 called call VERB iajs-1919 41 14 quasi quasi ADJ iajs-1919 41 15 - - ADJ iajs-1919 41 16 invertible invertible ADJ iajs-1919 41 17 [ [ X iajs-1919 41 18 7 7 NUM iajs-1919 41 19 ] ] PUNCT iajs-1919 41 20 . . PUNCT iajs-1919 42 1 firstly firstly ADV iajs-1919 42 2 , , PUNCT iajs-1919 42 3 we we PRON iajs-1919 42 4 are be AUX iajs-1919 42 5 now now ADV iajs-1919 42 6 in in ADP iajs-1919 42 7 a a DET iajs-1919 42 8 position position NOUN iajs-1919 42 9 to to PART iajs-1919 42 10 give give VERB iajs-1919 42 11 a a DET iajs-1919 42 12 characterization characterization NOUN iajs-1919 42 13 the the DET iajs-1919 42 14 notion notion NOUN iajs-1919 42 15 of of ADP iajs-1919 42 16 strongly strongly ADV iajs-1919 42 17 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 42 18 modules module NOUN iajs-1919 42 19 . . PUNCT iajs-1919 43 1 theorem theorem NOUN iajs-1919 43 2 3 3 NUM iajs-1919 43 3 . . PUNCT iajs-1919 44 1 a a DET iajs-1919 44 2 module module NOUN iajs-1919 44 3 m m NOUN iajs-1919 44 4 is be AUX iajs-1919 44 5 strongly strongly ADV iajs-1919 44 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 44 7 if if SCONJ iajs-1919 44 8 and and CCONJ iajs-1919 44 9 only only ADV iajs-1919 44 10 if if SCONJ iajs-1919 44 11 all all DET iajs-1919 44 12 its its PRON iajs-1919 44 13 s s ADJ iajs-1919 44 14 - - ADJ iajs-1919 44 15 essential essential ADJ iajs-1919 44 16 submodules submodule NOUN iajs-1919 44 17 are be AUX iajs-1919 44 18 quasi quasi ADJ iajs-1919 44 19 - - ADJ iajs-1919 44 20 invertible invertible ADJ iajs-1919 44 21 . . PUNCT iajs-1919 45 1 proof proof NOUN iajs-1919 45 2 . . PUNCT iajs-1919 46 1 assume assume VERB iajs-1919 46 2 m m PROPN iajs-1919 46 3 is be AUX iajs-1919 46 4 a a DET iajs-1919 46 5 strongly strongly ADV iajs-1919 46 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 46 7 r r NOUN iajs-1919 46 8 - - PUNCT iajs-1919 46 9 module module NOUN iajs-1919 46 10 . . PUNCT iajs-1919 47 1 let let VERB iajs-1919 47 2 𝑁 𝑁 PROPN iajs-1919 47 3 ⊴ ⊴ ADP iajs-1919 47 4 𝑀 𝑀 PROPN iajs-1919 47 5 and and CCONJ iajs-1919 47 6 n n PROPN iajs-1919 47 7 is be AUX iajs-1919 47 8 not not PART iajs-1919 47 9 quasiinvertible quasiinvertible ADJ iajs-1919 47 10 , , PUNCT iajs-1919 47 11 i.e. i.e. X iajs-1919 47 12 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 47 13 , , PUNCT iajs-1919 47 14 𝑀 𝑀 PROPN iajs-1919 47 15 0 0 NUM iajs-1919 47 16 , , PUNCT iajs-1919 47 17 so so SCONJ iajs-1919 47 18 there there PRON iajs-1919 47 19 exists exist VERB iajs-1919 47 20 0 0 NUM iajs-1919 47 21 𝜑 𝜑 NOUN iajs-1919 47 22 : : PUNCT iajs-1919 47 23 → → SYM iajs-1919 47 24 𝑀. 𝑀. NOUN iajs-1919 47 25 consider consider VERB iajs-1919 47 26 𝜓 𝜓 PRON iajs-1919 47 27 𝜑 𝜑 X iajs-1919 47 28 ∘ ∘ X iajs-1919 47 29 𝜋 𝜋 ADP iajs-1919 47 30 ∈ ∈ PROPN iajs-1919 47 31 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 47 32 𝑀 𝑀 PROPN iajs-1919 47 33 , , PUNCT iajs-1919 47 34 where where SCONJ iajs-1919 47 35 𝜋 𝜋 NOUN iajs-1919 47 36 is be AUX iajs-1919 47 37 a a DET iajs-1919 47 38 natural natural ADJ iajs-1919 47 39 epimorphism epimorphism NOUN iajs-1919 47 40 map map NOUN iajs-1919 47 41 . . PUNCT iajs-1919 48 1 it it PRON iajs-1919 48 2 is be AUX iajs-1919 48 3 clear clear ADJ iajs-1919 48 4 that that SCONJ iajs-1919 48 5 𝑁 𝑁 PROPN iajs-1919 48 6 ⊆ ⊆ NUM iajs-1919 48 7 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 NOUN iajs-1919 48 8 , , PUNCT iajs-1919 48 9 but but CCONJ iajs-1919 48 10 𝑁 𝑁 PROPN iajs-1919 48 11 ⊴ ⊴ NUM iajs-1919 48 12 𝑀 𝑀 PROPN iajs-1919 48 13 , , PUNCT iajs-1919 48 14 this this PRON iajs-1919 48 15 implies imply VERB iajs-1919 48 16 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 PROPN iajs-1919 48 17 ⊴ ⊴ PROPN iajs-1919 48 18 𝑀 𝑀 PROPN iajs-1919 48 19 , , PUNCT iajs-1919 48 20 and and CCONJ iajs-1919 48 21 hence hence ADV iajs-1919 48 22 𝜓 𝜓 PROPN iajs-1919 48 23 0 0 NUM iajs-1919 48 24 , , PUNCT iajs-1919 48 25 as as SCONJ iajs-1919 48 26 m m PROPN iajs-1919 48 27 is be AUX iajs-1919 48 28 strongly strongly ADV iajs-1919 48 29 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 48 30 , , PUNCT iajs-1919 48 31 thus thus ADV iajs-1919 48 32 𝜑 𝜑 PROPN iajs-1919 48 33 0 0 NUM iajs-1919 48 34 , , PUNCT iajs-1919 48 35 a a DET iajs-1919 48 36 contradiction contradiction NOUN iajs-1919 48 37 . . PUNCT iajs-1919 49 1 therefore therefore ADV iajs-1919 49 2 𝑁 𝑁 PROPN iajs-1919 49 3 ⊴ ⊴ ADP iajs-1919 49 4 𝑀 𝑀 PROPN iajs-1919 49 5 and and CCONJ iajs-1919 49 6 n n PROPN iajs-1919 49 7 is be AUX iajs-1919 49 8 quasi quasi ADJ iajs-1919 49 9 - - ADJ iajs-1919 49 10 invertible invertible ADJ iajs-1919 49 11 . . PUNCT iajs-1919 50 1 conversely conversely ADV iajs-1919 50 2 , , PUNCT iajs-1919 50 3 let let VERB iajs-1919 50 4 0 0 NUM iajs-1919 50 5 𝑓 𝑓 DET iajs-1919 50 6 ∈ ∈ PROPN iajs-1919 50 7 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 50 8 𝑀 𝑀 PROPN iajs-1919 50 9 . . PUNCT iajs-1919 51 1 if if SCONJ iajs-1919 51 2 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 PROPN iajs-1919 51 3 ⊴ ⊴ PROPN iajs-1919 51 4 𝑀 𝑀 PROPN iajs-1919 51 5 , , PUNCT iajs-1919 51 6 so so ADV iajs-1919 51 7 by by ADP iajs-1919 51 8 hypothesis hypothesis NOUN iajs-1919 51 9 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 51 10 is be AUX iajs-1919 51 11 quasi quasi ADJ iajs-1919 51 12 - - ADJ iajs-1919 51 13 invertible invertible ADJ iajs-1919 51 14 . . PUNCT iajs-1919 52 1 but but CCONJ iajs-1919 52 2 , , PUNCT iajs-1919 52 3 we we PRON iajs-1919 52 4 can can AUX iajs-1919 52 5 define define VERB iajs-1919 52 6 a a DET iajs-1919 52 7 homomorphism homomorphism NOUN iajs-1919 52 8 ℎ ℎ ADP iajs-1919 52 9 : : PUNCT iajs-1919 52 10 → → SYM iajs-1919 52 11 𝑀 𝑀 PROPN iajs-1919 52 12 by by ADP iajs-1919 52 13 ℎ ℎ X iajs-1919 52 14 𝑚 𝑚 PROPN iajs-1919 52 15 𝐾𝑒𝑟𝑓 𝐾𝑒𝑟𝑓 PROPN iajs-1919 52 16 𝑓 𝑓 DET iajs-1919 52 17 𝑚 𝑚 NOUN iajs-1919 52 18 for for ADP iajs-1919 52 19 all all DET iajs-1919 52 20 𝑚 𝑚 ADP iajs-1919 52 21 ∈ ∈ PROPN iajs-1919 52 22 𝑀. 𝑀. NOUN iajs-1919 52 23 so so ADV iajs-1919 52 24 ℎ ℎ NOUN iajs-1919 52 25 0 0 PUNCT iajs-1919 52 26 and and CCONJ iajs-1919 52 27 hence hence ADV iajs-1919 52 28 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 52 29 , , PUNCT iajs-1919 52 30 𝑀 𝑀 PROPN iajs-1919 52 31 0 0 NUM iajs-1919 52 32 which which PRON iajs-1919 52 33 is be AUX iajs-1919 52 34 a a DET iajs-1919 52 35 contradiction contradiction NOUN iajs-1919 52 36 with with ADP iajs-1919 52 37 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 52 38 is be AUX iajs-1919 52 39 quasi quasi ADJ iajs-1919 52 40 - - ADJ iajs-1919 52 41 invertible invertible ADJ iajs-1919 52 42 . . PUNCT iajs-1919 53 1 therefore therefore ADV iajs-1919 53 2 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 PROPN iajs-1919 53 3 ⋬ ⋬ AUX iajs-1919 53 4 𝑀 𝑀 PROPN iajs-1919 54 1 and and CCONJ iajs-1919 54 2 m m PROPN iajs-1919 54 3 is be AUX iajs-1919 54 4 a a DET iajs-1919 54 5 strongly strongly ADV iajs-1919 54 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 54 7 r r NOUN iajs-1919 54 8 - - PUNCT iajs-1919 54 9 module module NOUN iajs-1919 54 10 . . PUNCT iajs-1919 55 1 ∎ ∎ PROPN iajs-1919 55 2 corollary corollary ADJ iajs-1919 55 3 4 4 NUM iajs-1919 55 4 . . PUNCT iajs-1919 56 1 let let VERB iajs-1919 56 2 m m PRON iajs-1919 56 3 be be AUX iajs-1919 56 4 a a DET iajs-1919 56 5 strongly strongly ADV iajs-1919 56 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 56 7 module module NOUN iajs-1919 56 8 . . PUNCT iajs-1919 57 1 if if SCONJ iajs-1919 57 2 𝑁 𝑁 PROPN iajs-1919 57 3 ⊴ ⊴ PROPN iajs-1919 57 4 𝑀 𝑀 PROPN iajs-1919 57 5 , , PUNCT iajs-1919 57 6 then then ADV iajs-1919 57 7 𝑟 𝑟 X iajs-1919 57 8 𝑁 𝑁 PROPN iajs-1919 57 9 𝑟 𝑟 PRON iajs-1919 57 10 𝑀 𝑀 PROPN iajs-1919 57 11 . . PUNCT iajs-1919 58 1 proof proof NOUN iajs-1919 58 2 . . PUNCT iajs-1919 59 1 assume assume VERB iajs-1919 59 2 𝑁 𝑁 PROPN iajs-1919 59 3 ⊴ ⊴ PROPN iajs-1919 59 4 𝑀 𝑀 PROPN iajs-1919 59 5 , , PUNCT iajs-1919 59 6 then then ADV iajs-1919 59 7 by by ADP iajs-1919 59 8 previous previous ADJ iajs-1919 59 9 theorem theorem NOUN iajs-1919 59 10 , , PUNCT iajs-1919 59 11 n n PRON iajs-1919 59 12 is be AUX iajs-1919 59 13 a a DET iajs-1919 59 14 quasi quasi ADJ iajs-1919 59 15 - - ADJ iajs-1919 59 16 invertible invertible ADJ iajs-1919 59 17 submodule submodule NOUN iajs-1919 59 18 , , PUNCT iajs-1919 59 19 and and CCONJ iajs-1919 59 20 so so ADV iajs-1919 59 21 𝑟 𝑟 X iajs-1919 59 22 𝑁 𝑁 PROPN iajs-1919 59 23 𝑟 𝑟 PRON iajs-1919 59 24 𝑀 𝑀 NOUN iajs-1919 59 25 by by ADP iajs-1919 59 26 [ [ X iajs-1919 59 27 7 7 NUM iajs-1919 59 28 , , PUNCT iajs-1919 59 29 prop prop NOUN iajs-1919 59 30 . . PUNCT iajs-1919 60 1 1.1.4 1.1.4 NUM iajs-1919 60 2 ] ] PUNCT iajs-1919 60 3 . . PUNCT iajs-1919 61 1 ∎ ∎ ADJ iajs-1919 61 2         SPACE iajs-1919 61 3 169 169 NUM iajs-1919 61 4     SPACE iajs-1919 61 5 ibn ibn PROPN iajs-1919 61 6 al al PROPN iajs-1919 61 7 - - PUNCT iajs-1919 61 8 haitham haitham PROPN iajs-1919 61 9 jour.for jour.for PROPN iajs-1919 61 10 pure&appl.sci pure&appl.sci PROPN iajs-1919 61 11 . . PUNCT iajs-1919 61 12 ihjpas ihjpas PROPN iajs-1919 61 13 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 61 14 vol vol NOUN iajs-1919 61 15 . . PUNCT iajs-1919 62 1 32 32 NUM iajs-1919 63 1 ( ( PUNCT iajs-1919 63 2 1 1 NUM iajs-1919 63 3 ) ) PUNCT iajs-1919 63 4 2019 2019 NUM iajs-1919 63 5 proposition proposition NOUN iajs-1919 63 6 5 5 NUM iajs-1919 63 7 . . PUNCT iajs-1919 64 1 let let VERB iajs-1919 64 2 m m PRON iajs-1919 64 3 be be AUX iajs-1919 64 4 an an DET iajs-1919 64 5 r r NOUN iajs-1919 64 6 - - PUNCT iajs-1919 64 7 module module NOUN iajs-1919 64 8 , , PUNCT iajs-1919 64 9 𝑅∗ 𝑅∗ NUM iajs-1919 64 10 𝑅 𝑅 NOUN iajs-1919 64 11 𝐴⁄ 𝐴⁄ PROPN iajs-1919 64 12 and and CCONJ iajs-1919 64 13 𝐴 𝐴 PROPN iajs-1919 64 14 ⊆ ⊆ NUM iajs-1919 64 15 𝑟 𝑟 DET iajs-1919 64 16 𝑀 𝑀 PROPN iajs-1919 64 17 . . PUNCT iajs-1919 65 1 then then ADV iajs-1919 65 2 m m PROPN iajs-1919 65 3 is be AUX iajs-1919 65 4 a a DET iajs-1919 65 5 strongly strongly ADV iajs-1919 65 6 𝒦nonsingular 𝒦nonsingular ADJ iajs-1919 65 7 r r NOUN iajs-1919 65 8 - - PUNCT iajs-1919 65 9 module module NOUN iajs-1919 65 10 if if SCONJ iajs-1919 65 11 and and CCONJ iajs-1919 65 12 only only ADV iajs-1919 65 13 if if SCONJ iajs-1919 65 14 m m NOUN iajs-1919 65 15 is be AUX iajs-1919 65 16 a a DET iajs-1919 65 17 strongly strongly ADV iajs-1919 65 18 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 65 19 𝑅∗-module 𝑅∗-module NOUN iajs-1919 65 20 . . PUNCT iajs-1919 66 1 proof proof NOUN iajs-1919 66 2 . . PUNCT iajs-1919 67 1 assume assume VERB iajs-1919 67 2 𝜋 𝜋 NOUN iajs-1919 67 3 : : PUNCT iajs-1919 67 4 𝑅 𝑅 PROPN iajs-1919 67 5 → → SYM iajs-1919 67 6 𝑅∗ 𝑅∗ NUM iajs-1919 67 7 is be AUX iajs-1919 67 8 a a DET iajs-1919 67 9 natural natural ADJ iajs-1919 67 10 epimorphism epimorphism NOUN iajs-1919 67 11 , , PUNCT iajs-1919 67 12 so so ADV iajs-1919 67 13 by by ADP iajs-1919 67 14 [ [ X iajs-1919 67 15 8 8 NUM iajs-1919 67 16 , , PUNCT iajs-1919 67 17 ex ex NOUN iajs-1919 67 18 . . PUNCT iajs-1919 67 19 p.51 p.51 NOUN iajs-1919 67 20 ] ] X iajs-1919 67 21 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 67 22 , , PUNCT iajs-1919 67 23 𝑀 𝑀 PROPN iajs-1919 67 24 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 67 25 ∗ ∗ NOUN iajs-1919 67 26 , , PUNCT iajs-1919 67 27 𝑀 𝑀 PROPN iajs-1919 67 28 for for ADP iajs-1919 67 29 each each DET iajs-1919 67 30 submodule submodule NOUN iajs-1919 67 31 n n PROPN iajs-1919 67 32 of of ADP iajs-1919 67 33 m. m. NOUN iajs-1919 67 34 so so ADV iajs-1919 67 35 , , PUNCT iajs-1919 67 36 the the DET iajs-1919 67 37 result result NOUN iajs-1919 67 38 is be AUX iajs-1919 67 39 follow follow VERB iajs-1919 67 40 . . PUNCT iajs-1919 68 1 ∎ ∎ NOUN iajs-1919 68 2 proposition proposition NOUN iajs-1919 68 3 6 6 NUM iajs-1919 68 4 . . PUNCT iajs-1919 69 1 let let VERB iajs-1919 69 2 m m PRON iajs-1919 69 3 be be AUX iajs-1919 69 4 a a DET iajs-1919 69 5 strongly strongly ADV iajs-1919 69 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 69 7 module module NOUN iajs-1919 69 8 with with ADP iajs-1919 69 9 𝑀 𝑀 PROPN iajs-1919 69 10 𝑋⁄ 𝑋⁄ PROPN iajs-1919 69 11 is be AUX iajs-1919 69 12 a a DET iajs-1919 69 13 projective projective ADJ iajs-1919 69 14 module module NOUN iajs-1919 69 15 for for ADP iajs-1919 69 16 all all DET iajs-1919 69 17 𝑋 𝑋 PROPN iajs-1919 70 1 ⊴ ⊴ ADP iajs-1919 70 2 𝑀. 𝑀. PROPN iajs-1919 70 3 then then ADV iajs-1919 70 4 𝑀 𝑀 PROPN iajs-1919 70 5 𝐴⁄ 𝐴⁄ PROPN iajs-1919 70 6 is be AUX iajs-1919 70 7 a a DET iajs-1919 70 8 strongly strongly ADV iajs-1919 70 9 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 70 10 module module NOUN iajs-1919 70 11 , , PUNCT iajs-1919 70 12 for for ADP iajs-1919 70 13 all all DET iajs-1919 70 14 𝐴 𝐴 PROPN iajs-1919 70 15 ⊴ ⊴ ADP iajs-1919 70 16 𝑀. 𝑀. NOUN iajs-1919 70 17 proof proof NOUN iajs-1919 70 18 . . PUNCT iajs-1919 71 1 for for SCONJ iajs-1919 71 2 𝐵 𝐵 PROPN iajs-1919 71 3 𝐴 𝐴 PROPN iajs-1919 71 4 ⊴ ⊴ ADP iajs-1919 71 5 𝑀 𝑀 PROPN iajs-1919 71 6 𝐴⁄⁄ 𝐴⁄⁄ VERB iajs-1919 71 7 , , PUNCT iajs-1919 71 8 to to PART iajs-1919 71 9 prove prove VERB iajs-1919 71 10 that that SCONJ iajs-1919 71 11 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 71 12 ⁄ ⁄ PROPN iajs-1919 71 13 ⁄ ⁄ PROPN iajs-1919 71 14 , , PUNCT iajs-1919 71 15 0 0 NUM iajs-1919 71 16 , , PUNCT iajs-1919 71 17 that that PRON iajs-1919 71 18 is be AUX iajs-1919 71 19 ; ; PUNCT iajs-1919 71 20 𝐻𝑜𝑚 𝐻𝑜𝑚 ADJ iajs-1919 71 21 , , PUNCT iajs-1919 71 22 0 0 NUM iajs-1919 71 23 . . PUNCT iajs-1919 72 1 if if SCONJ iajs-1919 72 2 false false ADJ iajs-1919 72 3 , , PUNCT iajs-1919 72 4 so so CCONJ iajs-1919 72 5 there there PRON iajs-1919 72 6 is be VERB iajs-1919 72 7 a a DET iajs-1919 72 8 nonzero nonzero ADJ iajs-1919 72 9 homomorphism homomorphism NOUN iajs-1919 72 10 𝜑 𝜑 X iajs-1919 72 11 : : PUNCT iajs-1919 72 12 → → PUNCT iajs-1919 72 13 . . PUNCT iajs-1919 73 1 note note VERB iajs-1919 73 2 that that SCONJ iajs-1919 73 3 𝐵 𝐵 PROPN iajs-1919 73 4 ⊴ ⊴ ADP iajs-1919 73 5 𝑀 𝑀 PROPN iajs-1919 73 6 ( ( PUNCT iajs-1919 73 7 in in ADP iajs-1919 73 8 fact fact NOUN iajs-1919 73 9 , , PUNCT iajs-1919 73 10 𝐴 𝐴 PROPN iajs-1919 73 11 ⊆ ⊆ NUM iajs-1919 73 12 𝐵 𝐵 PROPN iajs-1919 73 13 ⊆ ⊆ NUM iajs-1919 73 14 𝑀 𝑀 PROPN iajs-1919 73 15 with with ADP iajs-1919 73 16 𝐴 𝐴 PROPN iajs-1919 73 17 ⊴ ⊴ PROPN iajs-1919 73 18 𝑀 𝑀 PROPN iajs-1919 73 19 ) ) PUNCT iajs-1919 73 20 , , PUNCT iajs-1919 73 21 so so ADV iajs-1919 73 22 by by ADP iajs-1919 73 23 hypothesis hypothesis NOUN iajs-1919 73 24 𝑀 𝑀 PROPN iajs-1919 73 25 𝐵⁄ 𝐵⁄ PROPN iajs-1919 73 26 is be AUX iajs-1919 73 27 projective projective ADJ iajs-1919 73 28 , , PUNCT iajs-1919 73 29 hence hence ADV iajs-1919 73 30 there there PRON iajs-1919 73 31 is be VERB iajs-1919 73 32 a a DET iajs-1919 73 33 homomorphism homomorphism NOUN iajs-1919 73 34 𝜓 𝜓 NOUN iajs-1919 73 35 : : PUNCT iajs-1919 73 36 → → SYM iajs-1919 73 37 𝑀 𝑀 PROPN iajs-1919 73 38 such such ADJ iajs-1919 73 39 that that SCONJ iajs-1919 73 40 𝜑 𝜑 PROPN iajs-1919 73 41 𝜋 𝜋 PROPN iajs-1919 73 42 ∘ ∘ PROPN iajs-1919 73 43 𝜓. 𝜓. PROPN iajs-1919 74 1 it it PRON iajs-1919 74 2 is be AUX iajs-1919 74 3 clear clear ADJ iajs-1919 74 4 𝜓 𝜓 ADP iajs-1919 74 5 0 0 NUM iajs-1919 74 6 , , PUNCT iajs-1919 74 7 this this PRON iajs-1919 74 8 implies imply VERB iajs-1919 74 9 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 74 10 , , PUNCT iajs-1919 74 11 𝑀 𝑀 PROPN iajs-1919 74 12 0 0 NUM iajs-1919 74 13 with with ADP iajs-1919 74 14 𝐵 𝐵 PROPN iajs-1919 74 15 ⊴ ⊴ ADP iajs-1919 74 16 𝑀 𝑀 PROPN iajs-1919 74 17 , , PUNCT iajs-1919 74 18 is be AUX iajs-1919 74 19 a a DET iajs-1919 74 20 contradiction contradiction NOUN iajs-1919 74 21 with with ADP iajs-1919 74 22 m m PROPN iajs-1919 74 23 is be AUX iajs-1919 74 24 strongly strongly ADV iajs-1919 74 25 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 74 26 . . PUNCT iajs-1919 75 1 thus thus ADV iajs-1919 75 2 𝜑 𝜑 PROPN iajs-1919 75 3 0 0 NUM iajs-1919 75 4 and and CCONJ iajs-1919 75 5 𝑀 𝑀 PROPN iajs-1919 75 6 𝐴⁄ 𝐴⁄ NOUN iajs-1919 75 7 is be AUX iajs-1919 75 8 a a DET iajs-1919 75 9 strongly strongly ADV iajs-1919 75 10 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 75 11 r r NOUN iajs-1919 75 12 - - PUNCT iajs-1919 75 13 module module NOUN iajs-1919 75 14 . . PUNCT iajs-1919 76 1 ∎ ∎ NOUN iajs-1919 76 2 definition definition NOUN iajs-1919 76 3 7 7 NUM iajs-1919 76 4 . . PUNCT iajs-1919 77 1 let let VERB iajs-1919 77 2 m m PRON iajs-1919 77 3 be be AUX iajs-1919 77 4 a a DET iajs-1919 77 5 module module NOUN iajs-1919 77 6 , , PUNCT iajs-1919 77 7 define define VERB iajs-1919 77 8 the the DET iajs-1919 77 9 𝑠-𝒦-nonsigular 𝑠-𝒦-nonsigular ADJ iajs-1919 77 10 submodule submodule NOUN iajs-1919 77 11 of of ADP iajs-1919 77 12 m m PRON iajs-1919 77 13 by by ADP iajs-1919 77 14 𝑍𝒦 𝑍𝒦 PROPN iajs-1919 77 15 𝑀 𝑀 PROPN iajs-1919 77 16 ∑ ∑ ADV iajs-1919 77 17 𝐼𝑚𝜑∈ 𝐼𝑚𝜑∈ NOUN iajs-1919 77 18 , , PUNCT iajs-1919 77 19 where where SCONJ iajs-1919 77 20 𝑆 𝑆 PROPN iajs-1919 77 21 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 77 22 𝑀 𝑀 PROPN iajs-1919 77 23 and and CCONJ iajs-1919 77 24 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 77 25 ⊴ ⊴ ADP iajs-1919 77 26 𝑀. 𝑀. PROPN iajs-1919 77 27 now now ADV iajs-1919 77 28 , , PUNCT iajs-1919 77 29 we we PRON iajs-1919 77 30 will will AUX iajs-1919 77 31 give give VERB iajs-1919 77 32 another another DET iajs-1919 77 33 characterization characterization NOUN iajs-1919 77 34 for for ADP iajs-1919 77 35 a a DET iajs-1919 77 36 strongly strongly ADV iajs-1919 77 37 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 77 38 module module NOUN iajs-1919 77 39 as as SCONJ iajs-1919 77 40 follows follow VERB iajs-1919 77 41 . . PUNCT iajs-1919 78 1 proposition proposition NOUN iajs-1919 78 2 8 8 NUM iajs-1919 78 3 . . PUNCT iajs-1919 79 1 let let VERB iajs-1919 79 2 m m PRON iajs-1919 79 3 be be AUX iajs-1919 79 4 a a DET iajs-1919 79 5 module module NOUN iajs-1919 79 6 . . PUNCT iajs-1919 80 1 then then ADV iajs-1919 80 2 m m PROPN iajs-1919 80 3 is be AUX iajs-1919 80 4 strongly strongly ADV iajs-1919 80 5 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 80 6 if if SCONJ iajs-1919 80 7 and and CCONJ iajs-1919 80 8 only only ADV iajs-1919 80 9 if if SCONJ iajs-1919 80 10 𝑍𝒦 𝑍𝒦 PROPN iajs-1919 80 11 𝑀 𝑀 PROPN iajs-1919 80 12 0 0 NUM iajs-1919 80 13 . . PUNCT iajs-1919 81 1 proof proof NOUN iajs-1919 81 2 . . PUNCT iajs-1919 82 1 if if SCONJ iajs-1919 82 2 m m NOUN iajs-1919 82 3 is be AUX iajs-1919 82 4 a a DET iajs-1919 82 5 strongly strongly ADV iajs-1919 82 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 82 7 module module NOUN iajs-1919 82 8 , , PUNCT iajs-1919 82 9 then then ADV iajs-1919 82 10 for for ADP iajs-1919 82 11 all all PRON iajs-1919 82 12 𝜑 𝜑 PRON iajs-1919 82 13 ∈ ∈ PROPN iajs-1919 82 14 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 82 15 𝑀 𝑀 PROPN iajs-1919 82 16 with with ADP iajs-1919 82 17 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 82 18 ⊴ ⊴ PROPN iajs-1919 82 19 𝑀 𝑀 PROPN iajs-1919 82 20 , , PUNCT iajs-1919 82 21 implies imply VERB iajs-1919 82 22 𝐼𝑚𝜑 𝐼𝑚𝜑 PROPN iajs-1919 82 23 0 0 NUM iajs-1919 82 24 , , PUNCT iajs-1919 82 25 and and CCONJ iajs-1919 82 26 hence hence ADV iajs-1919 82 27 𝑍𝒦 𝑍𝒦 PROPN iajs-1919 82 28 𝑀 𝑀 PROPN iajs-1919 82 29 ∑ ∑ PUNCT iajs-1919 82 30 𝐼𝑚𝜑 𝐼𝑚𝜑 PROPN iajs-1919 82 31 0∈ 0∈ NOUN iajs-1919 82 32 , , PUNCT iajs-1919 82 33 where where SCONJ iajs-1919 82 34 𝑆 𝑆 PROPN iajs-1919 82 35 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 82 36 𝑀 𝑀 PROPN iajs-1919 82 37 and and CCONJ iajs-1919 82 38 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 NOUN iajs-1919 82 39 ⊴ ⊴ ADP iajs-1919 82 40 𝑀. 𝑀. NOUN iajs-1919 82 41 conversely conversely ADV iajs-1919 82 42 , , PUNCT iajs-1919 82 43 assume assume VERB iajs-1919 82 44 𝑍𝒦 𝑍𝒦 PROPN iajs-1919 82 45 𝑀 𝑀 PROPN iajs-1919 82 46 0 0 X iajs-1919 82 47 . . PUNCT iajs-1919 83 1 let let VERB iajs-1919 83 2 𝜓 𝜓 PRON iajs-1919 83 3 ∈ ∈ PROPN iajs-1919 83 4 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 83 5 𝑀 𝑀 PROPN iajs-1919 83 6 such such ADJ iajs-1919 83 7 that that DET iajs-1919 83 8 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 PROPN iajs-1919 83 9 ⊴ ⊴ PROPN iajs-1919 83 10 𝑀 𝑀 PROPN iajs-1919 83 11 , , PUNCT iajs-1919 83 12 then then ADV iajs-1919 83 13 𝐼𝑚𝜓 𝐼𝑚𝜓 PROPN iajs-1919 83 14 ⊆ ⊆ NUM iajs-1919 83 15 𝑍𝒦 𝑍𝒦 PROPN iajs-1919 83 16 𝑀 𝑀 PROPN iajs-1919 83 17 and and CCONJ iajs-1919 83 18 so so ADV iajs-1919 83 19 𝜓 𝜓 PROPN iajs-1919 83 20 0 0 X iajs-1919 83 21 . . PUNCT iajs-1919 84 1 hence hence ADV iajs-1919 84 2 m m PROPN iajs-1919 84 3 is be AUX iajs-1919 84 4 a a DET iajs-1919 84 5 strongly strongly ADV iajs-1919 84 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 84 7 module module NOUN iajs-1919 84 8 . . PUNCT iajs-1919 85 1 ∎ ∎ PROPN iajs-1919 85 2 let let VERB iajs-1919 85 3 m m PRON iajs-1919 85 4 be be AUX iajs-1919 85 5 a a DET iajs-1919 85 6 module module NOUN iajs-1919 85 7 , , PUNCT iajs-1919 85 8 recall recall VERB iajs-1919 85 9 that that SCONJ iajs-1919 85 10 a a DET iajs-1919 85 11 submodule submodule NOUN iajs-1919 85 12 n n PRON iajs-1919 85 13 is be AUX iajs-1919 85 14 supplement supplement NOUN iajs-1919 85 15 of of ADP iajs-1919 85 16 𝐾 𝐾 PROPN iajs-1919 85 17 𝑀 𝑀 PROPN iajs-1919 85 18 if if SCONJ iajs-1919 85 19 , , PUNCT iajs-1919 85 20 n n PRON iajs-1919 85 21 is be AUX iajs-1919 85 22 a a DET iajs-1919 85 23 minimal minimal ADJ iajs-1919 85 24 in in ADP iajs-1919 85 25 the the DET iajs-1919 85 26 set set NOUN iajs-1919 85 27 of of ADP iajs-1919 85 28 submodules submodule NOUN iajs-1919 85 29 𝐿 𝐿 PROPN iajs-1919 85 30 𝑀 𝑀 PROPN iajs-1919 85 31 with with ADP iajs-1919 85 32 𝐾 𝐾 PROPN iajs-1919 85 33 𝐿 𝐿 PROPN iajs-1919 85 34 𝑀 𝑀 PROPN iajs-1919 85 35 ( ( PUNCT iajs-1919 85 36 equivalently equivalently ADV iajs-1919 85 37 , , PUNCT iajs-1919 85 38 n n PRON iajs-1919 85 39 is be AUX iajs-1919 85 40 supplement supplement NOUN iajs-1919 85 41 of of ADP iajs-1919 85 42 𝐾 𝐾 PROPN iajs-1919 85 43 𝑀 𝑀 PROPN iajs-1919 86 1 if if SCONJ iajs-1919 86 2 and and CCONJ iajs-1919 86 3 only only ADV iajs-1919 86 4 if if SCONJ iajs-1919 86 5 𝐾 𝐾 PROPN iajs-1919 86 6 𝑁 𝑁 PROPN iajs-1919 86 7 𝑀 𝑀 PROPN iajs-1919 86 8 and and CCONJ iajs-1919 86 9 𝐾 𝐾 PROPN iajs-1919 86 10 ∩ ∩ NOUN iajs-1919 86 11 𝑁 𝑁 PROPN iajs-1919 86 12 ≪ ≪ ADJ iajs-1919 86 13 𝑁 𝑁 NOUN iajs-1919 86 14 ) ) PUNCT iajs-1919 87 1 [ [ X iajs-1919 87 2 9 9 NUM iajs-1919 87 3 ] ] PUNCT iajs-1919 87 4 . . PUNCT iajs-1919 88 1 we we PRON iajs-1919 88 2 say say VERB iajs-1919 88 3 that that SCONJ iajs-1919 88 4 a a DET iajs-1919 88 5 submodule submodule NOUN iajs-1919 88 6 n n PROPN iajs-1919 88 7 of of ADP iajs-1919 88 8 a a DET iajs-1919 88 9 module module NOUN iajs-1919 88 10 m m NOUN iajs-1919 88 11 is be AUX iajs-1919 88 12 a a DET iajs-1919 88 13 supplement supplement NOUN iajs-1919 88 14 if if SCONJ iajs-1919 88 15 it it PRON iajs-1919 88 16 is be AUX iajs-1919 88 17 a a DET iajs-1919 88 18 supplement supplement NOUN iajs-1919 88 19 for for ADP iajs-1919 88 20 some some DET iajs-1919 88 21 submodule submodule NOUN iajs-1919 88 22 l l NOUN iajs-1919 88 23 of of ADP iajs-1919 88 24 m. m. NOUN iajs-1919 88 25 the the DET iajs-1919 88 26 transitive transitive ADJ iajs-1919 88 27 property property NOUN iajs-1919 88 28 of of ADP iajs-1919 88 29 s s NOUN iajs-1919 88 30 - - ADJ iajs-1919 88 31 essential essential ADJ iajs-1919 88 32 submodules submodule NOUN iajs-1919 88 33 need need AUX iajs-1919 88 34 not not PART iajs-1919 88 35 be be AUX iajs-1919 88 36 hold hold VERB iajs-1919 88 37 , , PUNCT iajs-1919 88 38 see see VERB iajs-1919 88 39 [ [ X iajs-1919 88 40 4 4 NUM iajs-1919 88 41 , , PUNCT iajs-1919 88 42 ex ex NOUN iajs-1919 88 43 . . NOUN iajs-1919 88 44 2.8 2.8 NUM iajs-1919 88 45 ] ] PUNCT iajs-1919 88 46 . . PUNCT iajs-1919 89 1 so so ADV iajs-1919 89 2 , , PUNCT iajs-1919 89 3 we we PRON iajs-1919 89 4 will will AUX iajs-1919 89 5 give give VERB iajs-1919 89 6 a a DET iajs-1919 89 7 condition condition NOUN iajs-1919 89 8 for for ADP iajs-1919 89 9 which which PRON iajs-1919 89 10 the the DET iajs-1919 89 11 transitive transitive ADJ iajs-1919 89 12 property property NOUN iajs-1919 89 13 is be AUX iajs-1919 89 14 hold hold NOUN iajs-1919 89 15 of of ADP iajs-1919 89 16 s s NOUN iajs-1919 89 17 - - ADJ iajs-1919 89 18 essential essential ADJ iajs-1919 89 19 submodules submodule NOUN iajs-1919 89 20 . . PUNCT iajs-1919 90 1 lemma lemma PROPN iajs-1919 90 2 9 9 NUM iajs-1919 90 3 . . PUNCT iajs-1919 91 1 let let VERB iajs-1919 91 2 m m PRON iajs-1919 91 3 be be AUX iajs-1919 91 4 a a DET iajs-1919 91 5 module module NOUN iajs-1919 91 6 , , PUNCT iajs-1919 91 7 and and CCONJ iajs-1919 91 8 let let VERB iajs-1919 91 9 n n PRON iajs-1919 91 10 is be AUX iajs-1919 91 11 a a DET iajs-1919 91 12 supplement supplement NOUN iajs-1919 91 13 submodule submodule NOUN iajs-1919 91 14 in in ADP iajs-1919 91 15 m m PROPN iajs-1919 91 16 with with ADP iajs-1919 91 17 𝐾 𝐾 PROPN iajs-1919 91 18 ⊆ ⊆ NUM iajs-1919 91 19 𝑁 𝑁 PROPN iajs-1919 91 20 ⊆ ⊆ NUM iajs-1919 91 21 𝑀. 𝑀. NOUN iajs-1919 91 22 if if SCONJ iajs-1919 91 23 𝐾 𝐾 PROPN iajs-1919 91 24 ⊴ ⊴ ADP iajs-1919 91 25 𝑁 𝑁 PROPN iajs-1919 91 26 and and CCONJ iajs-1919 91 27 𝑁 𝑁 PROPN iajs-1919 91 28 ⊴ ⊴ NUM iajs-1919 91 29 𝑀 𝑀 PROPN iajs-1919 91 30 , , PUNCT iajs-1919 91 31 then then ADV iajs-1919 91 32 𝐾 𝐾 PROPN iajs-1919 91 33 ⊴ ⊴ ADP iajs-1919 91 34 𝑀. 𝑀. NOUN iajs-1919 91 35 proof proof NOUN iajs-1919 91 36 . . PUNCT iajs-1919 92 1 assume assume VERB iajs-1919 92 2 𝐿 𝐿 PROPN iajs-1919 92 3 ≪ ≪ PROPN iajs-1919 92 4 𝑀 𝑀 PROPN iajs-1919 92 5 with with ADP iajs-1919 92 6 𝐾 𝐾 PROPN iajs-1919 92 7 ∩ ∩ ADJ iajs-1919 92 8 𝐿 𝐿 PROPN iajs-1919 92 9 0 0 NUM iajs-1919 92 10 . . PUNCT iajs-1919 93 1 if if SCONJ iajs-1919 93 2 𝐿 𝐿 PROPN iajs-1919 93 3 ⊆ ⊆ NUM iajs-1919 93 4 𝑁 𝑁 PROPN iajs-1919 93 5 , , PUNCT iajs-1919 93 6 but but CCONJ iajs-1919 93 7 n n PRON iajs-1919 93 8 is be AUX iajs-1919 93 9 a a DET iajs-1919 93 10 supplement supplement NOUN iajs-1919 93 11 in in ADP iajs-1919 93 12 m m PROPN iajs-1919 93 13 , , PUNCT iajs-1919 93 14 then then ADV iajs-1919 93 15 by by ADP iajs-1919 93 16 [ [ X iajs-1919 93 17 10 10 NUM iajs-1919 93 18 , , PUNCT iajs-1919 93 19 prop prop NOUN iajs-1919 93 20 . . PUNCT iajs-1919 94 1 20.2 20.2 NUM iajs-1919 94 2 ] ] X iajs-1919 94 3 𝐿 𝐿 PROPN iajs-1919 94 4 ≪ ≪ PUNCT iajs-1919 94 5 𝑁 𝑁 PROPN iajs-1919 94 6 , , PUNCT iajs-1919 94 7 and and CCONJ iajs-1919 94 8 hence hence ADV iajs-1919 94 9 𝐿 𝐿 PROPN iajs-1919 94 10 0 0 NUM iajs-1919 94 11 , , PUNCT iajs-1919 94 12 since since SCONJ iajs-1919 94 13 𝐾 𝐾 PROPN iajs-1919 94 14 ⊴ ⊴ ADP iajs-1919 94 15 𝑁. 𝑁. PROPN iajs-1919 94 16 now now ADV iajs-1919 94 17 , , PUNCT iajs-1919 94 18 if if SCONJ iajs-1919 94 19 𝐿 𝐿 PROPN iajs-1919 94 20 ⊈ ⊈ PROPN iajs-1919 94 21 𝑁. 𝑁. PROPN iajs-1919 94 22 we we PRON iajs-1919 94 23 have have VERB iajs-1919 94 24 𝐿 𝐿 NOUN iajs-1919 94 25 ∩ ∩ NOUN iajs-1919 94 26 𝑁 𝑁 PROPN iajs-1919 94 27 ⊆ ⊆ NUM iajs-1919 94 28 𝑁 𝑁 PROPN iajs-1919 94 29 ⊆ ⊆ NUM iajs-1919 94 30 𝑀 𝑀 PROPN iajs-1919 94 31 , , PUNCT iajs-1919 94 32 but but CCONJ iajs-1919 94 33 ( ( PUNCT iajs-1919 94 34 𝐿 𝐿 PROPN iajs-1919 94 35 ≪ ≪ X iajs-1919 94 36 𝑀 𝑀 PROPN iajs-1919 94 37 implies imply VERB iajs-1919 94 38 𝐿 𝐿 PROPN iajs-1919 94 39 ∩ ∩ NOUN iajs-1919 94 40 𝑁 𝑁 PROPN iajs-1919 94 41 ≪ ≪ ADJ iajs-1919 94 42 𝑀 𝑀 PROPN iajs-1919 94 43 ) ) PUNCT iajs-1919 94 44 , , PUNCT iajs-1919 94 45 thus thus ADV iajs-1919 94 46 again again ADV iajs-1919 94 47 by by ADP iajs-1919 94 48 [ [ X iajs-1919 94 49 10 10 NUM iajs-1919 94 50 , , PUNCT iajs-1919 94 51 prop prop NOUN iajs-1919 94 52 . . PUNCT iajs-1919 95 1 20.2 20.2 NUM iajs-1919 95 2 ] ] PUNCT iajs-1919 95 3 𝐿 𝐿 PROPN iajs-1919 95 4 ∩ ∩ NOUN iajs-1919 95 5 𝑁 𝑁 PROPN iajs-1919 95 6 ≪ ≪ ADJ iajs-1919 95 7 𝑁 𝑁 PROPN iajs-1919 95 8 , , PUNCT iajs-1919 95 9 since since SCONJ iajs-1919 95 10 n n NUM iajs-1919 95 11 is be AUX iajs-1919 95 12 a a DET iajs-1919 95 13 supplement supplement NOUN iajs-1919 95 14 in in ADP iajs-1919 95 15 m. m. NOUN iajs-1919 95 16 but but CCONJ iajs-1919 95 17 𝐾 𝐾 PROPN iajs-1919 95 18 ∩ ∩ ADJ iajs-1919 95 19 𝐿 𝐿 NOUN iajs-1919 95 20 ∩ ∩ NOUN iajs-1919 95 21 𝑁 𝑁 PROPN iajs-1919 95 22 𝐾 𝐾 PROPN iajs-1919 95 23 ∩ ∩ ADJ iajs-1919 95 24 𝐿 𝐿 NOUN iajs-1919 95 25 0 0 NUM iajs-1919 95 26 and and CCONJ iajs-1919 95 27 𝐾 𝐾 PROPN iajs-1919 95 28 ⊴ ⊴ ADP iajs-1919 95 29 𝑁 𝑁 PROPN iajs-1919 95 30 , , PUNCT iajs-1919 95 31 this this PRON iajs-1919 95 32 implies imply VERB iajs-1919 95 33 𝐿 𝐿 PROPN iajs-1919 95 34 ∩ ∩ NOUN iajs-1919 95 35 𝑁 𝑁 PROPN iajs-1919 95 36 0 0 NUM iajs-1919 95 37 , , PUNCT iajs-1919 95 38 and and CCONJ iajs-1919 95 39 hence hence ADV iajs-1919 95 40 𝐿 𝐿 PROPN iajs-1919 95 41 0 0 NUM iajs-1919 95 42 , , PUNCT iajs-1919 95 43 as as ADP iajs-1919 95 44 𝑁 𝑁 PROPN iajs-1919 95 45 ⊴ ⊴ ADP iajs-1919 95 46 𝑀 𝑀 PROPN iajs-1919 95 47 . . PUNCT iajs-1919 96 1 ∎ ∎ PROPN iajs-1919 96 2 now now ADV iajs-1919 96 3 , , PUNCT iajs-1919 96 4 we we PRON iajs-1919 96 5 present present VERB iajs-1919 96 6 the the DET iajs-1919 96 7 following follow VERB iajs-1919 96 8 proposition proposition NOUN iajs-1919 96 9 . . PUNCT iajs-1919 96 10         SPACE iajs-1919 97 1 170 170 NUM iajs-1919 97 2     SPACE iajs-1919 97 3 ibn ibn PROPN iajs-1919 97 4 al al PROPN iajs-1919 97 5 - - PUNCT iajs-1919 97 6 haitham haitham PROPN iajs-1919 97 7 jour.for jour.for PROPN iajs-1919 97 8 pure&appl.sci pure&appl.sci PROPN iajs-1919 97 9 . . PUNCT iajs-1919 97 10 ihjpas ihjpas PROPN iajs-1919 97 11 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 97 12 vol vol NOUN iajs-1919 97 13 . . PUNCT iajs-1919 98 1 32 32 NUM iajs-1919 99 1 ( ( PUNCT iajs-1919 99 2 1 1 NUM iajs-1919 99 3 ) ) PUNCT iajs-1919 99 4 2019 2019 NUM iajs-1919 99 5 proposition proposition NOUN iajs-1919 99 6 10 10 NUM iajs-1919 99 7 . . PUNCT iajs-1919 100 1 let let VERB iajs-1919 100 2 m m PRON iajs-1919 100 3 be be AUX iajs-1919 100 4 a a DET iajs-1919 100 5 quasi quasi ADJ iajs-1919 100 6 - - ADJ iajs-1919 100 7 injective injective ADJ iajs-1919 100 8 r r NOUN iajs-1919 100 9 - - PUNCT iajs-1919 100 10 module module NOUN iajs-1919 100 11 , , PUNCT iajs-1919 100 12 and and CCONJ iajs-1919 100 13 let let VERB iajs-1919 100 14 n n PRON iajs-1919 100 15 is be AUX iajs-1919 100 16 a a DET iajs-1919 100 17 s s NOUN iajs-1919 100 18 - - ADJ iajs-1919 100 19 essential essential ADJ iajs-1919 100 20 and and CCONJ iajs-1919 100 21 supplement supplement VERB iajs-1919 100 22 submodule submodule NOUN iajs-1919 100 23 in in ADP iajs-1919 100 24 m. m. NOUN iajs-1919 100 25 if if SCONJ iajs-1919 100 26 m m NOUN iajs-1919 100 27 is be AUX iajs-1919 100 28 a a DET iajs-1919 100 29 strongly strongly ADV iajs-1919 100 30 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 100 31 r r NOUN iajs-1919 100 32 - - PUNCT iajs-1919 100 33 module module NOUN iajs-1919 100 34 , , PUNCT iajs-1919 100 35 then then ADV iajs-1919 100 36 so so ADV iajs-1919 100 37 is be AUX iajs-1919 100 38 n. n. NOUN iajs-1919 100 39 proof proof NOUN iajs-1919 100 40 . . PUNCT iajs-1919 101 1 let let VERB iajs-1919 101 2 0 0 NUM iajs-1919 102 1 𝑓 𝑓 X iajs-1919 102 2 : : PUNCT iajs-1919 102 3 𝑁 𝑁 PROPN iajs-1919 102 4 → → SYM iajs-1919 102 5 𝑁 𝑁 PROPN iajs-1919 102 6 be be AUX iajs-1919 102 7 a a DET iajs-1919 102 8 homomrphism homomrphism NOUN iajs-1919 102 9 . . PUNCT iajs-1919 103 1 since since SCONJ iajs-1919 103 2 m m PROPN iajs-1919 103 3 is be AUX iajs-1919 103 4 a a DET iajs-1919 103 5 quasi quasi ADJ iajs-1919 103 6 - - ADJ iajs-1919 103 7 injective injective ADJ iajs-1919 103 8 module module NOUN iajs-1919 103 9 , , PUNCT iajs-1919 103 10 there there PRON iajs-1919 103 11 exists exist VERB iajs-1919 103 12 0 0 NUM iajs-1919 103 13 𝜑 𝜑 NOUN iajs-1919 103 14 ∈ ∈ PROPN iajs-1919 103 15 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 103 16 𝑀 𝑀 PROPN iajs-1919 103 17 such such ADJ iajs-1919 103 18 that that SCONJ iajs-1919 103 19 𝑖 𝑖 ADP iajs-1919 103 20 ∘ ∘ NOUN iajs-1919 103 21 𝑓 𝑓 PRON iajs-1919 103 22 𝜑 𝜑 PROPN iajs-1919 103 23 ∘ ∘ NUM iajs-1919 103 24 𝑖 𝑖 SYM iajs-1919 103 25 , , PUNCT iajs-1919 103 26 where where SCONJ iajs-1919 103 27 𝑖 𝑖 VERB iajs-1919 103 28 : : PUNCT iajs-1919 103 29 𝑁 𝑁 PROPN iajs-1919 103 30 → → SYM iajs-1919 103 31 𝑀 𝑀 PROPN iajs-1919 103 32 is be AUX iajs-1919 103 33 an an DET iajs-1919 103 34 inclusion inclusion NOUN iajs-1919 103 35 map map NOUN iajs-1919 103 36 . . PUNCT iajs-1919 104 1 as as SCONJ iajs-1919 104 2 m m PROPN iajs-1919 104 3 is be AUX iajs-1919 104 4 strongly strongly ADV iajs-1919 104 5 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 104 6 , , PUNCT iajs-1919 104 7 we we PRON iajs-1919 104 8 get get VERB iajs-1919 104 9 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 ADJ iajs-1919 104 10 ⋬ ⋬ X iajs-1919 105 1 𝑀. 𝑀. NOUN iajs-1919 105 2 clearly clearly ADV iajs-1919 105 3 , , PUNCT iajs-1919 105 4 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 PROPN iajs-1919 105 5 ⊆ ⊆ NUM iajs-1919 105 6 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 ADV iajs-1919 105 7 then then ADV iajs-1919 105 8 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 PROPN iajs-1919 105 9 ⋬ ⋬ AUX iajs-1919 106 1 𝑀. 𝑀. PROPN iajs-1919 106 2 if if SCONJ iajs-1919 106 3 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 106 4 ⊴ ⊴ ADP iajs-1919 106 5 𝑁 𝑁 PROPN iajs-1919 106 6 , , PUNCT iajs-1919 106 7 and and CCONJ iajs-1919 106 8 since since SCONJ iajs-1919 106 9 𝑁(supplement 𝑁(supplement NOUN iajs-1919 106 10 ) ) PUNCT iajs-1919 106 11 ⊴ ⊴ ADP iajs-1919 106 12 𝑀 𝑀 PROPN iajs-1919 106 13 , , PUNCT iajs-1919 106 14 so so ADV iajs-1919 106 15 by by ADP iajs-1919 106 16 previous previous ADJ iajs-1919 106 17 lemma lemma PROPN iajs-1919 106 18 , , PUNCT iajs-1919 106 19 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 PROPN iajs-1919 106 20 ⊴ ⊴ PROPN iajs-1919 106 21 𝑀 𝑀 PROPN iajs-1919 106 22 , , PUNCT iajs-1919 106 23 is be AUX iajs-1919 106 24 a a DET iajs-1919 106 25 contradiction contradiction NOUN iajs-1919 106 26 . . PUNCT iajs-1919 107 1 therefore therefore ADV iajs-1919 107 2 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 PROPN iajs-1919 107 3 ⋬ ⋬ AUX iajs-1919 108 1 𝑁 𝑁 NOUN iajs-1919 108 2 , , PUNCT iajs-1919 108 3 and and CCONJ iajs-1919 108 4 n n PRON iajs-1919 108 5 is be AUX iajs-1919 108 6 a a DET iajs-1919 108 7 strongly strongly ADV iajs-1919 108 8 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 108 9 module module NOUN iajs-1919 108 10 . . PUNCT iajs-1919 109 1 ∎ ∎ NOUN iajs-1919 109 2 a a DET iajs-1919 109 3 quasi quasi ADJ iajs-1919 109 4 - - ADJ iajs-1919 109 5 injective injective ADJ iajs-1919 109 6 module module NOUN iajs-1919 109 7 𝑀 𝑀 PROPN iajs-1919 109 8 is be AUX iajs-1919 109 9 called call VERB iajs-1919 109 10 quasi quasi ADJ iajs-1919 109 11 - - ADJ iajs-1919 109 12 injective injective ADJ iajs-1919 109 13 hull hull NOUN iajs-1919 109 14 of of ADP iajs-1919 109 15 a a DET iajs-1919 109 16 module module NOUN iajs-1919 109 17 m m VERB iajs-1919 109 18 if if SCONJ iajs-1919 109 19 , , PUNCT iajs-1919 109 20 there there PRON iajs-1919 109 21 exists exist VERB iajs-1919 109 22 a a DET iajs-1919 109 23 monomorphism monomorphism NOUN iajs-1919 109 24 𝜑 𝜑 X iajs-1919 109 25 : : PUNCT iajs-1919 109 26 𝑀 𝑀 PROPN iajs-1919 109 27 → → SYM iajs-1919 109 28 𝑀 𝑀 PROPN iajs-1919 109 29 with with ADP iajs-1919 109 30 𝐼𝑚𝜑 𝐼𝑚𝜑 PROPN iajs-1919 109 31 ⊴ ⊴ ADP iajs-1919 109 32 𝑀 𝑀 PROPN iajs-1919 110 1 [ [ X iajs-1919 110 2 11 11 NUM iajs-1919 110 3 ] ] PUNCT iajs-1919 110 4 . . PUNCT iajs-1919 111 1 corollary corollary ADJ iajs-1919 111 2 11 11 NUM iajs-1919 111 3 . . PUNCT iajs-1919 112 1 let let VERB iajs-1919 112 2 𝑀 𝑀 PROPN iajs-1919 112 3 be be AUX iajs-1919 112 4 a a DET iajs-1919 112 5 strongly strongly ADV iajs-1919 112 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 112 7 module module NOUN iajs-1919 112 8 . . PUNCT iajs-1919 113 1 if if SCONJ iajs-1919 113 2 m m NOUN iajs-1919 113 3 is be AUX iajs-1919 113 4 a a DET iajs-1919 113 5 supplement supplement NOUN iajs-1919 113 6 in in ADP iajs-1919 113 7 𝑀 𝑀 PROPN iajs-1919 113 8 , , PUNCT iajs-1919 113 9 then then ADV iajs-1919 113 10 m m VERB iajs-1919 113 11 is be AUX iajs-1919 113 12 strongly strongly ADV iajs-1919 113 13 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 113 14 . . PUNCT iajs-1919 114 1 next next ADV iajs-1919 114 2 , , PUNCT iajs-1919 114 3 we we PRON iajs-1919 114 4 will will AUX iajs-1919 114 5 study study VERB iajs-1919 114 6 the the DET iajs-1919 114 7 behavior behavior NOUN iajs-1919 114 8 of of ADP iajs-1919 114 9 s s NOUN iajs-1919 114 10 - - ADJ iajs-1919 114 11 essential essential ADJ iajs-1919 114 12 submodule submodule NOUN iajs-1919 114 13 and and CCONJ iajs-1919 114 14 strongly strongly ADV iajs-1919 114 15 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 114 16 module module NOUN iajs-1919 114 17 under under ADP iajs-1919 114 18 localization localization NOUN iajs-1919 114 19 . . PUNCT iajs-1919 115 1 firstly firstly ADV iajs-1919 115 2 , , PUNCT iajs-1919 115 3 we we PRON iajs-1919 115 4 have have VERB iajs-1919 115 5 the the DET iajs-1919 115 6 following follow VERB iajs-1919 115 7 lemma lemma PROPN iajs-1919 115 8 . . PUNCT iajs-1919 116 1 lemma lemma PROPN iajs-1919 116 2 12 12 NUM iajs-1919 116 3 . . PUNCT iajs-1919 117 1 let let AUX iajs-1919 117 2 m m PRON iajs-1919 117 3 be be AUX iajs-1919 117 4 a a DET iajs-1919 117 5 module module NOUN iajs-1919 117 6 , , PUNCT iajs-1919 117 7 𝑁 𝑁 PROPN iajs-1919 117 8 𝐾 𝐾 PROPN iajs-1919 117 9 𝑀 𝑀 PROPN iajs-1919 117 10 and and CCONJ iajs-1919 117 11 let let VERB iajs-1919 117 12 s s PRON iajs-1919 117 13 is be AUX iajs-1919 117 14 a a DET iajs-1919 117 15 multiplicative multiplicative ADJ iajs-1919 117 16 closed close VERB iajs-1919 117 17 subset subset NOUN iajs-1919 117 18 of of ADP iajs-1919 117 19 r r NOUN iajs-1919 117 20 , , PUNCT iajs-1919 117 21 provided provide VERB iajs-1919 117 22 𝑆 𝑆 PROPN iajs-1919 117 23 𝐿 𝐿 PROPN iajs-1919 117 24 𝑆 𝑆 PROPN iajs-1919 117 25 𝐿 𝐿 PROPN iajs-1919 117 26 iff iff PROPN iajs-1919 117 27 𝐿 𝐿 PROPN iajs-1919 117 28 𝐿 𝐿 PROPN iajs-1919 117 29 for for ADP iajs-1919 117 30 all all DET iajs-1919 117 31 𝐿 𝐿 PROPN iajs-1919 117 32 , , PUNCT iajs-1919 117 33 𝐿 𝐿 PROPN iajs-1919 117 34 𝑀. 𝑀. PROPN iajs-1919 117 35 then then ADV iajs-1919 117 36 the the DET iajs-1919 117 37 following follow VERB iajs-1919 117 38 hold hold NOUN iajs-1919 117 39 . . PUNCT iajs-1919 118 1 𝑖 𝑖 X iajs-1919 119 1 𝑁 𝑁 NOUN iajs-1919 119 2 ≪ ≪ PUNCT iajs-1919 119 3 𝐾 𝐾 PROPN iajs-1919 119 4 in in ADP iajs-1919 119 5 m m NOUN iajs-1919 119 6 as as ADP iajs-1919 119 7 r r NOUN iajs-1919 119 8 - - PUNCT iajs-1919 119 9 module module NOUN iajs-1919 119 10 if if SCONJ iajs-1919 119 11 and and CCONJ iajs-1919 119 12 only only ADV iajs-1919 119 13 if if SCONJ iajs-1919 119 14 𝑆 𝑆 PROPN iajs-1919 119 15 𝑁 𝑁 PROPN iajs-1919 119 16 ≪ ≪ PUNCT iajs-1919 119 17 𝑆 𝑆 PROPN iajs-1919 119 18 𝐾 𝐾 PROPN iajs-1919 119 19 in in ADP iajs-1919 119 20 𝑆 𝑆 PROPN iajs-1919 119 21 𝑀 𝑀 PROPN iajs-1919 119 22 as as ADP iajs-1919 119 23 𝑆 𝑆 PROPN iajs-1919 119 24 𝑅-module 𝑅-module PROPN iajs-1919 119 25 . . PUNCT iajs-1919 119 26 𝑖𝑖 𝑖𝑖 X iajs-1919 120 1 𝑁 𝑁 PROPN iajs-1919 120 2 ⊴ ⊴ ADP iajs-1919 120 3 𝐾 𝐾 PROPN iajs-1919 120 4 in in ADP iajs-1919 120 5 m m NOUN iajs-1919 120 6 as as ADP iajs-1919 120 7 r r NOUN iajs-1919 120 8 - - PUNCT iajs-1919 120 9 module module NOUN iajs-1919 120 10 if if SCONJ iajs-1919 120 11 and and CCONJ iajs-1919 120 12 only only ADV iajs-1919 120 13 if if SCONJ iajs-1919 120 14 𝑆 𝑆 PROPN iajs-1919 120 15 𝑁 𝑁 PROPN iajs-1919 120 16 ⊴ ⊴ ADP iajs-1919 120 17 𝑆 𝑆 PROPN iajs-1919 120 18 𝐾 𝐾 PROPN iajs-1919 120 19 in in ADP iajs-1919 120 20 𝑆 𝑆 PROPN iajs-1919 120 21 𝑀 𝑀 PROPN iajs-1919 120 22 as as ADP iajs-1919 120 23 𝑆 𝑆 PROPN iajs-1919 120 24 𝑅-module 𝑅-module PROPN iajs-1919 120 25 . . PUNCT iajs-1919 121 1 proof proof NOUN iajs-1919 121 2 . . PUNCT iajs-1919 122 1 𝑖 𝑖 PRON iajs-1919 122 2 assume assume VERB iajs-1919 122 3 𝑁 𝑁 PROPN iajs-1919 122 4 ≪ ≪ PUNCT iajs-1919 122 5 𝐾 𝐾 PROPN iajs-1919 122 6 𝑀. 𝑀. PROPN iajs-1919 122 7 let let VERB iajs-1919 122 8 𝑆 𝑆 PROPN iajs-1919 122 9 𝐿 𝐿 PROPN iajs-1919 122 10 𝑆 𝑆 PROPN iajs-1919 122 11 𝐾 𝐾 PROPN iajs-1919 122 12 with with ADP iajs-1919 122 13 𝑆 𝑆 PROPN iajs-1919 122 14 𝑁 𝑁 PROPN iajs-1919 122 15 𝑆 𝑆 PROPN iajs-1919 122 16 𝐿 𝐿 PROPN iajs-1919 122 17 𝑆 𝑆 PROPN iajs-1919 122 18 𝐾 𝐾 PROPN iajs-1919 122 19 , , PUNCT iajs-1919 122 20 where where SCONJ iajs-1919 122 21 𝐿 𝐿 PROPN iajs-1919 122 22 𝐾. 𝐾. PROPN iajs-1919 122 23 but but CCONJ iajs-1919 122 24 we we PRON iajs-1919 122 25 have have VERB iajs-1919 122 26 𝑆 𝑆 PROPN iajs-1919 122 27 𝑁 𝑁 PROPN iajs-1919 122 28 𝑆 𝑆 PROPN iajs-1919 122 29 𝐿 𝐿 PROPN iajs-1919 122 30 𝑆 𝑆 PROPN iajs-1919 122 31 𝑁 𝑁 PROPN iajs-1919 122 32 𝐿 𝐿 PROPN iajs-1919 122 33 , , PUNCT iajs-1919 122 34 so so ADV iajs-1919 122 35 𝑆 𝑆 PROPN iajs-1919 122 36 𝑁 𝑁 PROPN iajs-1919 122 37 𝐿 𝐿 PROPN iajs-1919 122 38 𝑆 𝑆 PROPN iajs-1919 122 39 𝐾 𝐾 PROPN iajs-1919 122 40 , , PUNCT iajs-1919 122 41 and and CCONJ iajs-1919 122 42 hence hence ADV iajs-1919 122 43 𝑁 𝑁 PROPN iajs-1919 122 44 𝐿 𝐿 PROPN iajs-1919 122 45 𝐾 𝐾 PROPN iajs-1919 122 46 by by ADP iajs-1919 122 47 hypothesis hypothesis NOUN iajs-1919 122 48 , , PUNCT iajs-1919 122 49 thus thus ADV iajs-1919 122 50 𝐿 𝐿 PROPN iajs-1919 122 51 𝐾 𝐾 PROPN iajs-1919 122 52 , , PUNCT iajs-1919 122 53 as as ADP iajs-1919 122 54 𝑁 𝑁 PROPN iajs-1919 122 55 ≪ ≪ ADJ iajs-1919 122 56 𝐾. 𝐾. PROPN iajs-1919 122 57 therefore therefore ADV iajs-1919 122 58 𝑆 𝑆 PROPN iajs-1919 122 59 𝐿 𝐿 PROPN iajs-1919 122 60 𝑆 𝑆 PROPN iajs-1919 122 61 𝐾 𝐾 PROPN iajs-1919 122 62 , , PUNCT iajs-1919 122 63 and and CCONJ iajs-1919 122 64 so so ADV iajs-1919 122 65 𝑆 𝑆 PROPN iajs-1919 122 66 𝑁 𝑁 PROPN iajs-1919 122 67 ≪ ≪ PUNCT iajs-1919 122 68 𝑆 𝑆 PROPN iajs-1919 122 69 𝐾 𝐾 PROPN iajs-1919 122 70 in in ADP iajs-1919 122 71 𝑆 𝑆 PROPN iajs-1919 122 72 𝑀. 𝑀. PROPN iajs-1919 122 73 conversely conversely ADV iajs-1919 122 74 , , PUNCT iajs-1919 122 75 if if SCONJ iajs-1919 122 76 𝑁 𝑁 PROPN iajs-1919 122 77 𝐿 𝐿 PROPN iajs-1919 122 78 𝐾 𝐾 PROPN iajs-1919 122 79 where where SCONJ iajs-1919 122 80 𝐿 𝐿 PROPN iajs-1919 122 81 𝐾. 𝐾. PROPN iajs-1919 122 82 then then ADV iajs-1919 122 83 𝑆 𝑆 PROPN iajs-1919 122 84 𝑁 𝑁 PROPN iajs-1919 122 85 𝑆 𝑆 PROPN iajs-1919 122 86 𝐿 𝐿 PROPN iajs-1919 122 87 𝑆 𝑆 PROPN iajs-1919 122 88 𝑁 𝑁 PROPN iajs-1919 122 89 𝐿 𝐿 PROPN iajs-1919 122 90 𝑆 𝑆 PROPN iajs-1919 122 91 𝐾 𝐾 PROPN iajs-1919 122 92 , , PUNCT iajs-1919 122 93 and and CCONJ iajs-1919 122 94 hence hence ADV iajs-1919 122 95 𝑆 𝑆 PROPN iajs-1919 122 96 𝐿 𝐿 PROPN iajs-1919 122 97 𝑆 𝑆 PROPN iajs-1919 122 98 𝐾 𝐾 PROPN iajs-1919 122 99 , , PUNCT iajs-1919 122 100 as as SCONJ iajs-1919 122 101 𝑆 𝑆 PROPN iajs-1919 122 102 𝑁 𝑁 PROPN iajs-1919 122 103 ≪ ≪ PUNCT iajs-1919 122 104 𝑆 𝑆 PROPN iajs-1919 122 105 𝐾. 𝐾. PROPN iajs-1919 122 106 by by ADP iajs-1919 122 107 hypothesis hypothesis NOUN iajs-1919 122 108 , , PUNCT iajs-1919 122 109 𝐿 𝐿 PROPN iajs-1919 122 110 𝐾 𝐾 PROPN iajs-1919 122 111 , , PUNCT iajs-1919 122 112 and and CCONJ iajs-1919 122 113 so so ADV iajs-1919 122 114 𝑁 𝑁 PROPN iajs-1919 122 115 ≪ ≪ ADJ iajs-1919 122 116 𝐾 𝐾 PROPN iajs-1919 122 117 in in ADP iajs-1919 122 118 m. m. NOUN iajs-1919 122 119 𝑖𝑖 𝑖𝑖 PUNCT iajs-1919 122 120 if if SCONJ iajs-1919 122 121 𝑁 𝑁 PROPN iajs-1919 122 122 ⊴ ⊴ ADP iajs-1919 122 123 𝐾 𝐾 PROPN iajs-1919 122 124 𝑀. 𝑀. PROPN iajs-1919 122 125 let let VERB iajs-1919 122 126 𝑆 𝑆 PROPN iajs-1919 122 127 𝐿 𝐿 PROPN iajs-1919 122 128 ≪ ≪ PUNCT iajs-1919 122 129 𝑆 𝑆 PROPN iajs-1919 122 130 𝐾 𝐾 PROPN iajs-1919 122 131 such such ADJ iajs-1919 122 132 that that SCONJ iajs-1919 122 133 𝑆 𝑆 PROPN iajs-1919 122 134 𝑁 𝑁 PROPN iajs-1919 122 135 ∩ ∩ NOUN iajs-1919 122 136 𝑆 𝑆 PROPN iajs-1919 122 137 𝐿 𝐿 PROPN iajs-1919 122 138 𝑆 𝑆 PROPN iajs-1919 122 139 0 0 NUM iajs-1919 122 140 , , PUNCT iajs-1919 122 141 where where SCONJ iajs-1919 122 142 𝐿 𝐿 PROPN iajs-1919 122 143 𝐾. 𝐾. PROPN iajs-1919 122 144 by by ADP iajs-1919 122 145 𝑖 𝑖 X iajs-1919 122 146 , , PUNCT iajs-1919 122 147 𝐿 𝐿 PROPN iajs-1919 122 148 ≪ ≪ NOUN iajs-1919 122 149 𝐾. 𝐾. PROPN iajs-1919 122 150 but but CCONJ iajs-1919 122 151 , , PUNCT iajs-1919 122 152 we we PRON iajs-1919 122 153 have have VERB iajs-1919 122 154 𝑆 𝑆 PROPN iajs-1919 122 155 𝑁 𝑁 PROPN iajs-1919 122 156 ∩ ∩ ADJ iajs-1919 122 157 𝐿 𝐿 PROPN iajs-1919 122 158 𝑆 𝑆 PROPN iajs-1919 122 159 𝑁 𝑁 PROPN iajs-1919 122 160 ∩ ∩ ADJ iajs-1919 122 161 𝑆 𝑆 PROPN iajs-1919 122 162 𝐿 𝐿 PROPN iajs-1919 122 163 𝑆 𝑆 PROPN iajs-1919 122 164 0 0 NUM iajs-1919 122 165 , , PUNCT iajs-1919 122 166 𝑁 𝑁 PROPN iajs-1919 122 167 ∩ ∩ ADJ iajs-1919 122 168 𝐿 𝐿 NOUN iajs-1919 122 169 0 0 NUM iajs-1919 122 170 by by ADP iajs-1919 122 171 hypothesis hypothesis NOUN iajs-1919 122 172 . . PUNCT iajs-1919 123 1 as as SCONJ iajs-1919 123 2 𝑁 𝑁 PROPN iajs-1919 123 3 ⊴ ⊴ ADP iajs-1919 123 4 𝐾 𝐾 PROPN iajs-1919 123 5 and and CCONJ iajs-1919 123 6 𝐿 𝐿 PROPN iajs-1919 123 7 ≪ ≪ PUNCT iajs-1919 123 8 𝐾 𝐾 PROPN iajs-1919 123 9 implies imply VERB iajs-1919 123 10 𝐿 𝐿 PROPN iajs-1919 123 11 0 0 PROPN iajs-1919 123 12 , , PUNCT iajs-1919 123 13 thus thus ADV iajs-1919 123 14 𝑆 𝑆 PROPN iajs-1919 123 15 𝐿 𝐿 PROPN iajs-1919 123 16 𝑆 𝑆 PROPN iajs-1919 123 17 0 0 NUM iajs-1919 123 18 . . PUNCT iajs-1919 124 1 conversely conversely ADV iajs-1919 124 2 , , PUNCT iajs-1919 124 3 suppose suppose VERB iajs-1919 124 4 𝑁 𝑁 PROPN iajs-1919 124 5 ∩ ∩ ADJ iajs-1919 124 6 𝐿 𝐿 NOUN iajs-1919 124 7 0 0 NUM iajs-1919 124 8 where where SCONJ iajs-1919 124 9 𝐿 𝐿 PROPN iajs-1919 124 10 ≪ ≪ PUNCT iajs-1919 124 11 𝐾 𝐾 PROPN iajs-1919 124 12 , , PUNCT iajs-1919 124 13 implies imply VERB iajs-1919 124 14 𝑆 𝑆 PROPN iajs-1919 124 15 𝐿 𝐿 PROPN iajs-1919 124 16 ≪ ≪ PUNCT iajs-1919 124 17 𝑆 𝑆 PROPN iajs-1919 124 18 𝐾 𝐾 PROPN iajs-1919 124 19 , , PUNCT iajs-1919 124 20 by by ADP iajs-1919 124 21 𝑖 𝑖 X iajs-1919 124 22 . . PUNCT iajs-1919 125 1 so so ADV iajs-1919 125 2 𝑆 𝑆 PROPN iajs-1919 125 3 𝑁 𝑁 PROPN iajs-1919 125 4 ∩ ∩ NOUN iajs-1919 125 5 𝑆 𝑆 PROPN iajs-1919 125 6 𝐿 𝐿 PROPN iajs-1919 125 7 𝑆 𝑆 PROPN iajs-1919 125 8 𝑁 𝑁 PROPN iajs-1919 125 9 ∩ ∩ ADJ iajs-1919 125 10 𝐿 𝐿 PROPN iajs-1919 125 11 𝑆 𝑆 PROPN iajs-1919 125 12 0 0 NUM iajs-1919 125 13 , , PUNCT iajs-1919 125 14 thus thus ADV iajs-1919 125 15 𝑆 𝑆 PROPN iajs-1919 125 16 𝐿 𝐿 PROPN iajs-1919 125 17 𝑆 𝑆 PROPN iajs-1919 125 18 0 0 NUM iajs-1919 125 19 , , PUNCT iajs-1919 125 20 as as SCONJ iajs-1919 125 21 𝑆 𝑆 PROPN iajs-1919 125 22 𝑁 𝑁 PROPN iajs-1919 125 23 ⊴ ⊴ ADP iajs-1919 125 24 𝑆 𝑆 PROPN iajs-1919 125 25 𝐾. 𝐾. PROPN iajs-1919 125 26 by by ADP iajs-1919 125 27 hypothesis hypothesis NOUN iajs-1919 125 28 , , PUNCT iajs-1919 125 29 𝐿 𝐿 PROPN iajs-1919 125 30 0 0 NUM iajs-1919 125 31 . . PUNCT iajs-1919 126 1 ∎ ∎ PROPN iajs-1919 126 2 however however ADV iajs-1919 126 3 , , PUNCT iajs-1919 126 4 we we PRON iajs-1919 126 5 get get VERB iajs-1919 126 6 the the DET iajs-1919 126 7 following follow VERB iajs-1919 126 8 result result NOUN iajs-1919 126 9 . . PUNCT iajs-1919 127 1 proposition proposition NOUN iajs-1919 127 2 13 13 NUM iajs-1919 127 3 . . PUNCT iajs-1919 128 1 let let VERB iajs-1919 128 2 m m PRON iajs-1919 128 3 be be AUX iajs-1919 128 4 an an DET iajs-1919 128 5 r r NOUN iajs-1919 128 6 - - PUNCT iajs-1919 128 7 module module NOUN iajs-1919 128 8 , , PUNCT iajs-1919 128 9 and and CCONJ iajs-1919 128 10 let let VERB iajs-1919 128 11 s s PRON iajs-1919 128 12 is be AUX iajs-1919 128 13 a a DET iajs-1919 128 14 multiplicative multiplicative ADJ iajs-1919 128 15 closed close VERB iajs-1919 128 16 subset subset NOUN iajs-1919 128 17 of of ADP iajs-1919 128 18 r r NOUN iajs-1919 128 19 such such ADJ iajs-1919 128 20 that that SCONJ iajs-1919 128 21 𝑆 𝑆 PROPN iajs-1919 128 22 𝐿 𝐿 PROPN iajs-1919 128 23 𝑆 𝑆 PROPN iajs-1919 128 24 𝐾 𝐾 PROPN iajs-1919 128 25 iff iff PROPN iajs-1919 128 26 𝐿 𝐿 PROPN iajs-1919 128 27 𝐾 𝐾 PROPN iajs-1919 128 28 for for ADP iajs-1919 128 29 all all DET iajs-1919 128 30 𝐿 𝐿 PROPN iajs-1919 128 31 , , PUNCT iajs-1919 128 32 𝐾 𝐾 PROPN iajs-1919 128 33 𝑀. 𝑀. PROPN iajs-1919 128 34 then then ADV iajs-1919 128 35 m m VERB iajs-1919 128 36 is be AUX iajs-1919 128 37 a a DET iajs-1919 128 38 strongly strongly ADV iajs-1919 128 39 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 128 40 r r NOUN iajs-1919 128 41 - - PUNCT iajs-1919 128 42 module module NOUN iajs-1919 128 43 , , PUNCT iajs-1919 128 44 whenever whenever SCONJ iajs-1919 128 45 𝑆 𝑆 PROPN iajs-1919 128 46 𝑀 𝑀 PROPN iajs-1919 128 47 is be AUX iajs-1919 128 48 a a DET iajs-1919 128 49 strongly strongly ADV iajs-1919 128 50 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 128 51 𝑆 𝑆 PROPN iajs-1919 128 52 𝑅-module 𝑅-module PROPN iajs-1919 128 53 . . PUNCT iajs-1919 129 1 proof proof NOUN iajs-1919 129 2 . . PUNCT iajs-1919 130 1 assume assume VERB iajs-1919 130 2 0 0 NUM iajs-1919 131 1 𝑔 𝑔 PROPN iajs-1919 131 2 ∈ ∈ PROPN iajs-1919 131 3 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 131 4 𝑀 𝑀 PROPN iajs-1919 131 5 . . PUNCT iajs-1919 132 1 we we PRON iajs-1919 132 2 can can AUX iajs-1919 132 3 define define VERB iajs-1919 132 4 an an DET iajs-1919 132 5 𝑆 𝑆 PROPN iajs-1919 132 6 𝑅-homomorphism 𝑅-homomorphism PROPN iajs-1919 132 7 𝑆 𝑆 PROPN iajs-1919 132 8 𝑔 𝑔 NOUN iajs-1919 132 9 : : PUNCT iajs-1919 132 10 𝑆 𝑆 PROPN iajs-1919 132 11 𝑀 𝑀 PROPN iajs-1919 132 12 → → SYM iajs-1919 132 13 𝑆 𝑆 PROPN iajs-1919 132 14 𝑀 𝑀 PROPN iajs-1919 132 15 such such ADJ iajs-1919 132 16 that that SCONJ iajs-1919 132 17 𝑆 𝑆 PROPN iajs-1919 132 18 𝑔 𝑔 PROPN iajs-1919 132 19 for for ADP iajs-1919 132 20 each each DET iajs-1919 132 21 𝑚 𝑚 PROPN iajs-1919 132 22 ∈ ∈ PROPN iajs-1919 132 23 𝑀 𝑀 PROPN iajs-1919 132 24 , , PUNCT iajs-1919 132 25 𝑠 𝑠 PROPN iajs-1919 132 26 ∈ ∈ PROPN iajs-1919 132 27 𝑆. 𝑆. PROPN iajs-1919 132 28 it it PRON iajs-1919 132 29 is be AUX iajs-1919 132 30 clear clear ADJ iajs-1919 132 31 𝑆 𝑆 PROPN iajs-1919 132 32 𝑔 𝑔 PROPN iajs-1919 132 33 0 0 NUM iajs-1919 132 34 , , PUNCT iajs-1919 132 35 so so ADV iajs-1919 132 36 𝑘𝑒𝑟 𝑘𝑒𝑟 VERB iajs-1919 132 37 𝑆 𝑆 PROPN iajs-1919 132 38 𝑔 𝑔 PROPN iajs-1919 132 39 ⋬ ⋬ X iajs-1919 132 40 𝑆 𝑆 PROPN iajs-1919 132 41 𝑀 𝑀 PROPN iajs-1919 132 42 , , PUNCT iajs-1919 132 43 as as SCONJ iajs-1919 132 44 𝑆 𝑆 PROPN iajs-1919 132 45 𝑀 𝑀 PROPN iajs-1919 132 46 is be AUX iajs-1919 132 47 strongly strongly ADV iajs-1919 132 48 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 132 49 . . PUNCT iajs-1919 133 1 also also ADV iajs-1919 133 2 , , PUNCT iajs-1919 133 3 it it PRON iajs-1919 133 4 is be AUX iajs-1919 133 5 easy easy ADJ iajs-1919 133 6 to to PART iajs-1919 133 7 see see VERB iajs-1919 133 8 that that DET iajs-1919 133 9 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 133 10 𝑆 𝑆 PROPN iajs-1919 133 11 𝑔 𝑔 PROPN iajs-1919 133 12 𝑆 𝑆 PROPN iajs-1919 133 13 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 133 14 , , PUNCT iajs-1919 133 15 this this PRON iajs-1919 133 16 implies imply VERB iajs-1919 133 17 that that SCONJ iajs-1919 133 18 𝑆 𝑆 PROPN iajs-1919 133 19 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 133 20 ⋬ ⋬ X iajs-1919 133 21 𝑆 𝑆 PROPN iajs-1919 133 22 𝑀 𝑀 PROPN iajs-1919 133 23 , , PUNCT iajs-1919 133 24 and and CCONJ iajs-1919 133 25 hence hence ADV iajs-1919 133 26 by by ADP iajs-1919 133 27 lemma lemma PROPN iajs-1919 133 28 12 12 NUM iajs-1919 133 29 𝑖𝑖 𝑖𝑖 NOUN iajs-1919 133 30 , , PUNCT iajs-1919 133 31 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 133 32 ⋬ ⋬ NOUN iajs-1919 134 1 𝑀. 𝑀. NOUN iajs-1919 134 2 ∎ ∎ NOUN iajs-1919 134 3         SPACE iajs-1919 135 1 171 171 NUM iajs-1919 135 2     SPACE iajs-1919 135 3 ibn ibn PROPN iajs-1919 135 4 al al PROPN iajs-1919 135 5 - - PUNCT iajs-1919 135 6 haitham haitham PROPN iajs-1919 135 7 jour.for jour.for PROPN iajs-1919 135 8 pure&appl.sci pure&appl.sci PROPN iajs-1919 135 9 . . PUNCT iajs-1919 136 1 ihjpas ihjpas PROPN iajs-1919 136 2 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 136 3 vol vol NOUN iajs-1919 136 4 . . PUNCT iajs-1919 136 5 32 32 NUM iajs-1919 136 6 ( ( PUNCT iajs-1919 136 7 1 1 NUM iajs-1919 136 8 ) ) SYM iajs-1919 136 9 2019 2019 NUM iajs-1919 136 10 proposition proposition NOUN iajs-1919 136 11 14 14 NUM iajs-1919 136 12 . . PUNCT iajs-1919 137 1 let let VERB iajs-1919 137 2 m m PRON iajs-1919 137 3 be be AUX iajs-1919 137 4 an an DET iajs-1919 137 5 r r NOUN iajs-1919 137 6 - - PUNCT iajs-1919 137 7 module module NOUN iajs-1919 137 8 , , PUNCT iajs-1919 137 9 and and CCONJ iajs-1919 137 10 let let VERB iajs-1919 137 11 p p NOUN iajs-1919 137 12 is be AUX iajs-1919 137 13 a a DET iajs-1919 137 14 maximal maximal ADJ iajs-1919 137 15 ideal ideal NOUN iajs-1919 137 16 of of ADP iajs-1919 137 17 r. r. PROPN iajs-1919 137 18 if if SCONJ iajs-1919 137 19 𝑀 𝑀 PROPN iajs-1919 137 20 is be AUX iajs-1919 137 21 a a DET iajs-1919 137 22 strongly strongly ADV iajs-1919 137 23 𝒦nonsigular 𝒦nonsigular PROPN iajs-1919 137 24 𝑅 𝑅 PROPN iajs-1919 137 25 -module -module NOUN iajs-1919 137 26 , , PUNCT iajs-1919 137 27 then then ADV iajs-1919 137 28 m m VERB iajs-1919 137 29 is be AUX iajs-1919 137 30 a a DET iajs-1919 137 31 strongly strongly ADV iajs-1919 137 32 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 137 33 r r NOUN iajs-1919 137 34 - - PUNCT iajs-1919 137 35 module module NOUN iajs-1919 137 36 . . PUNCT iajs-1919 138 1 recall recall VERB iajs-1919 138 2 that that SCONJ iajs-1919 138 3 an an DET iajs-1919 138 4 r r NOUN iajs-1919 138 5 - - PUNCT iajs-1919 138 6 module module NOUN iajs-1919 138 7 m m NOUN iajs-1919 138 8 is be AUX iajs-1919 138 9 called call VERB iajs-1919 138 10 multiplication multiplication NOUN iajs-1919 138 11 if if SCONJ iajs-1919 138 12 for for ADP iajs-1919 138 13 each each DET iajs-1919 138 14 submodule submodule NOUN iajs-1919 138 15 n n PROPN iajs-1919 138 16 of of ADP iajs-1919 138 17 m m PRON iajs-1919 138 18 , , PUNCT iajs-1919 138 19 𝑁 𝑁 PROPN iajs-1919 138 20 𝑀𝐼 𝑀𝐼 PROPN iajs-1919 138 21 for for ADP iajs-1919 138 22 some some DET iajs-1919 138 23 ideal ideal ADJ iajs-1919 138 24 i i PRON iajs-1919 138 25 of of ADP iajs-1919 138 26 r r NOUN iajs-1919 138 27 ( ( PUNCT iajs-1919 138 28 equivalently equivalently ADV iajs-1919 138 29 , , PUNCT iajs-1919 138 30 m m VERB iajs-1919 138 31 a a DET iajs-1919 138 32 multiplication multiplication NOUN iajs-1919 138 33 if if SCONJ iajs-1919 138 34 and and CCONJ iajs-1919 138 35 only only ADV iajs-1919 138 36 if if SCONJ iajs-1919 138 37 𝑁 𝑁 PROPN iajs-1919 138 38 𝑀. 𝑀. PROPN iajs-1919 138 39 𝑁 𝑁 PROPN iajs-1919 138 40 : : PUNCT iajs-1919 138 41 𝑀 𝑀 PROPN iajs-1919 138 42 ) ) PUNCT iajs-1919 139 1 [ [ X iajs-1919 139 2 12 12 NUM iajs-1919 139 3 ] ] PUNCT iajs-1919 139 4 . . PUNCT iajs-1919 140 1 if if SCONJ iajs-1919 140 2 𝑟 𝑟 PRON iajs-1919 140 3 𝑀 𝑀 PROPN iajs-1919 140 4 0 0 NUM iajs-1919 140 5 , , PUNCT iajs-1919 140 6 then then ADV iajs-1919 140 7 m m VERB iajs-1919 140 8 is be AUX iajs-1919 140 9 called call VERB iajs-1919 140 10 a a DET iajs-1919 140 11 faithful faithful ADJ iajs-1919 140 12 r r NOUN iajs-1919 140 13 - - PUNCT iajs-1919 140 14 module module NOUN iajs-1919 140 15 . . PUNCT iajs-1919 141 1 an an DET iajs-1919 141 2 r r NOUN iajs-1919 141 3 - - PUNCT iajs-1919 141 4 module module NOUN iajs-1919 141 5 m m NOUN iajs-1919 141 6 is be AUX iajs-1919 141 7 said say VERB iajs-1919 141 8 to to PART iajs-1919 141 9 be be AUX iajs-1919 141 10 scalar scalar ADJ iajs-1919 141 11 if if SCONJ iajs-1919 141 12 for for ADP iajs-1919 141 13 any any DET iajs-1919 141 14 𝜑 𝜑 NOUN iajs-1919 141 15 ∈ ∈ PROPN iajs-1919 141 16 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 141 17 𝑀 𝑀 PROPN iajs-1919 141 18 , , PUNCT iajs-1919 141 19 𝜑 𝜑 X iajs-1919 141 20 𝑚 𝑚 X iajs-1919 141 21 𝑚𝑟 𝑚𝑟 NOUN iajs-1919 141 22 for for ADP iajs-1919 141 23 some some DET iajs-1919 141 24 𝑟 𝑟 PRON iajs-1919 141 25 ∈ ∈ PROPN iajs-1919 141 26 𝑅 𝑅 PROPN iajs-1919 141 27 , , PUNCT iajs-1919 141 28 and and CCONJ iajs-1919 141 29 for for ADP iajs-1919 141 30 all all DET iajs-1919 141 31 𝑚 𝑚 ADP iajs-1919 141 32 ∈ ∈ PROPN iajs-1919 141 33 𝑀 𝑀 PROPN iajs-1919 142 1 [ [ X iajs-1919 142 2 13 13 NUM iajs-1919 142 3 ] ] PUNCT iajs-1919 142 4 . . PUNCT iajs-1919 143 1 now now ADV iajs-1919 143 2 , , PUNCT iajs-1919 143 3 we we PRON iajs-1919 143 4 will will AUX iajs-1919 143 5 studied study VERB iajs-1919 143 6 the the DET iajs-1919 143 7 strongly strongly ADV iajs-1919 143 8 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 143 9 property property NOUN iajs-1919 143 10 for for ADP iajs-1919 143 11 rings ring NOUN iajs-1919 143 12 and and CCONJ iajs-1919 143 13 modules module NOUN iajs-1919 143 14 . . PUNCT iajs-1919 144 1 but but CCONJ iajs-1919 144 2 , , PUNCT iajs-1919 144 3 in in ADP iajs-1919 144 4 a a DET iajs-1919 144 5 position position NOUN iajs-1919 144 6 we we PRON iajs-1919 144 7 need need VERB iajs-1919 144 8 the the DET iajs-1919 144 9 following follow VERB iajs-1919 144 10 lemma lemma PROPN iajs-1919 144 11 . . PUNCT iajs-1919 145 1 lemma lemma PROPN iajs-1919 145 2 15 15 NUM iajs-1919 145 3 . . PUNCT iajs-1919 146 1 the the DET iajs-1919 146 2 following follow VERB iajs-1919 146 3 holds hold NOUN iajs-1919 146 4 , , PUNCT iajs-1919 146 5 for for ADP iajs-1919 146 6 faithful faithful ADJ iajs-1919 146 7 multiplication multiplication NOUN iajs-1919 146 8 r r NOUN iajs-1919 146 9 - - PUNCT iajs-1919 146 10 module module NOUN iajs-1919 146 11 m. m. NOUN iajs-1919 146 12 𝑖 𝑖 ADP iajs-1919 147 1 𝑁 𝑁 PROPN iajs-1919 147 2 ≪ ≪ PUNCT iajs-1919 147 3 𝑀 𝑀 PROPN iajs-1919 147 4 if if SCONJ iajs-1919 147 5 and and CCONJ iajs-1919 147 6 only only ADV iajs-1919 147 7 if if SCONJ iajs-1919 147 8 𝐼 𝐼 PROPN iajs-1919 147 9 ≪ ≪ ADJ iajs-1919 147 10 𝑅 𝑅 NOUN iajs-1919 147 11 , , PUNCT iajs-1919 147 12 where where SCONJ iajs-1919 147 13 𝑁 𝑁 PROPN iajs-1919 147 14 𝑀𝐼. 𝑀𝐼. X iajs-1919 147 15 𝑖𝑖 𝑖𝑖 X iajs-1919 148 1 𝑁 𝑁 PROPN iajs-1919 148 2 ⊴ ⊴ ADP iajs-1919 148 3 𝑀 𝑀 PROPN iajs-1919 148 4 if if SCONJ iajs-1919 148 5 and and CCONJ iajs-1919 148 6 only only ADV iajs-1919 148 7 if if SCONJ iajs-1919 148 8 𝐼 𝐼 PROPN iajs-1919 148 9 ⊴ ⊴ PROPN iajs-1919 148 10 𝑅 𝑅 PROPN iajs-1919 148 11 , , PUNCT iajs-1919 148 12 where where SCONJ iajs-1919 148 13 𝑁 𝑁 PROPN iajs-1919 148 14 𝑀𝐼. 𝑀𝐼. NOUN iajs-1919 148 15 proof proof NOUN iajs-1919 148 16 . . PUNCT iajs-1919 149 1 𝑖 𝑖 PRON iajs-1919 149 2 assume assume VERB iajs-1919 149 3 that that SCONJ iajs-1919 149 4 𝑁 𝑁 PROPN iajs-1919 149 5 ≪ ≪ ADJ iajs-1919 149 6 𝑀. 𝑀. NOUN iajs-1919 149 7 let let VERB iajs-1919 149 8 j j NOUN iajs-1919 149 9 be be AUX iajs-1919 149 10 any any DET iajs-1919 149 11 ideal ideal NOUN iajs-1919 149 12 of of ADP iajs-1919 149 13 r r NOUN iajs-1919 149 14 with with ADP iajs-1919 149 15 𝐼 𝐼 PROPN iajs-1919 149 16 𝐽 𝐽 PROPN iajs-1919 149 17 𝑅 𝑅 PROPN iajs-1919 149 18 , , PUNCT iajs-1919 149 19 so so SCONJ iajs-1919 149 20 𝑀 𝑀 PROPN iajs-1919 149 21 𝐼 𝐼 PROPN iajs-1919 149 22 𝐽 𝐽 PROPN iajs-1919 149 23 𝑀𝑅 𝑀𝑅 PROPN iajs-1919 149 24 , , PUNCT iajs-1919 149 25 that that ADV iajs-1919 149 26 is be AUX iajs-1919 149 27 ; ; PUNCT iajs-1919 149 28 𝑁 𝑁 PROPN iajs-1919 149 29 𝑀𝐽 𝑀𝐽 PROPN iajs-1919 149 30 𝑀 𝑀 PROPN iajs-1919 149 31 , , PUNCT iajs-1919 149 32 but but CCONJ iajs-1919 149 33 𝑁 𝑁 PROPN iajs-1919 149 34 ≪ ≪ PUNCT iajs-1919 149 35 𝑀 𝑀 PROPN iajs-1919 149 36 implies imply VERB iajs-1919 149 37 𝑀𝐽 𝑀𝐽 PROPN iajs-1919 149 38 𝑀 𝑀 PROPN iajs-1919 149 39 , , PUNCT iajs-1919 149 40 and and CCONJ iajs-1919 149 41 so so ADV iajs-1919 149 42 𝐽 𝐽 PROPN iajs-1919 149 43 𝑅 𝑅 PROPN iajs-1919 149 44 , , PUNCT iajs-1919 149 45 since since SCONJ iajs-1919 149 46 m m PROPN iajs-1919 149 47 is be AUX iajs-1919 149 48 a a DET iajs-1919 149 49 faithful faithful ADJ iajs-1919 149 50 multiplication multiplication NOUN iajs-1919 149 51 r r NOUN iajs-1919 149 52 - - PUNCT iajs-1919 149 53 module module NOUN iajs-1919 149 54 . . PUNCT iajs-1919 150 1 thus thus ADV iajs-1919 150 2 𝐼 𝐼 ADP iajs-1919 150 3 ≪ ≪ ADJ iajs-1919 150 4 𝑅. 𝑅. NOUN iajs-1919 150 5 conversely conversely ADV iajs-1919 150 6 , , PUNCT iajs-1919 150 7 let let VERB iajs-1919 150 8 𝐾 𝐾 PROPN iajs-1919 150 9 𝑀 𝑀 PROPN iajs-1919 150 10 with with ADP iajs-1919 150 11 𝑁 𝑁 PROPN iajs-1919 150 12 𝐾 𝐾 PROPN iajs-1919 150 13 𝑀. 𝑀. PROPN iajs-1919 150 14 as as SCONJ iajs-1919 150 15 m m NOUN iajs-1919 150 16 is be AUX iajs-1919 150 17 multiplication multiplication NOUN iajs-1919 150 18 , , PUNCT iajs-1919 150 19 𝐾 𝐾 PROPN iajs-1919 150 20 𝑀𝐽 𝑀𝐽 VERB iajs-1919 150 21 for for ADP iajs-1919 150 22 some some DET iajs-1919 150 23 𝐽 𝐽 NOUN iajs-1919 150 24 𝑅. 𝑅. NOUN iajs-1919 150 25 hence hence ADV iajs-1919 150 26 𝑀 𝑀 NOUN iajs-1919 150 27 𝐼 𝐼 PROPN iajs-1919 150 28 𝐽 𝐽 NOUN iajs-1919 150 29 𝑁 𝑁 PROPN iajs-1919 150 30 𝐾 𝐾 PROPN iajs-1919 150 31 𝑀 𝑀 PROPN iajs-1919 150 32 𝑀𝑅 𝑀𝑅 PROPN iajs-1919 150 33 , , PUNCT iajs-1919 150 34 but but CCONJ iajs-1919 150 35 m m NOUN iajs-1919 150 36 is be AUX iajs-1919 150 37 a a DET iajs-1919 150 38 faithful faithful ADJ iajs-1919 150 39 multiplication multiplication NOUN iajs-1919 150 40 rmodule rmodule NOUN iajs-1919 150 41 , , PUNCT iajs-1919 150 42 so so SCONJ iajs-1919 150 43 𝐼 𝐼 PROPN iajs-1919 150 44 𝐽 𝐽 PROPN iajs-1919 150 45 𝑅 𝑅 PROPN iajs-1919 150 46 , , PUNCT iajs-1919 150 47 thus thus ADV iajs-1919 150 48 𝐽 𝐽 PROPN iajs-1919 150 49 𝑅 𝑅 PROPN iajs-1919 150 50 ( ( PUNCT iajs-1919 150 51 since since SCONJ iajs-1919 150 52 𝐼 𝐼 PROPN iajs-1919 150 53 ≪ ≪ ADJ iajs-1919 150 54 𝑅 𝑅 NOUN iajs-1919 150 55 ) ) PUNCT iajs-1919 150 56 . . PUNCT iajs-1919 151 1 therefore therefore ADV iajs-1919 151 2 , , PUNCT iajs-1919 151 3 𝐾 𝐾 PROPN iajs-1919 151 4 𝑀𝐽 𝑀𝐽 PROPN iajs-1919 151 5 𝑀𝑅 𝑀𝑅 PROPN iajs-1919 151 6 𝑀 𝑀 PROPN iajs-1919 151 7 , , PUNCT iajs-1919 151 8 and and CCONJ iajs-1919 151 9 hence hence ADV iajs-1919 151 10 𝑁 𝑁 PROPN iajs-1919 151 11 ≪ ≪ ADJ iajs-1919 151 12 𝑀. 𝑀. NOUN iajs-1919 151 13 𝑖𝑖 𝑖𝑖 NOUN iajs-1919 151 14 let let VERB iajs-1919 151 15 𝑁 𝑁 PROPN iajs-1919 151 16 ⊴ ⊴ ADP iajs-1919 151 17 𝑀. 𝑀. PROPN iajs-1919 151 18 suppose suppose VERB iajs-1919 151 19 that that SCONJ iajs-1919 151 20 𝐽 𝐽 PROPN iajs-1919 151 21 ≪ ≪ VERB iajs-1919 151 22 𝑅 𝑅 NOUN iajs-1919 151 23 with with ADP iajs-1919 151 24 𝐼 𝐼 PROPN iajs-1919 151 25 ∩ ∩ ADJ iajs-1919 151 26 𝐽 𝐽 PROPN iajs-1919 151 27 0 0 NUM iajs-1919 151 28 , , PUNCT iajs-1919 151 29 then then ADV iajs-1919 151 30 𝑁 𝑁 PROPN iajs-1919 151 31 ∩ ∩ ADJ iajs-1919 151 32 𝑀𝐽 𝑀𝐽 PROPN iajs-1919 151 33 𝑀𝐼 𝑀𝐼 PROPN iajs-1919 151 34 ∩ ∩ NOUN iajs-1919 151 35 𝑀𝐽 𝑀𝐽 PROPN iajs-1919 151 36 𝑀 𝑀 PROPN iajs-1919 151 37 𝐼 𝐼 PROPN iajs-1919 151 38 ∩ ∩ NOUN iajs-1919 151 39 𝐽 𝐽 PROPN iajs-1919 151 40 0 0 NUM iajs-1919 151 41 , , PUNCT iajs-1919 151 42 but but CCONJ iajs-1919 151 43 by by ADP iajs-1919 151 44 𝑖 𝑖 X iajs-1919 151 45 , , PUNCT iajs-1919 151 46 𝑀𝐽 𝑀𝐽 PROPN iajs-1919 151 47 ≪ ≪ PUNCT iajs-1919 151 48 𝑀 𝑀 PROPN iajs-1919 151 49 , , PUNCT iajs-1919 151 50 hence hence ADV iajs-1919 151 51 𝑀𝐽 𝑀𝐽 PROPN iajs-1919 151 52 0 0 NUM iajs-1919 151 53 , , PUNCT iajs-1919 151 54 implies imply VERB iajs-1919 151 55 𝐽 𝐽 PROPN iajs-1919 151 56 0 0 NUM iajs-1919 151 57 ( ( PUNCT iajs-1919 151 58 since since SCONJ iajs-1919 151 59 m m PROPN iajs-1919 151 60 is be AUX iajs-1919 151 61 faithful faithful ADJ iajs-1919 151 62 ) ) PUNCT iajs-1919 151 63 . . PUNCT iajs-1919 152 1 thus thus ADV iajs-1919 152 2 𝐼 𝐼 ADP iajs-1919 152 3 ⊴ ⊴ ADP iajs-1919 152 4 𝑅. 𝑅. NOUN iajs-1919 152 5 conversely conversely ADV iajs-1919 152 6 , , PUNCT iajs-1919 152 7 let let VERB iajs-1919 152 8 𝐾 𝐾 PROPN iajs-1919 152 9 ≪ ≪ VERB iajs-1919 152 10 𝑀 𝑀 PROPN iajs-1919 152 11 such such ADJ iajs-1919 152 12 that that SCONJ iajs-1919 152 13 𝑁 𝑁 PROPN iajs-1919 152 14 ∩ ∩ ADJ iajs-1919 152 15 𝐾 𝐾 PROPN iajs-1919 152 16 0 0 PROPN iajs-1919 152 17 . . PUNCT iajs-1919 153 1 since since SCONJ iajs-1919 153 2 m m PROPN iajs-1919 153 3 is be AUX iajs-1919 153 4 multiplication multiplication NOUN iajs-1919 153 5 , , PUNCT iajs-1919 153 6 then then ADV iajs-1919 153 7 there there PRON iajs-1919 153 8 is be VERB iajs-1919 153 9 a a DET iajs-1919 153 10 small small ADJ iajs-1919 153 11 ideal ideal NOUN iajs-1919 153 12 j j PROPN iajs-1919 153 13 of of ADP iajs-1919 153 14 r r NOUN iajs-1919 153 15 with with ADP iajs-1919 153 16 𝐾 𝐾 PROPN iajs-1919 153 17 𝑀𝐽 𝑀𝐽 PROPN iajs-1919 153 18 , , PUNCT iajs-1919 153 19 by by ADP iajs-1919 153 20 𝑖 𝑖 X iajs-1919 153 21 . . PUNCT iajs-1919 154 1 hence hence ADV iajs-1919 154 2 𝑀 𝑀 PROPN iajs-1919 154 3 𝐼 𝐼 PROPN iajs-1919 154 4 ∩ ∩ ADJ iajs-1919 154 5 𝐽 𝐽 PROPN iajs-1919 154 6 𝑀𝐼 𝑀𝐼 PROPN iajs-1919 154 7 ∩ ∩ NOUN iajs-1919 154 8 𝑀𝐽 𝑀𝐽 VERB iajs-1919 154 9 𝑁 𝑁 PROPN iajs-1919 154 10 ∩ ∩ ADJ iajs-1919 154 11 𝐾 𝐾 PROPN iajs-1919 154 12 0 0 NUM iajs-1919 154 13 , , PUNCT iajs-1919 154 14 so so ADV iajs-1919 154 15 by by ADP iajs-1919 154 16 faithfulty faithfulty NOUN iajs-1919 154 17 for for ADP iajs-1919 154 18 m m PROPN iajs-1919 154 19 , , PUNCT iajs-1919 154 20 we we PRON iajs-1919 154 21 get get VERB iajs-1919 154 22 𝐼 𝐼 ADP iajs-1919 154 23 ∩ ∩ NOUN iajs-1919 154 24 𝐽 𝐽 PROPN iajs-1919 154 25 0 0 NUM iajs-1919 154 26 , , PUNCT iajs-1919 154 27 then then ADV iajs-1919 154 28 𝐽 𝐽 PROPN iajs-1919 154 29 0 0 NUM iajs-1919 154 30 , , PUNCT iajs-1919 154 31 as as ADP iajs-1919 154 32 𝐽 𝐽 PROPN iajs-1919 154 33 ≪ ≪ PUNCT iajs-1919 154 34 𝑅 𝑅 PROPN iajs-1919 154 35 and and CCONJ iajs-1919 154 36 𝐼 𝐼 PROPN iajs-1919 154 37 ⊴ ⊴ ADP iajs-1919 154 38 𝑅. 𝑅. NOUN iajs-1919 154 39 thus thus ADV iajs-1919 154 40 𝐾 𝐾 PROPN iajs-1919 154 41 𝑀𝐽 𝑀𝐽 PROPN iajs-1919 154 42 0 0 NUM iajs-1919 154 43 , , PUNCT iajs-1919 154 44 and and CCONJ iajs-1919 154 45 so so ADV iajs-1919 154 46 𝑁 𝑁 PROPN iajs-1919 154 47 ⊴ ⊴ ADP iajs-1919 154 48 𝑀. 𝑀. PROPN iajs-1919 154 49 ∎ ∎ PROPN iajs-1919 154 50 proposition proposition NOUN iajs-1919 154 51 16 16 NUM iajs-1919 154 52 . . PUNCT iajs-1919 155 1 let let VERB iajs-1919 155 2 m m PRON iajs-1919 155 3 be be AUX iajs-1919 155 4 a a DET iajs-1919 155 5 faithful faithful ADJ iajs-1919 155 6 multiplication multiplication NOUN iajs-1919 155 7 r r NOUN iajs-1919 155 8 - - NOUN iajs-1919 155 9 module module NOUN iajs-1919 155 10 . . PUNCT iajs-1919 156 1 if if SCONJ iajs-1919 156 2 m m NOUN iajs-1919 156 3 is be AUX iajs-1919 156 4 a a DET iajs-1919 156 5 strongly strongly ADV iajs-1919 156 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 156 7 rmodule rmodule NOUN iajs-1919 156 8 , , PUNCT iajs-1919 156 9 then then ADV iajs-1919 156 10 r r NOUN iajs-1919 156 11 is be AUX iajs-1919 156 12 strongly strongly ADV iajs-1919 156 13 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 156 14 . . PUNCT iajs-1919 157 1 the the DET iajs-1919 157 2 converse converse NOUN iajs-1919 157 3 hold hold NOUN iajs-1919 157 4 , , PUNCT iajs-1919 157 5 whenever whenever SCONJ iajs-1919 157 6 m m VERB iajs-1919 157 7 is be AUX iajs-1919 157 8 finitely finitely ADV iajs-1919 157 9 generated generate VERB iajs-1919 157 10 . . PUNCT iajs-1919 158 1 proof proof NOUN iajs-1919 158 2 . . PUNCT iajs-1919 159 1 assume assume VERB iajs-1919 159 2 that that SCONJ iajs-1919 159 3 m m PROPN iajs-1919 159 4 is be AUX iajs-1919 159 5 a a DET iajs-1919 159 6 strongly strongly ADV iajs-1919 159 7 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 159 8 r r NOUN iajs-1919 159 9 - - PUNCT iajs-1919 159 10 module module NOUN iajs-1919 159 11 . . PUNCT iajs-1919 160 1 let let VERB iajs-1919 160 2 0 0 NUM iajs-1919 160 3 𝜑 𝜑 NOUN iajs-1919 160 4 ∈ ∈ PROPN iajs-1919 160 5 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 160 6 𝑅 𝑅 PROPN iajs-1919 160 7 . . PUNCT iajs-1919 161 1 for for ADP iajs-1919 161 2 𝑟 𝑟 DET iajs-1919 161 3 ∈ ∈ PROPN iajs-1919 161 4 𝑅 𝑅 PROPN iajs-1919 161 5 , , PUNCT iajs-1919 161 6 we we PRON iajs-1919 161 7 know know VERB iajs-1919 161 8 𝜑 𝜑 PRON iajs-1919 162 1 𝑎 𝑎 X iajs-1919 162 2 𝑎. 𝑎. NOUN iajs-1919 162 3 𝜑 𝜑 NOUN iajs-1919 162 4 1 1 X iajs-1919 162 5 . . PUNCT iajs-1919 163 1 we we PRON iajs-1919 163 2 can can AUX iajs-1919 163 3 define define VERB iajs-1919 163 4 𝜓 𝜓 NOUN iajs-1919 163 5 : : PUNCT iajs-1919 163 6 𝑀 𝑀 PROPN iajs-1919 163 7 → → SYM iajs-1919 163 8 𝑀 𝑀 PROPN iajs-1919 163 9 by by ADP iajs-1919 163 10 𝜓 𝜓 ADP iajs-1919 163 11 𝑚 𝑚 X iajs-1919 163 12 𝑚. 𝑚. NOUN iajs-1919 163 13 𝜑 𝜑 ADP iajs-1919 163 14 1 1 NUM iajs-1919 163 15 for for ADP iajs-1919 163 16 all all DET iajs-1919 163 17 𝑚 𝑚 ADP iajs-1919 163 18 ∈ ∈ NOUN iajs-1919 163 19 𝑀. 𝑀. PROPN iajs-1919 163 20 it it PRON iajs-1919 163 21 is be AUX iajs-1919 163 22 easy easy ADJ iajs-1919 163 23 to to PART iajs-1919 163 24 see see VERB iajs-1919 163 25 𝜓 𝜓 PROPN iajs-1919 163 26 is be AUX iajs-1919 163 27 well well ADV iajs-1919 163 28 - - PUNCT iajs-1919 163 29 defined define VERB iajs-1919 163 30 and and CCONJ iajs-1919 163 31 homomorphism homomorphism NOUN iajs-1919 163 32 . . PUNCT iajs-1919 164 1 if if SCONJ iajs-1919 164 2 𝜓 𝜓 PROPN iajs-1919 164 3 0 0 NUM iajs-1919 164 4 , , PUNCT iajs-1919 164 5 then then ADV iajs-1919 164 6 𝑀. 𝑀. NOUN iajs-1919 164 7 𝜑 𝜑 PROPN iajs-1919 164 8 1 1 NUM iajs-1919 164 9 0 0 NUM iajs-1919 164 10 , , PUNCT iajs-1919 164 11 hence hence ADV iajs-1919 164 12 𝜑 𝜑 PROPN iajs-1919 164 13 1 1 NUM iajs-1919 164 14 ∈ ∈ PROPN iajs-1919 164 15 𝑟 𝑟 X iajs-1919 164 16 𝑀 𝑀 PROPN iajs-1919 164 17 0 0 NUM iajs-1919 164 18 , , PUNCT iajs-1919 164 19 so so ADV iajs-1919 164 20 𝜑 𝜑 PROPN iajs-1919 164 21 0 0 NUM iajs-1919 164 22 which which PRON iajs-1919 164 23 is be AUX iajs-1919 164 24 a a DET iajs-1919 164 25 contradiction contradiction NOUN iajs-1919 164 26 . . PUNCT iajs-1919 165 1 hence hence ADV iajs-1919 165 2 0 0 NUM iajs-1919 165 3 𝜓 𝜓 PRON iajs-1919 165 4 ∈ ∈ PROPN iajs-1919 165 5 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 165 6 𝑀 𝑀 PROPN iajs-1919 165 7 , , PUNCT iajs-1919 165 8 and and CCONJ iajs-1919 165 9 so so ADV iajs-1919 165 10 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 PROPN iajs-1919 165 11 ⋬ ⋬ AUX iajs-1919 166 1 𝑀 𝑀 PROPN iajs-1919 166 2 , , PUNCT iajs-1919 166 3 as as SCONJ iajs-1919 166 4 m m PROPN iajs-1919 166 5 is be AUX iajs-1919 166 6 strongly strongly ADV iajs-1919 166 7 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 166 8 . . PUNCT iajs-1919 167 1 since since SCONJ iajs-1919 167 2 m m PROPN iajs-1919 167 3 is be AUX iajs-1919 167 4 a a DET iajs-1919 167 5 multiplication multiplication NOUN iajs-1919 167 6 r r NOUN iajs-1919 167 7 - - PUNCT iajs-1919 167 8 module module NOUN iajs-1919 167 9 , , PUNCT iajs-1919 167 10 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 NOUN iajs-1919 167 11 𝑀. 𝑀. PROPN iajs-1919 167 12 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 PROPN iajs-1919 167 13 : : PUNCT iajs-1919 167 14 𝑀 𝑀 PROPN iajs-1919 167 15 . . PUNCT iajs-1919 168 1 but but CCONJ iajs-1919 168 2 , , PUNCT iajs-1919 168 3 we we PRON iajs-1919 168 4 have have AUX iajs-1919 168 5 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 VERB iajs-1919 168 6 : : PUNCT iajs-1919 169 1 𝑀 𝑀 PROPN iajs-1919 169 2 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 169 3 , , PUNCT iajs-1919 169 4 to to PART iajs-1919 169 5 see see VERB iajs-1919 169 6 this this PRON iajs-1919 169 7 : : PUNCT iajs-1919 169 8 if if SCONJ iajs-1919 169 9 𝑟 𝑟 X iajs-1919 169 10 ∈ ∈ PROPN iajs-1919 169 11 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 NOUN iajs-1919 169 12 : : PUNCT iajs-1919 169 13 𝑀 𝑀 PROPN iajs-1919 169 14 , , PUNCT iajs-1919 169 15 𝑀𝑟 𝑀𝑟 PROPN iajs-1919 169 16 ⊆ ⊆ NUM iajs-1919 169 17 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 NOUN iajs-1919 169 18 , , PUNCT iajs-1919 169 19 so so SCONJ iajs-1919 169 20 𝜓 𝜓 ADP iajs-1919 169 21 𝑀𝑟 𝑀𝑟 PROPN iajs-1919 169 22 𝑀𝑟. 𝑀𝑟. PROPN iajs-1919 169 23 𝜑 𝜑 NOUN iajs-1919 169 24 1 1 NUM iajs-1919 169 25 𝑀. 𝑀. NOUN iajs-1919 169 26 𝜑 𝜑 NOUN iajs-1919 169 27 𝑟 𝑟 NOUN iajs-1919 169 28 0 0 NUM iajs-1919 169 29 , , PUNCT iajs-1919 169 30 hence hence ADV iajs-1919 169 31 𝜑 𝜑 NOUN iajs-1919 169 32 𝑟 𝑟 X iajs-1919 169 33 ∈ ∈ PROPN iajs-1919 169 34 𝑟 𝑟 X iajs-1919 169 35 𝑀 𝑀 PROPN iajs-1919 169 36 0 0 NUM iajs-1919 169 37 , , PUNCT iajs-1919 169 38 thus thus ADV iajs-1919 169 39 𝑟 𝑟 X iajs-1919 169 40 ∈ ∈ PROPN iajs-1919 169 41 𝑘𝑒𝑟𝜑. 𝑘𝑒𝑟𝜑. VERB iajs-1919 169 42 now now ADV iajs-1919 169 43 , , PUNCT iajs-1919 169 44 if if SCONJ iajs-1919 169 45 𝑥 𝑥 PROPN iajs-1919 169 46 ∈ ∈ PROPN iajs-1919 169 47 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 NOUN iajs-1919 169 48 , , PUNCT iajs-1919 169 49 𝜑 𝜑 PROPN iajs-1919 169 50 𝑥 𝑥 X iajs-1919 169 51 𝑥. 𝑥. NOUN iajs-1919 169 52 𝜑 𝜑 PROPN iajs-1919 169 53 1 1 NUM iajs-1919 169 54 0 0 NUM iajs-1919 169 55 hence hence ADV iajs-1919 169 56 𝑀𝑥. 𝑀𝑥. NOUN iajs-1919 169 57 𝜑 𝜑 NOUN iajs-1919 169 58 1 1 NUM iajs-1919 169 59 0 0 NUM iajs-1919 169 60 , , PUNCT iajs-1919 169 61 so so SCONJ iajs-1919 169 62 𝜓 𝜓 X iajs-1919 169 63 𝑀𝑥 𝑀𝑥 PROPN iajs-1919 169 64 0 0 NUM iajs-1919 169 65 implies imply VERB iajs-1919 169 66 𝑀𝑥 𝑀𝑥 PROPN iajs-1919 169 67 ⊆ ⊆ NUM iajs-1919 169 68 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 NOUN iajs-1919 169 69 , , PUNCT iajs-1919 169 70 thus thus ADV iajs-1919 169 71 𝑥 𝑥 PRON iajs-1919 169 72 ∈ ∈ PROPN iajs-1919 169 73 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 NOUN iajs-1919 169 74 : : PUNCT iajs-1919 169 75 𝑀 𝑀 PROPN iajs-1919 169 76 . . PUNCT iajs-1919 170 1 since since SCONJ iajs-1919 170 2 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 PROPN iajs-1919 170 3 ⋬ ⋬ AUX iajs-1919 170 4 𝑀 𝑀 PROPN iajs-1919 170 5 , , PUNCT iajs-1919 170 6 so so PROPN iajs-1919 170 7 𝑀. 𝑀. PROPN iajs-1919 170 8 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 PROPN iajs-1919 170 9 : : PUNCT iajs-1919 170 10 𝑀 𝑀 PROPN iajs-1919 170 11 ⋬ ⋬ AUX iajs-1919 170 12 𝑀 𝑀 PROPN iajs-1919 170 13 , , PUNCT iajs-1919 170 14 so so ADV iajs-1919 170 15 by by ADP iajs-1919 170 16 lemma lemma PROPN iajs-1919 170 17 15 15 NUM iajs-1919 170 18 𝑖𝑖 𝑖𝑖 NOUN iajs-1919 170 19 , , PUNCT iajs-1919 170 20 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 PROPN iajs-1919 170 21 : : PUNCT iajs-1919 170 22 𝑀 𝑀 PROPN iajs-1919 170 23 ⋬ ⋬ AUX iajs-1919 170 24 𝑅 𝑅 NOUN iajs-1919 170 25 , , PUNCT iajs-1919 170 26 which which PRON iajs-1919 170 27 hence hence ADV iajs-1919 170 28 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 170 29 ⋬ ⋬ PUNCT iajs-1919 171 1 𝑅 𝑅 NOUN iajs-1919 171 2 , , PUNCT iajs-1919 171 3 therefore therefore ADV iajs-1919 171 4 r r NOUN iajs-1919 171 5 is be AUX iajs-1919 171 6 strongly strongly ADV iajs-1919 171 7 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 171 8 . . PUNCT iajs-1919 172 1 conversely conversely ADV iajs-1919 172 2 , , PUNCT iajs-1919 172 3 let let VERB iajs-1919 172 4 0 0 NUM iajs-1919 172 5 𝑔 𝑔 PROPN iajs-1919 172 6 ∈ ∈ PROPN iajs-1919 172 7 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 172 8 𝑀 𝑀 PROPN iajs-1919 172 9 . . PUNCT iajs-1919 173 1 if if SCONJ iajs-1919 173 2 m m NOUN iajs-1919 173 3 is be AUX iajs-1919 173 4 finitely finitely ADV iajs-1919 173 5 generated generate VERB iajs-1919 173 6 multiplication multiplication NOUN iajs-1919 173 7 r r NOUN iajs-1919 173 8 - - PUNCT iajs-1919 173 9 module module NOUN iajs-1919 173 10 , , PUNCT iajs-1919 173 11 then then ADV iajs-1919 173 12 m m PROPN iajs-1919 173 13 is be AUX iajs-1919 173 14 a a DET iajs-1919 173 15 scalar scalar ADJ iajs-1919 173 16 r r NOUN iajs-1919 173 17 - - PUNCT iajs-1919 173 18 module module NOUN iajs-1919 173 19 , , PUNCT iajs-1919 173 20 by by ADP iajs-1919 173 21 [ [ X iajs-1919 173 22 14 14 NUM iajs-1919 173 23 , , PUNCT iajs-1919 173 24 th th NOUN iajs-1919 173 25 . . NOUN iajs-1919 173 26 2.3 2.3 NUM iajs-1919 173 27 ] ] PUNCT iajs-1919 173 28 . . PUNCT iajs-1919 174 1 hence hence ADV iajs-1919 174 2 𝑔 𝑔 X iajs-1919 174 3 𝑚 𝑚 X iajs-1919 174 4 𝑚𝑟 𝑚𝑟 NOUN iajs-1919 174 5 for for ADP iajs-1919 174 6 some some DET iajs-1919 174 7 𝑟 𝑟 PRON iajs-1919 174 8 ∈ ∈ PROPN iajs-1919 174 9 𝑅 𝑅 PROPN iajs-1919 174 10 , , PUNCT iajs-1919 174 11 and and CCONJ iajs-1919 174 12 for for ADP iajs-1919 174 13 all all DET iajs-1919 174 14 𝑚 𝑚 PART iajs-1919 174 15 ∈ ∈ NOUN iajs-1919 174 16 𝑀. 𝑀. PROPN iajs-1919 174 17 it it PRON iajs-1919 174 18 follows follow VERB iajs-1919 174 19 that that SCONJ iajs-1919 174 20 ℎ ℎ PROPN iajs-1919 174 21 ∈ ∈ PROPN iajs-1919 174 22 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 174 23 𝑅 𝑅 PROPN iajs-1919 174 24 defined define VERB iajs-1919 174 25 by by ADP iajs-1919 174 26 ℎ ℎ X iajs-1919 174 27 𝑥 𝑥 PROPN iajs-1919 174 28 𝑥𝑟 𝑥𝑟 PROPN iajs-1919 174 29 for for ADP iajs-1919 174 30 all all DET iajs-1919 174 31 𝑥 𝑥 PRON iajs-1919 174 32 ∈ ∈ NOUN iajs-1919 174 33 𝑅. 𝑅. NOUN iajs-1919 174 34 note note NOUN iajs-1919 174 35 ℎ ℎ PROPN iajs-1919 174 36 1 1 NUM iajs-1919 174 37 1 1 NUM iajs-1919 174 38 . . PUNCT iajs-1919 175 1 𝑟 𝑟 PRON iajs-1919 175 2 𝑟 𝑟 NOUN iajs-1919 175 3 0 0 PUNCT iajs-1919 176 1 ( ( PUNCT iajs-1919 176 2 in in ADP iajs-1919 176 3 fact fact NOUN iajs-1919 176 4 , , PUNCT iajs-1919 176 5 if if SCONJ iajs-1919 176 6 𝑟 𝑟 NOUN iajs-1919 176 7 0 0 NUM iajs-1919 176 8 implies imply VERB iajs-1919 176 9 𝑔 𝑔 PROPN iajs-1919 176 10 0 0 NUM iajs-1919 176 11 ) ) PUNCT iajs-1919 176 12 , , PUNCT iajs-1919 176 13 and and CCONJ iajs-1919 176 14 hence hence ADV iajs-1919 176 15 0 0 NUM iajs-1919 176 16 ℎ ℎ PROPN iajs-1919 176 17 ∈ ∈ PROPN iajs-1919 176 18 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 176 19 𝑅 𝑅 PROPN iajs-1919 176 20 , , PUNCT iajs-1919 176 21 but but CCONJ iajs-1919 176 22 r r NOUN iajs-1919 176 23 is be AUX iajs-1919 176 24 strongly strongly ADV iajs-1919 176 25 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 176 26 , , PUNCT iajs-1919 176 27 then then ADV iajs-1919 176 28 𝑘𝑒𝑟ℎ 𝑘𝑒𝑟ℎ X iajs-1919 176 29 ⋬ ⋬ X iajs-1919 176 30 𝑅. 𝑅. NOUN iajs-1919 176 31 on on ADP iajs-1919 176 32 the the DET iajs-1919 176 33 other other ADJ iajs-1919 176 34 hand hand NOUN iajs-1919 176 35 , , PUNCT iajs-1919 176 36 we we PRON iajs-1919 176 37 have have VERB iajs-1919 176 38         SPACE iajs-1919 176 39 172 172 NUM iajs-1919 176 40     SPACE iajs-1919 176 41 ibn ibn PROPN iajs-1919 176 42 al al PROPN iajs-1919 176 43 - - PUNCT iajs-1919 176 44 haitham haitham PROPN iajs-1919 176 45 jour.for jour.for PROPN iajs-1919 176 46 pure&appl.sci pure&appl.sci PROPN iajs-1919 176 47 . . PUNCT iajs-1919 177 1 ihjpas ihjpas PROPN iajs-1919 177 2 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 177 3 vol vol NOUN iajs-1919 177 4 . . PUNCT iajs-1919 177 5 32 32 NUM iajs-1919 177 6 ( ( PUNCT iajs-1919 177 7 1 1 NUM iajs-1919 177 8 ) ) PUNCT iajs-1919 177 9 2019 2019 NUM iajs-1919 177 10 𝑘𝑒𝑟ℎ 𝑘𝑒𝑟ℎ NOUN iajs-1919 177 11 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 177 12 : : PUNCT iajs-1919 177 13 𝑀 𝑀 PROPN iajs-1919 177 14 which which PRON iajs-1919 177 15 implies imply VERB iajs-1919 177 16 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 177 17 : : PUNCT iajs-1919 177 18 𝑀 𝑀 PROPN iajs-1919 177 19 ⋬ ⋬ AUX iajs-1919 178 1 𝑅 𝑅 NOUN iajs-1919 178 2 , , PUNCT iajs-1919 178 3 and and CCONJ iajs-1919 178 4 hence hence ADV iajs-1919 178 5 𝑀. 𝑀. PROPN iajs-1919 178 6 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 178 7 : : PUNCT iajs-1919 179 1 𝑀 𝑀 PROPN iajs-1919 179 2 ⋬ ⋬ AUX iajs-1919 179 3 𝑀 𝑀 PROPN iajs-1919 179 4 , , PUNCT iajs-1919 179 5 by by ADP iajs-1919 179 6 lemma lemma PROPN iajs-1919 179 7 15 15 NUM iajs-1919 179 8 𝑖𝑖 𝑖𝑖 NOUN iajs-1919 179 9 , , PUNCT iajs-1919 179 10 thus thus ADV iajs-1919 179 11 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 179 12 ⋬ ⋬ X iajs-1919 179 13 𝑀 𝑀 PROPN iajs-1919 179 14 , , PUNCT iajs-1919 179 15 and and CCONJ iajs-1919 179 16 m m PROPN iajs-1919 179 17 is be AUX iajs-1919 179 18 a a DET iajs-1919 179 19 strongly strongly ADV iajs-1919 179 20 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 179 21 r r NOUN iajs-1919 179 22 - - PUNCT iajs-1919 179 23 module module NOUN iajs-1919 179 24 . . PUNCT iajs-1919 180 1 ∎ ∎ PROPN iajs-1919 180 2 next next ADV iajs-1919 180 3 , , PUNCT iajs-1919 180 4 proved prove VERB iajs-1919 180 5 that that SCONJ iajs-1919 180 6 the the DET iajs-1919 180 7 property property NOUN iajs-1919 180 8 of of ADP iajs-1919 180 9 strongly strongly ADV iajs-1919 180 10 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 180 11 of of ADP iajs-1919 180 12 modules module NOUN iajs-1919 180 13 is be AUX iajs-1919 180 14 inherited inherit VERB iajs-1919 180 15 by by ADP iajs-1919 180 16 isomorphism isomorphism NOUN iajs-1919 180 17 . . PUNCT iajs-1919 181 1 proposition proposition NOUN iajs-1919 181 2 17 17 NUM iajs-1919 181 3 . . PUNCT iajs-1919 182 1 for for ADP iajs-1919 182 2 two two NUM iajs-1919 182 3 modules module NOUN iajs-1919 182 4 𝑀 𝑀 PROPN iajs-1919 182 5 and and CCONJ iajs-1919 182 6 𝑀 𝑀 PROPN iajs-1919 182 7 , , PUNCT iajs-1919 182 8 if if SCONJ iajs-1919 182 9 𝑀 𝑀 PROPN iajs-1919 182 10 ≅ ≅ PROPN iajs-1919 182 11 𝑀 𝑀 PROPN iajs-1919 182 12 then then ADV iajs-1919 182 13 𝑀 𝑀 PROPN iajs-1919 182 14 is be AUX iajs-1919 182 15 a a DET iajs-1919 182 16 strongly strongly ADV iajs-1919 182 17 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 182 18 module module NOUN iajs-1919 182 19 , , PUNCT iajs-1919 182 20 whenever whenever SCONJ iajs-1919 182 21 𝑀 𝑀 PROPN iajs-1919 182 22 is be AUX iajs-1919 182 23 strongly strongly ADV iajs-1919 182 24 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 182 25 . . PUNCT iajs-1919 183 1 proof proof NOUN iajs-1919 183 2 . . PUNCT iajs-1919 184 1 since since SCONJ iajs-1919 184 2 𝑀 𝑀 PROPN iajs-1919 184 3 ≅ ≅ PROPN iajs-1919 184 4 𝑀 𝑀 PROPN iajs-1919 184 5 , , PUNCT iajs-1919 184 6 there there PRON iajs-1919 184 7 exists exist VERB iajs-1919 184 8 an an DET iajs-1919 184 9 isomorphism isomorphism NOUN iajs-1919 184 10 𝑓 𝑓 PRON iajs-1919 184 11 : : PUNCT iajs-1919 184 12 𝑀 𝑀 PROPN iajs-1919 184 13 ⟶ ⟶ PROPN iajs-1919 184 14 𝑀 𝑀 PROPN iajs-1919 184 15 . . PUNCT iajs-1919 185 1 assume assume VERB iajs-1919 185 2 𝑀 𝑀 PROPN iajs-1919 185 3 is be AUX iajs-1919 185 4 a a DET iajs-1919 185 5 strongly strongly ADV iajs-1919 185 6 𝒦nonsigular 𝒦nonsigular ADJ iajs-1919 185 7 module module NOUN iajs-1919 185 8 . . PUNCT iajs-1919 186 1 let let VERB iajs-1919 186 2 𝑔 𝑔 PROPN iajs-1919 186 3 ∈ ∈ PROPN iajs-1919 186 4 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 186 5 𝑀 𝑀 PROPN iajs-1919 186 6 such such ADJ iajs-1919 186 7 that that DET iajs-1919 186 8 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 186 9 ⊴ ⊴ ADP iajs-1919 186 10 𝑀 𝑀 PROPN iajs-1919 186 11 . . PUNCT iajs-1919 187 1 consider consider VERB iajs-1919 187 2 𝜓 𝜓 PRON iajs-1919 187 3 𝑓 𝑓 PRON iajs-1919 187 4 ∘ ∘ PROPN iajs-1919 187 5 𝑔 𝑔 PROPN iajs-1919 187 6 ∘ ∘ PROPN iajs-1919 187 7 𝑓 𝑓 DET iajs-1919 187 8 ∈ ∈ PROPN iajs-1919 187 9 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 187 10 𝑀 𝑀 PROPN iajs-1919 187 11 , , PUNCT iajs-1919 187 12 where where SCONJ iajs-1919 187 13 𝑓 𝑓 X iajs-1919 187 14 : : PUNCT iajs-1919 187 15 𝑀 𝑀 PROPN iajs-1919 187 16 ⟶ ⟶ NOUN iajs-1919 187 17 𝑀 𝑀 PROPN iajs-1919 187 18 isomorphism isomorphism NOUN iajs-1919 187 19 . . PUNCT iajs-1919 188 1 now now ADV iajs-1919 188 2 , , PUNCT iajs-1919 188 3 we we PRON iajs-1919 188 4 have have AUX iajs-1919 188 5 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 VERB iajs-1919 188 6 𝑓 𝑓 DET iajs-1919 188 7 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 188 8 , , PUNCT iajs-1919 188 9 to to PART iajs-1919 188 10 see see VERB iajs-1919 188 11 this this PRON iajs-1919 188 12 : : PUNCT iajs-1919 188 13 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 NOUN iajs-1919 188 14 𝑥 𝑥 X iajs-1919 188 15 ∈ ∈ PROPN iajs-1919 188 16 𝑀 𝑀 PROPN iajs-1919 189 1 | | ADV iajs-1919 189 2 𝑓 𝑓 DET iajs-1919 189 3 ∘ ∘ X iajs-1919 189 4 𝑔 𝑔 PROPN iajs-1919 189 5 ∘ ∘ NOUN iajs-1919 189 6 𝑓 𝑓 PRON iajs-1919 189 7 𝑥 𝑥 NOUN iajs-1919 189 8 0 0 PUNCT iajs-1919 189 9 𝑥 𝑥 PRON iajs-1919 189 10 ∈ ∈ PROPN iajs-1919 189 11 𝑀 𝑀 NOUN iajs-1919 190 1 | | ADV iajs-1919 190 2 𝑔 𝑔 X iajs-1919 190 3 ∘ ∘ X iajs-1919 190 4 𝑓 𝑓 PRON iajs-1919 190 5 𝑥 𝑥 PRON iajs-1919 190 6 ∈ ∈ ADJ iajs-1919 190 7 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 190 8 0 0 PUNCT iajs-1919 191 1 𝑥 𝑥 PRON iajs-1919 191 2 ∈ ∈ PROPN iajs-1919 191 3 𝑀 𝑀 NOUN iajs-1919 191 4 | | ADV iajs-1919 191 5 𝑓 𝑓 ADV iajs-1919 191 6 𝑥 𝑥 X iajs-1919 191 7 ∈ ∈ ADJ iajs-1919 191 8 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 191 9 𝑥 𝑥 NOUN iajs-1919 191 10 ∈ ∈ NOUN iajs-1919 191 11 𝑀 𝑀 NOUN iajs-1919 192 1 | | ADV iajs-1919 192 2 𝑥 𝑥 X iajs-1919 192 3 ∈ ∈ PROPN iajs-1919 192 4 𝑓 𝑓 DET iajs-1919 192 5 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 192 6 𝑓 𝑓 DET iajs-1919 192 7 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 192 8 . . PUNCT iajs-1919 193 1 by by ADP iajs-1919 193 2 proposition proposition NOUN iajs-1919 193 3 1.1(2 1.1(2 NUM iajs-1919 193 4 ) ) PUNCT iajs-1919 193 5 , , PUNCT iajs-1919 193 6 we we PRON iajs-1919 193 7 get get VERB iajs-1919 193 8 𝑓 𝑓 DET iajs-1919 193 9 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 193 10 ⊴ ⊴ ADP iajs-1919 193 11 𝑀 𝑀 PROPN iajs-1919 193 12 , , PUNCT iajs-1919 193 13 ( ( PUNCT iajs-1919 193 14 since since SCONJ iajs-1919 193 15 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 193 16 ⊴ ⊴ ADP iajs-1919 193 17 𝑀 𝑀 PROPN iajs-1919 193 18 ) ) PUNCT iajs-1919 193 19 , , PUNCT iajs-1919 193 20 this this PRON iajs-1919 193 21 implies imply VERB iajs-1919 193 22 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 VERB iajs-1919 193 23 ⊴ ⊴ ADP iajs-1919 193 24 𝑀 𝑀 PROPN iajs-1919 193 25 and and CCONJ iajs-1919 193 26 hence hence ADV iajs-1919 193 27 𝜓 𝜓 PROPN iajs-1919 193 28 0 0 NUM iajs-1919 193 29 , , PUNCT iajs-1919 193 30 as as SCONJ iajs-1919 193 31 𝑀 𝑀 PROPN iajs-1919 193 32 is be AUX iajs-1919 193 33 strongly strongly ADV iajs-1919 193 34 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 193 35 . . PUNCT iajs-1919 194 1 thus thus ADV iajs-1919 194 2 , , PUNCT iajs-1919 194 3 0 0 NUM iajs-1919 194 4 𝑓 𝑓 DET iajs-1919 194 5 ∘ ∘ PROPN iajs-1919 194 6 𝑔 𝑔 PROPN iajs-1919 194 7 𝐼𝑚𝑓 𝐼𝑚𝑓 PROPN iajs-1919 194 8 𝑓 𝑓 DET iajs-1919 194 9 ∘ ∘ PROPN iajs-1919 194 10 𝑔 𝑔 PROPN iajs-1919 194 11 𝑀 𝑀 PROPN iajs-1919 194 12 , , PUNCT iajs-1919 194 13 thus thus ADV iajs-1919 194 14 𝐼𝑚𝑔 𝐼𝑚𝑔 VERB iajs-1919 194 15 ⊆ ⊆ NUM iajs-1919 194 16 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 194 17 0 0 NUM iajs-1919 194 18 . . PUNCT iajs-1919 195 1 therefore therefore ADV iajs-1919 195 2 𝑔 𝑔 PROPN iajs-1919 195 3 0 0 PROPN iajs-1919 195 4 . . PUNCT iajs-1919 196 1 ∎ ∎ PROPN iajs-1919 196 2 proposition proposition NOUN iajs-1919 196 3 18 18 NUM iajs-1919 196 4 . . PUNCT iajs-1919 197 1 let let VERB iajs-1919 197 2 m m PRON iajs-1919 197 3 be be AUX iajs-1919 197 4 a a DET iajs-1919 197 5 faithful faithful ADJ iajs-1919 197 6 scalar scalar ADJ iajs-1919 197 7 r r NOUN iajs-1919 197 8 - - PUNCT iajs-1919 197 9 module module NOUN iajs-1919 197 10 . . PUNCT iajs-1919 198 1 then then ADV iajs-1919 198 2 r r NOUN iajs-1919 198 3 is be AUX iajs-1919 198 4 strongly strongly ADV iajs-1919 198 5 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 198 6 if if SCONJ iajs-1919 198 7 and and CCONJ iajs-1919 198 8 only only ADV iajs-1919 198 9 if if SCONJ iajs-1919 198 10 𝑆 𝑆 PROPN iajs-1919 198 11 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 198 12 𝑀 𝑀 PROPN iajs-1919 198 13 is be AUX iajs-1919 198 14 strongly strongly ADV iajs-1919 198 15 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 198 16 . . PUNCT iajs-1919 199 1 proof proof NOUN iajs-1919 199 2 . . PUNCT iajs-1919 200 1 since since SCONJ iajs-1919 200 2 m m PROPN iajs-1919 200 3 is be AUX iajs-1919 200 4 a a DET iajs-1919 200 5 scalar scalar ADJ iajs-1919 200 6 r r NOUN iajs-1919 200 7 - - PUNCT iajs-1919 200 8 module module NOUN iajs-1919 200 9 , , PUNCT iajs-1919 200 10 then then ADV iajs-1919 200 11 by by ADP iajs-1919 200 12 [ [ X iajs-1919 200 13 15 15 NUM iajs-1919 200 14 , , PUNCT iajs-1919 200 15 lemma lemma PROPN iajs-1919 200 16 3.6.2 3.6.2 X iajs-1919 200 17 ] ] X iajs-1919 200 18 𝑆 𝑆 PROPN iajs-1919 200 19 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 200 20 𝑀 𝑀 PROPN iajs-1919 200 21 ≅ ≅ PROPN iajs-1919 200 22 𝑅 𝑅 PROPN iajs-1919 200 23 𝑟 𝑟 PROPN iajs-1919 200 24 𝑀⁄ 𝑀⁄ PROPN iajs-1919 200 25 , , PUNCT iajs-1919 200 26 but but CCONJ iajs-1919 200 27 m m NOUN iajs-1919 200 28 is be AUX iajs-1919 200 29 faithful faithful ADJ iajs-1919 200 30 , , PUNCT iajs-1919 200 31 hence hence ADV iajs-1919 200 32 𝑆 𝑆 PROPN iajs-1919 200 33 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 200 34 𝑀 𝑀 PROPN iajs-1919 200 35 ≅ ≅ NOUN iajs-1919 200 36 𝑅. 𝑅. ADV iajs-1919 200 37 by by ADP iajs-1919 200 38 proposition proposition NOUN iajs-1919 200 39 17 17 NUM iajs-1919 200 40 , , PUNCT iajs-1919 200 41 the the DET iajs-1919 200 42 result result NOUN iajs-1919 200 43 is be AUX iajs-1919 200 44 follow follow VERB iajs-1919 200 45 . . PUNCT iajs-1919 201 1 ∎ ∎ PROPN iajs-1919 201 2 proposition proposition NOUN iajs-1919 201 3 19 19 NUM iajs-1919 201 4 . . PUNCT iajs-1919 202 1 let let VERB iajs-1919 202 2 m m PRON iajs-1919 202 3 be be AUX iajs-1919 202 4 a a DET iajs-1919 202 5 faithful faithful ADJ iajs-1919 202 6 multiplication multiplication NOUN iajs-1919 202 7 r r NOUN iajs-1919 202 8 - - NOUN iajs-1919 202 9 module module NOUN iajs-1919 202 10 . . PUNCT iajs-1919 203 1 if if SCONJ iajs-1919 203 2 r r NOUN iajs-1919 203 3 is be AUX iajs-1919 203 4 strongly strongly ADV iajs-1919 203 5 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 203 6 , , PUNCT iajs-1919 203 7 then then ADV iajs-1919 203 8 𝑟 𝑟 X iajs-1919 203 9 𝑁 𝑁 PROPN iajs-1919 203 10 𝑟 𝑟 PRON iajs-1919 203 11 𝑀 𝑀 PROPN iajs-1919 203 12 for for ADP iajs-1919 203 13 all all DET iajs-1919 203 14 𝑁 𝑁 PROPN iajs-1919 203 15 ⊴ ⊴ NOUN iajs-1919 203 16 𝑀. 𝑀. NOUN iajs-1919 203 17 proof proof NOUN iajs-1919 203 18 . . PUNCT iajs-1919 204 1 as as SCONJ iajs-1919 204 2 m m PROPN iajs-1919 204 3 is be AUX iajs-1919 204 4 a a DET iajs-1919 204 5 faithful faithful ADJ iajs-1919 204 6 multiplication multiplication NOUN iajs-1919 204 7 r r NOUN iajs-1919 204 8 - - NOUN iajs-1919 204 9 module module NOUN iajs-1919 204 10 , , PUNCT iajs-1919 204 11 if if SCONJ iajs-1919 204 12 𝑁 𝑁 PROPN iajs-1919 204 13 ⊴ ⊴ NUM iajs-1919 204 14 𝑀 𝑀 PROPN iajs-1919 204 15 , , PUNCT iajs-1919 204 16 there there PRON iajs-1919 204 17 is be VERB iajs-1919 204 18 𝐼 𝐼 PROPN iajs-1919 204 19 ⊴ ⊴ ADP iajs-1919 204 20 𝑅 𝑅 PROPN iajs-1919 204 21 with with ADP iajs-1919 204 22 𝑁 𝑁 PROPN iajs-1919 204 23 𝑀𝐼 𝑀𝐼 PROPN iajs-1919 204 24 , , PUNCT iajs-1919 204 25 by by ADP iajs-1919 204 26 lemma lemma PROPN iajs-1919 204 27 15 15 NUM iajs-1919 204 28 𝑖𝑖 𝑖𝑖 NOUN iajs-1919 204 29 . . PUNCT iajs-1919 205 1 for for ADP iajs-1919 205 2 𝑟 𝑟 DET iajs-1919 205 3 ∈ ∈ NOUN iajs-1919 205 4 𝑟 𝑟 NOUN iajs-1919 205 5 𝑁 𝑁 PROPN iajs-1919 205 6 , , PUNCT iajs-1919 205 7 𝑁𝑟 𝑁𝑟 PROPN iajs-1919 205 8 0 0 NUM iajs-1919 205 9 , , PUNCT iajs-1919 205 10 then then ADV iajs-1919 205 11 𝑀𝐼. 𝑀𝐼. VERB iajs-1919 205 12 𝑟 𝑟 X iajs-1919 205 13 0 0 NUM iajs-1919 205 14 , , PUNCT iajs-1919 205 15 hence hence ADV iajs-1919 205 16 𝐼𝑟 𝐼𝑟 ADP iajs-1919 205 17 ⊆ ⊆ NUM iajs-1919 205 18 𝑟 𝑟 SYM iajs-1919 205 19 𝑀 𝑀 PROPN iajs-1919 205 20 0 0 NUM iajs-1919 205 21 , , PUNCT iajs-1919 205 22 so so SCONJ iajs-1919 205 23 𝑟 𝑟 X iajs-1919 205 24 ∈ ∈ NOUN iajs-1919 205 25 𝑟 𝑟 NOUN iajs-1919 205 26 𝐼 𝐼 PROPN iajs-1919 205 27 implies imply VERB iajs-1919 205 28 𝑟 𝑟 NOUN iajs-1919 205 29 𝑁 𝑁 PROPN iajs-1919 205 30 𝑟 𝑟 NOUN iajs-1919 205 31 𝐼 𝐼 PROPN iajs-1919 205 32 . . PUNCT iajs-1919 206 1 since since SCONJ iajs-1919 206 2 r r NOUN iajs-1919 206 3 is be AUX iajs-1919 206 4 strongly strongly ADV iajs-1919 206 5 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 206 6 with with ADP iajs-1919 206 7 𝐼 𝐼 PROPN iajs-1919 206 8 ⊴ ⊴ PROPN iajs-1919 206 9 𝑅 𝑅 PROPN iajs-1919 206 10 , , PUNCT iajs-1919 206 11 then then ADV iajs-1919 206 12 i i PRON iajs-1919 206 13 is be AUX iajs-1919 206 14 a a DET iajs-1919 206 15 quasi quasi ADJ iajs-1919 206 16 - - ADJ iajs-1919 206 17 invertible invertible ADJ iajs-1919 206 18 ideal ideal NOUN iajs-1919 206 19 ( ( PUNCT iajs-1919 206 20 by by ADP iajs-1919 206 21 theorem theorem NOUN iajs-1919 206 22 2.2 2.2 NUM iajs-1919 206 23 ) ) PUNCT iajs-1919 206 24 , , PUNCT iajs-1919 206 25 so so CCONJ iajs-1919 206 26 𝑟 𝑟 X iajs-1919 206 27 𝐼 𝐼 ADP iajs-1919 206 28 𝑟 𝑟 DET iajs-1919 206 29 𝑅 𝑅 PROPN iajs-1919 206 30 0 0 NUM iajs-1919 206 31 by by ADP iajs-1919 206 32 [ [ X iajs-1919 206 33 7 7 NUM iajs-1919 206 34 , , PUNCT iajs-1919 206 35 prop prop NOUN iajs-1919 206 36 . . PUNCT iajs-1919 207 1 1.1.4 1.1.4 NUM iajs-1919 207 2 ] ] PUNCT iajs-1919 207 3 . . PUNCT iajs-1919 208 1 hence hence ADV iajs-1919 208 2 𝑟 𝑟 DET iajs-1919 208 3 𝑁 𝑁 PROPN iajs-1919 208 4 0 0 NUM iajs-1919 208 5 𝑟 𝑟 PRON iajs-1919 208 6 𝑀 𝑀 PROPN iajs-1919 208 7 . . PUNCT iajs-1919 209 1 ∎ ∎ PROPN iajs-1919 209 2 3 3 X iajs-1919 209 3 . . X iajs-1919 209 4 direct direct ADJ iajs-1919 209 5 summand summand NOUN iajs-1919 209 6 and and CCONJ iajs-1919 209 7 direct direct ADJ iajs-1919 209 8 sums sum NOUN iajs-1919 209 9 we we PRON iajs-1919 209 10 start start VERB iajs-1919 209 11 with with ADP iajs-1919 209 12 following follow VERB iajs-1919 209 13 result result NOUN iajs-1919 209 14 . . PUNCT iajs-1919 210 1 proposition proposition NOUN iajs-1919 210 2 20 20 NUM iajs-1919 210 3 . . PUNCT iajs-1919 211 1 let let VERB iajs-1919 211 2 m m PRON iajs-1919 211 3 be be AUX iajs-1919 211 4 a a DET iajs-1919 211 5 strongly strongly ADV iajs-1919 211 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 211 7 module module NOUN iajs-1919 211 8 , , PUNCT iajs-1919 211 9 and and CCONJ iajs-1919 211 10 𝐴 𝐴 PROPN iajs-1919 211 11 𝑀. 𝑀. PROPN iajs-1919 211 12 if if SCONJ iajs-1919 211 13 𝐴 𝐴 PROPN iajs-1919 211 14 ⊴ ⊴ ADP iajs-1919 211 15 𝐵 𝐵 PROPN iajs-1919 211 16 ⨁ ⨁ PROPN iajs-1919 211 17 𝑀 𝑀 PROPN iajs-1919 211 18 , , PUNCT iajs-1919 211 19 then then ADV iajs-1919 211 20 𝐵 𝐵 PROPN iajs-1919 211 21 𝐵 𝐵 NOUN iajs-1919 211 22 for for ADP iajs-1919 211 23 𝑖 𝑖 DET iajs-1919 211 24 ∈ ∈ PROPN iajs-1919 211 25 1,2 1,2 NUM iajs-1919 211 26 . . PUNCT iajs-1919 212 1 proof proof NOUN iajs-1919 212 2 . . PUNCT iajs-1919 213 1 consider consider VERB iajs-1919 213 2 𝜌 𝜌 X iajs-1919 213 3 : : PUNCT iajs-1919 213 4 𝑀 𝑀 PROPN iajs-1919 213 5 ⟶ ⟶ NOUN iajs-1919 213 6 𝐵 𝐵 PROPN iajs-1919 213 7 is be AUX iajs-1919 213 8 the the DET iajs-1919 213 9 canonical canonical ADJ iajs-1919 213 10 projection projection NOUN iajs-1919 213 11 map map NOUN iajs-1919 213 12 , , PUNCT iajs-1919 213 13 for for ADP iajs-1919 213 14 𝑖 𝑖 ADP iajs-1919 213 15 1,2 1,2 NUM iajs-1919 213 16 . . PUNCT iajs-1919 214 1 we we PRON iajs-1919 214 2 have have VERB iajs-1919 214 3 𝜌 𝜌 ADP iajs-1919 214 4 𝐴 𝐴 PROPN iajs-1919 214 5 𝐴 𝐴 PROPN iajs-1919 214 6 𝜌 𝜌 PROPN iajs-1919 214 7 𝐴 𝐴 PROPN iajs-1919 214 8 . . PUNCT iajs-1919 215 1 since since SCONJ iajs-1919 215 2 1 1 NUM iajs-1919 215 3 𝜌 𝜌 X iajs-1919 215 4 𝜌 𝜌 X iajs-1919 215 5 ∈ ∈ PROPN iajs-1919 215 6 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 215 7 𝑀 𝑀 PROPN iajs-1919 215 8 , , PUNCT iajs-1919 215 9 so so ADV iajs-1919 215 10 we we PRON iajs-1919 215 11 have have VERB iajs-1919 215 12 1 1 NUM iajs-1919 215 13 𝜌 𝜌 PART iajs-1919 215 14 𝜌 𝜌 X iajs-1919 215 15 𝐴 𝐴 PROPN iajs-1919 215 16 1 1 NUM iajs-1919 215 17 𝜌 𝜌 ADP iajs-1919 215 18 𝜌 𝜌 X iajs-1919 215 19 𝐴 𝐴 PROPN iajs-1919 215 20 1 1 NUM iajs-1919 215 21 𝜌 𝜌 ADP iajs-1919 215 22 𝜌 𝜌 X iajs-1919 215 23 𝐴 𝐴 PROPN iajs-1919 215 24 1 1 NUM iajs-1919 215 25 𝜌 𝜌 X iajs-1919 215 26 𝜌 𝜌 X iajs-1919 215 27 𝐴 𝐴 PROPN iajs-1919 215 28 0 0 PUNCT iajs-1919 215 29 ( ( PUNCT iajs-1919 215 30 since since SCONJ iajs-1919 215 31 𝜌 𝜌 PRON iajs-1919 215 32 is be AUX iajs-1919 215 33 an an DET iajs-1919 215 34 idempotent idempotent NOUN iajs-1919 215 35 ) ) PUNCT iajs-1919 215 36 , , PUNCT iajs-1919 215 37 then then ADV iajs-1919 215 38 𝐴 𝐴 PROPN iajs-1919 215 39 ⊆ ⊆ NUM iajs-1919 215 40 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 215 41 1 1 NUM iajs-1919 215 42 𝜌 𝜌 PRON iajs-1919 215 43 𝜌 𝜌 X iajs-1919 215 44 . . PUNCT iajs-1919 216 1 now now ADV iajs-1919 216 2 , , PUNCT iajs-1919 216 3 𝐵 𝐵 PROPN iajs-1919 216 4 ⨁ ⨁ PROPN iajs-1919 216 5 𝑀 𝑀 PROPN iajs-1919 216 6 , , PUNCT iajs-1919 216 7 so so ADV iajs-1919 216 8 𝑀 𝑀 PROPN iajs-1919 216 9 𝐵 𝐵 PROPN iajs-1919 216 10 ⨁𝐵 ⨁𝐵 NOUN iajs-1919 216 11 for for ADP iajs-1919 216 12 some some DET iajs-1919 216 13 𝐵 𝐵 NOUN iajs-1919 216 14 𝑀. 𝑀. PROPN iajs-1919 216 15 hence hence ADV iajs-1919 216 16 1 1 NUM iajs-1919 216 17 𝜌 𝜌 X iajs-1919 216 18 𝜌 𝜌 X iajs-1919 216 19 𝐵 𝐵 NOUN iajs-1919 216 20 1 1 NUM iajs-1919 216 21 𝜌 𝜌 X iajs-1919 216 22 𝜌 𝜌 X iajs-1919 216 23 𝐵 𝐵 NOUN iajs-1919 216 24 1 1 NUM iajs-1919 216 25 𝜌 𝜌 ADP iajs-1919 216 26 0 0 NUM iajs-1919 216 27 0 0 NUM iajs-1919 216 28 , , PUNCT iajs-1919 216 29 thus thus ADV iajs-1919 216 30 𝐵 𝐵 NOUN iajs-1919 216 31 ⊆ ⊆ NUM iajs-1919 216 32 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 216 33 1 1 NUM iajs-1919 216 34 𝜌 𝜌 PRON iajs-1919 216 35 𝜌 𝜌 X iajs-1919 216 36 . . PUNCT iajs-1919 217 1 therefore therefore ADV iajs-1919 217 2 𝐵 𝐵 PROPN iajs-1919 217 3 ⨁𝐴 ⨁𝐴 PROPN iajs-1919 217 4 ⊆ ⊆ NUM iajs-1919 217 5 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 217 6 1 1 NUM iajs-1919 217 7 𝜌 𝜌 PRON iajs-1919 217 8 𝜌 𝜌 X iajs-1919 217 9 . . PUNCT iajs-1919 218 1 on on ADP iajs-1919 218 2 the the DET iajs-1919 218 3 other other ADJ iajs-1919 218 4 hand hand NOUN iajs-1919 218 5 , , PUNCT iajs-1919 218 6 𝐵 𝐵 NOUN iajs-1919 218 7 ⊴ ⊴ ADP iajs-1919 218 8 𝐵 𝐵 PROPN iajs-1919 218 9 and and CCONJ iajs-1919 218 10 𝐴 𝐴 PROPN iajs-1919 218 11 ⊴ ⊴ ADP iajs-1919 218 12 𝐵 𝐵 PROPN iajs-1919 218 13 , , PUNCT iajs-1919 218 14 then then ADV iajs-1919 218 15 𝐵 𝐵 PROPN iajs-1919 218 16 ⨁𝐴 ⨁𝐴 PROPN iajs-1919 218 17 ⊴ ⊴ NUM iajs-1919 218 18 𝐵 𝐵 PROPN iajs-1919 218 19 ⨁𝐵 ⨁𝐵 PROPN iajs-1919 218 20 𝑀 𝑀 PROPN iajs-1919 218 21 by by ADP iajs-1919 218 22 proposition proposition NOUN iajs-1919 218 23 1 1 NUM iajs-1919 218 24 ( ( PUNCT iajs-1919 218 25 3 3 NUM iajs-1919 218 26 ) ) PUNCT iajs-1919 218 27 , , PUNCT iajs-1919 218 28 and and CCONJ iajs-1919 218 29         SPACE iajs-1919 218 30 173 173 NUM iajs-1919 218 31     SPACE iajs-1919 218 32 ibn ibn PROPN iajs-1919 218 33 al al PROPN iajs-1919 218 34 - - PUNCT iajs-1919 218 35 haitham haitham PROPN iajs-1919 218 36 jour.for jour.for PROPN iajs-1919 218 37 pure&appl.sci pure&appl.sci PROPN iajs-1919 218 38 . . PUNCT iajs-1919 218 39 ihjpas ihjpas PROPN iajs-1919 218 40 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 218 41 vol vol NOUN iajs-1919 218 42 . . PUNCT iajs-1919 219 1 32 32 NUM iajs-1919 220 1 ( ( PUNCT iajs-1919 220 2 1 1 NUM iajs-1919 220 3 ) ) PUNCT iajs-1919 220 4 2019 2019 NUM iajs-1919 221 1 so so ADV iajs-1919 221 2 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 221 3 1 1 NUM iajs-1919 221 4 𝜌 𝜌 PRON iajs-1919 221 5 𝜌 𝜌 X iajs-1919 221 6 ⊴ ⊴ ADP iajs-1919 221 7 𝑀 𝑀 PROPN iajs-1919 221 8 which which PRON iajs-1919 221 9 implies imply VERB iajs-1919 221 10 1 1 NUM iajs-1919 221 11 𝜌 𝜌 X iajs-1919 221 12 𝜌 𝜌 X iajs-1919 221 13 0 0 NUM iajs-1919 221 14 , , PUNCT iajs-1919 221 15 as as SCONJ iajs-1919 221 16 m m PROPN iajs-1919 221 17 is be AUX iajs-1919 221 18 strongly strongly ADV iajs-1919 221 19 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 221 20 . . PUNCT iajs-1919 222 1 hence hence ADV iajs-1919 222 2 𝜌 𝜌 X iajs-1919 222 3 𝜌 𝜌 X iajs-1919 222 4 𝜌 𝜌 X iajs-1919 222 5 , , PUNCT iajs-1919 222 6 so so ADV iajs-1919 222 7 𝐵 𝐵 NOUN iajs-1919 222 8 𝜌 𝜌 ADP iajs-1919 222 9 𝐵 𝐵 NOUN iajs-1919 222 10 𝜌 𝜌 X iajs-1919 222 11 𝜌 𝜌 ADP iajs-1919 222 12 𝐵 𝐵 NOUN iajs-1919 222 13 𝜌 𝜌 X iajs-1919 222 14 𝜌 𝜌 ADP iajs-1919 222 15 𝐵 𝐵 NOUN iajs-1919 222 16 𝜌 𝜌 ADP iajs-1919 222 17 𝐵 𝐵 PROPN iajs-1919 222 18 ⊆ ⊆ NUM iajs-1919 222 19 𝐵 𝐵 NOUN iajs-1919 222 20 ⇒ ⇒ VERB iajs-1919 222 21 𝐵 𝐵 PROPN iajs-1919 222 22 ⊆ ⊆ NUM iajs-1919 222 23 𝐵 𝐵 NOUN iajs-1919 222 24 . . PUNCT iajs-1919 223 1 similarly similarly ADV iajs-1919 223 2 , , PUNCT iajs-1919 223 3 taking take VERB iajs-1919 223 4 1 1 NUM iajs-1919 223 5 𝜌 𝜌 X iajs-1919 223 6 𝜌 𝜌 X iajs-1919 223 7 ∈ ∈ PROPN iajs-1919 223 8 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 223 9 𝑀 𝑀 PROPN iajs-1919 223 10 , , PUNCT iajs-1919 223 11 and and CCONJ iajs-1919 223 12 we we PRON iajs-1919 223 13 get get VERB iajs-1919 223 14 𝐵 𝐵 NOUN iajs-1919 223 15 ⊆ ⊆ NUM iajs-1919 223 16 𝐵 𝐵 NOUN iajs-1919 223 17 . . PUNCT iajs-1919 224 1 ∎ ∎ PROPN iajs-1919 224 2 based base VERB iajs-1919 224 3 on on ADP iajs-1919 224 4 our our PRON iajs-1919 224 5 result result NOUN iajs-1919 224 6 , , PUNCT iajs-1919 224 7 we we PRON iajs-1919 224 8 prove prove VERB iajs-1919 224 9 that that SCONJ iajs-1919 224 10 direct direct ADJ iajs-1919 224 11 summands summand NOUN iajs-1919 224 12 of of ADP iajs-1919 224 13 a a DET iajs-1919 224 14 strongly strongly ADV iajs-1919 224 15 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 224 16 module module NOUN iajs-1919 224 17 inherit inherit VERB iajs-1919 224 18 the the DET iajs-1919 224 19 property property NOUN iajs-1919 224 20 . . PUNCT iajs-1919 225 1 proposition proposition NOUN iajs-1919 225 2 21 21 NUM iajs-1919 225 3 . . PUNCT iajs-1919 226 1 a a DET iajs-1919 226 2 direct direct ADJ iajs-1919 226 3 summand summand NOUN iajs-1919 226 4 of of ADP iajs-1919 226 5 a a DET iajs-1919 226 6 strongly strongly ADV iajs-1919 226 7 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 226 8 module module NOUN iajs-1919 226 9 is be AUX iajs-1919 226 10 strongly strongly ADV iajs-1919 226 11 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 226 12 . . PUNCT iajs-1919 227 1 proof proof NOUN iajs-1919 227 2 . . PUNCT iajs-1919 228 1 let let VERB iajs-1919 228 2 𝑀 𝑀 PRON iajs-1919 228 3 be be AUX iajs-1919 228 4 a a DET iajs-1919 228 5 strongly strongly ADV iajs-1919 228 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 228 7 module module NOUN iajs-1919 228 8 , , PUNCT iajs-1919 228 9 and and CCONJ iajs-1919 228 10 𝐴 𝐴 PROPN iajs-1919 228 11 ⨁ ⨁ PROPN iajs-1919 228 12 𝑀 𝑀 PROPN iajs-1919 228 13 , , PUNCT iajs-1919 228 14 so so SCONJ iajs-1919 228 15 𝑀 𝑀 PROPN iajs-1919 228 16 𝐴⨁𝐵 𝐴⨁𝐵 NOUN iajs-1919 228 17 for for ADP iajs-1919 228 18 some some DET iajs-1919 228 19 𝐵 𝐵 NOUN iajs-1919 228 20 𝑀. 𝑀. PROPN iajs-1919 228 21 assume assume VERB iajs-1919 228 22 that that SCONJ iajs-1919 228 23 𝑓 𝑓 DET iajs-1919 228 24 ∈ ∈ PROPN iajs-1919 228 25 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 228 26 𝐴 𝐴 PROPN iajs-1919 228 27 such such ADJ iajs-1919 228 28 that that DET iajs-1919 228 29 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 228 30 ⊴ ⊴ ADP iajs-1919 228 31 𝐴. 𝐴. PROPN iajs-1919 228 32 consider consider VERB iajs-1919 228 33 ℎ ℎ PROPN iajs-1919 228 34 𝑖 𝑖 X iajs-1919 228 35 ∘ ∘ PROPN iajs-1919 228 36 𝑓 𝑓 DET iajs-1919 228 37 ∘ ∘ X iajs-1919 228 38 𝜌 𝜌 ADP iajs-1919 228 39 ∈ ∈ PROPN iajs-1919 228 40 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 228 41 𝑀 𝑀 PROPN iajs-1919 228 42 , , PUNCT iajs-1919 228 43 where where SCONJ iajs-1919 228 44 𝜌 𝜌 PRON iajs-1919 228 45 is be AUX iajs-1919 228 46 the the DET iajs-1919 228 47 canonical canonical ADJ iajs-1919 228 48 projection projection NOUN iajs-1919 228 49 map map NOUN iajs-1919 228 50 onto onto ADP iajs-1919 228 51 𝐴 𝐴 PROPN iajs-1919 228 52 , , PUNCT iajs-1919 228 53 and and CCONJ iajs-1919 228 54 i i PRON iajs-1919 228 55 is be AUX iajs-1919 228 56 the the DET iajs-1919 228 57 inclusion inclusion NOUN iajs-1919 228 58 map map NOUN iajs-1919 228 59 from from ADP iajs-1919 228 60 𝐴 𝐴 PROPN iajs-1919 228 61 to to ADP iajs-1919 228 62 𝑀. 𝑀. PROPN iajs-1919 228 63 so so ADV iajs-1919 228 64 , , PUNCT iajs-1919 228 65 we we PRON iajs-1919 228 66 have have VERB iajs-1919 228 67 𝐾𝑒𝑟ℎ 𝐾𝑒𝑟ℎ PROPN iajs-1919 228 68 𝐾𝑒𝑟𝑓⨁𝐵 𝐾𝑒𝑟𝑓⨁𝐵 PROPN iajs-1919 228 69 , , PUNCT iajs-1919 228 70 to to PART iajs-1919 228 71 see see VERB iajs-1919 228 72 this this PRON iajs-1919 228 73 : : PUNCT iajs-1919 228 74 for for ADP iajs-1919 228 75 𝑥 𝑥 PROPN iajs-1919 228 76 ∈ ∈ PROPN iajs-1919 228 77 𝑘𝑒𝑟ℎ 𝑘𝑒𝑟ℎ NOUN iajs-1919 228 78 , , PUNCT iajs-1919 228 79 𝑥 𝑥 PROPN iajs-1919 228 80 𝑎 𝑎 NOUN iajs-1919 228 81 𝑏 𝑏 NOUN iajs-1919 228 82 where where SCONJ iajs-1919 228 83 𝑎 𝑎 PROPN iajs-1919 228 84 ∈ ∈ PROPN iajs-1919 228 85 𝐴 𝐴 NOUN iajs-1919 228 86 and and CCONJ iajs-1919 228 87 𝑏 𝑏 DET iajs-1919 228 88 ∈ ∈ PROPN iajs-1919 228 89 𝐵 𝐵 NOUN iajs-1919 228 90 with with ADP iajs-1919 228 91 ℎ ℎ X iajs-1919 228 92 𝑥 𝑥 PROPN iajs-1919 228 93 0 0 NUM iajs-1919 228 94 , , PUNCT iajs-1919 228 95 so so ADV iajs-1919 229 1 𝑓 𝑓 DET iajs-1919 229 2 𝑎 𝑎 NOUN iajs-1919 229 3 𝑖 𝑖 NOUN iajs-1919 229 4 ∘ ∘ NOUN iajs-1919 229 5 𝑓 𝑓 PRON iajs-1919 229 6 𝑎 𝑎 NOUN iajs-1919 229 7 𝑖 𝑖 NOUN iajs-1919 229 8 ∘ ∘ NOUN iajs-1919 229 9 𝑓 𝑓 PRON iajs-1919 229 10 𝜌 𝜌 X iajs-1919 229 11 𝑥 𝑥 ADP iajs-1919 229 12 ℎ ℎ PART iajs-1919 229 13 𝑥 𝑥 PROPN iajs-1919 229 14 0 0 NUM iajs-1919 229 15 , , PUNCT iajs-1919 229 16 then then ADV iajs-1919 229 17 𝑎 𝑎 PROPN iajs-1919 229 18 ∈ ∈ ADJ iajs-1919 229 19 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 229 20 , , PUNCT iajs-1919 229 21 and and CCONJ iajs-1919 229 22 hence hence ADV iajs-1919 229 23 𝑥 𝑥 X iajs-1919 229 24 𝑎 𝑎 X iajs-1919 229 25 𝑏 𝑏 NOUN iajs-1919 229 26 ∈ ∈ PROPN iajs-1919 229 27 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 229 28 𝐵 𝐵 PROPN iajs-1919 229 29 , , PUNCT iajs-1919 229 30 that that PRON iajs-1919 229 31 is be AUX iajs-1919 229 32 ; ; PUNCT iajs-1919 229 33 𝑘𝑒𝑟ℎ 𝑘𝑒𝑟ℎ PROPN iajs-1919 229 34 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 229 35 𝐵. 𝐵. PROPN iajs-1919 229 36 on on ADP iajs-1919 229 37 the the DET iajs-1919 229 38 other other ADJ iajs-1919 229 39 hand hand NOUN iajs-1919 229 40 , , PUNCT iajs-1919 229 41 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 229 42 ∩ ∩ PROPN iajs-1919 229 43 𝐵 𝐵 PROPN iajs-1919 229 44 ⊆ ⊆ NUM iajs-1919 229 45 𝐴 𝐴 PROPN iajs-1919 229 46 ∩ ∩ ADJ iajs-1919 229 47 𝐵 𝐵 NOUN iajs-1919 229 48 0 0 NUM iajs-1919 229 49 , , PUNCT iajs-1919 229 50 which which PRON iajs-1919 229 51 implies imply VERB iajs-1919 229 52 𝑘𝑒𝑟ℎ 𝑘𝑒𝑟ℎ NOUN iajs-1919 229 53 𝑘𝑒𝑟𝑓⨁𝐵. 𝑘𝑒𝑟𝑓⨁𝐵. PROPN iajs-1919 229 54 since since SCONJ iajs-1919 229 55 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 229 56 ⊴ ⊴ ADP iajs-1919 229 57 𝐴 𝐴 PROPN iajs-1919 229 58 and and CCONJ iajs-1919 229 59 𝐵 𝐵 PROPN iajs-1919 229 60 ⊴ ⊴ ADP iajs-1919 229 61 𝐵 𝐵 PROPN iajs-1919 229 62 , , PUNCT iajs-1919 229 63 then then ADV iajs-1919 229 64 𝑘𝑒𝑟ℎ 𝑘𝑒𝑟ℎ PROPN iajs-1919 229 65 𝑘𝑒𝑟𝑓⨁𝐵 𝑘𝑒𝑟𝑓⨁𝐵 VERB iajs-1919 229 66 ⊴ ⊴ PROPN iajs-1919 229 67 𝐴⨁𝐵 𝐴⨁𝐵 PROPN iajs-1919 229 68 𝑀 𝑀 PROPN iajs-1919 229 69 by by ADP iajs-1919 229 70 proposition proposition NOUN iajs-1919 229 71 1.1(3 1.1(3 NUM iajs-1919 229 72 ) ) PUNCT iajs-1919 229 73 . . PUNCT iajs-1919 230 1 thus thus ADV iajs-1919 230 2 ℎ ℎ PROPN iajs-1919 230 3 0 0 NUM iajs-1919 230 4 , , PUNCT iajs-1919 230 5 as as ADP iajs-1919 230 6 m m VERB iajs-1919 230 7 strongly strongly ADV iajs-1919 230 8 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 230 9 . . PUNCT iajs-1919 230 10 hence hence ADV iajs-1919 230 11 𝐼𝑚𝑓 𝐼𝑚𝑓 PROPN iajs-1919 230 12 𝑓 𝑓 PROPN iajs-1919 230 13 𝐴 𝐴 PROPN iajs-1919 230 14 𝑖 𝑖 X iajs-1919 230 15 ∘ ∘ X iajs-1919 230 16 𝑓 𝑓 DET iajs-1919 230 17 𝐴 𝐴 PROPN iajs-1919 230 18 𝑖 𝑖 X iajs-1919 230 19 ∘ ∘ NOUN iajs-1919 230 20 𝑓 𝑓 PRON iajs-1919 230 21 𝜌 𝜌 X iajs-1919 230 22 𝑀 𝑀 PROPN iajs-1919 230 23 ℎ ℎ PROPN iajs-1919 230 24 𝑀 𝑀 PROPN iajs-1919 230 25 0 0 NUM iajs-1919 230 26 . . PUNCT iajs-1919 231 1 therfore therfore ADJ iajs-1919 231 2 𝑓 𝑓 DET iajs-1919 231 3 0 0 NUM iajs-1919 231 4 and and CCONJ iajs-1919 231 5 𝐴 𝐴 PROPN iajs-1919 231 6 is be AUX iajs-1919 231 7 strongly strongly ADV iajs-1919 231 8 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 231 9 . . PUNCT iajs-1919 232 1 ∎ ∎ PROPN iajs-1919 232 2 definition definition NOUN iajs-1919 232 3 22 22 NUM iajs-1919 232 4 . . PUNCT iajs-1919 233 1 let let VERB iajs-1919 233 2 m m PRON iajs-1919 233 3 and and CCONJ iajs-1919 233 4 n n CCONJ iajs-1919 233 5 be be VERB iajs-1919 233 6 two two NUM iajs-1919 233 7 r r NOUN iajs-1919 233 8 - - PUNCT iajs-1919 233 9 modules module NOUN iajs-1919 233 10 . . PUNCT iajs-1919 234 1 then then ADV iajs-1919 234 2 m m PROPN iajs-1919 234 3 is be AUX iajs-1919 234 4 called call VERB iajs-1919 234 5 strongly strongly ADV iajs-1919 234 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 234 7 relative relative NOUN iajs-1919 234 8 to to ADP iajs-1919 234 9 n n PRON iajs-1919 234 10 if if SCONJ iajs-1919 234 11 , , PUNCT iajs-1919 234 12 every every DET iajs-1919 234 13 𝜑 𝜑 PROPN iajs-1919 234 14 ∈ ∈ PROPN iajs-1919 234 15 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 234 16 𝑀 𝑀 PROPN iajs-1919 234 17 , , PUNCT iajs-1919 234 18 𝑁 𝑁 PROPN iajs-1919 234 19 such such ADJ iajs-1919 234 20 that that DET iajs-1919 234 21 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 234 22 ⊴ ⊴ PROPN iajs-1919 234 23 𝑀 𝑀 PROPN iajs-1919 234 24 , , PUNCT iajs-1919 234 25 implies imply VERB iajs-1919 234 26 𝜑 𝜑 PROPN iajs-1919 234 27 0 0 X iajs-1919 234 28 . . PUNCT iajs-1919 235 1 obviously obviously ADV iajs-1919 235 2 , , PUNCT iajs-1919 235 3 m m VERB iajs-1919 235 4 is be AUX iajs-1919 235 5 strongly strongly ADV iajs-1919 235 6 𝒦nonsigular 𝒦nonsigular ADJ iajs-1919 235 7 if if SCONJ iajs-1919 236 1 and and CCONJ iajs-1919 236 2 only only ADV iajs-1919 236 3 if if SCONJ iajs-1919 236 4 m m NOUN iajs-1919 236 5 is be AUX iajs-1919 236 6 strongly strongly ADV iajs-1919 236 7 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 236 8 relative relative NOUN iajs-1919 236 9 to to ADP iajs-1919 236 10 m. m. NOUN iajs-1919 236 11 proposition proposition NOUN iajs-1919 236 12 23 23 NUM iajs-1919 236 13 . . PUNCT iajs-1919 237 1 if if SCONJ iajs-1919 237 2 m m NOUN iajs-1919 237 3 is be AUX iajs-1919 237 4 a a DET iajs-1919 237 5 strongly strongly ADV iajs-1919 237 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 237 7 module module NOUN iajs-1919 237 8 . . PUNCT iajs-1919 238 1 for for ADP iajs-1919 238 2 𝑁 𝑁 PROPN iajs-1919 238 3 𝑀 𝑀 PROPN iajs-1919 238 4 , , PUNCT iajs-1919 238 5 m m VERB iajs-1919 238 6 is be AUX iajs-1919 238 7 strongly strongly ADV iajs-1919 238 8 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 238 9 relative relative NOUN iajs-1919 238 10 to to ADP iajs-1919 238 11 n. n. NOUN iajs-1919 238 12 proof proof NOUN iajs-1919 238 13 . . PUNCT iajs-1919 239 1 if if SCONJ iajs-1919 239 2 𝑁 𝑁 PROPN iajs-1919 239 3 𝑀 𝑀 PROPN iajs-1919 239 4 , , PUNCT iajs-1919 239 5 clear clear ADJ iajs-1919 239 6 that that SCONJ iajs-1919 239 7 m m NOUN iajs-1919 239 8 is be AUX iajs-1919 239 9 strongly strongly ADV iajs-1919 239 10 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 239 11 relative relative NOUN iajs-1919 239 12 to to ADP iajs-1919 239 13 n. n. NOUN iajs-1919 239 14 assume assume VERB iajs-1919 239 15 that that SCONJ iajs-1919 239 16 𝑁 𝑁 PROPN iajs-1919 239 17 𝑀 𝑀 PROPN iajs-1919 239 18 , , PUNCT iajs-1919 239 19 if if SCONJ iajs-1919 239 20 𝜓 𝜓 PROPN iajs-1919 239 21 ∈ ∈ PROPN iajs-1919 239 22 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 239 23 𝑀 𝑀 PROPN iajs-1919 239 24 , , PUNCT iajs-1919 239 25 𝑁 𝑁 PROPN iajs-1919 239 26 with with ADP iajs-1919 239 27 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 NOUN iajs-1919 239 28 ⊴ ⊴ ADP iajs-1919 239 29 𝑀. 𝑀. PROPN iajs-1919 239 30 consider consider VERB iajs-1919 239 31 ℎ ℎ PART iajs-1919 239 32 𝑖 𝑖 X iajs-1919 239 33 ∘ ∘ PROPN iajs-1919 239 34 𝜓 𝜓 PROPN iajs-1919 239 35 , , PUNCT iajs-1919 239 36 where where SCONJ iajs-1919 239 37 𝑖 𝑖 PRON iajs-1919 239 38 is be AUX iajs-1919 239 39 the the DET iajs-1919 239 40 inclusion inclusion NOUN iajs-1919 239 41 map map NOUN iajs-1919 239 42 from from ADP iajs-1919 239 43 n n PRON iajs-1919 239 44 to to PART iajs-1919 239 45 m. m. VERB iajs-1919 239 46 so so SCONJ iajs-1919 239 47 ℎ ℎ ADP iajs-1919 239 48 ∈ ∈ PROPN iajs-1919 239 49 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 239 50 𝑀 𝑀 PROPN iajs-1919 239 51 such such ADJ iajs-1919 239 52 that that DET iajs-1919 239 53 𝑘𝑒𝑟ℎ 𝑘𝑒𝑟ℎ PROPN iajs-1919 239 54 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 PROPN iajs-1919 239 55 ⊴ ⊴ PROPN iajs-1919 239 56 𝑀 𝑀 PROPN iajs-1919 239 57 , , PUNCT iajs-1919 239 58 then then ADV iajs-1919 239 59 ℎ ℎ PROPN iajs-1919 239 60 0 0 NUM iajs-1919 239 61 , , PUNCT iajs-1919 239 62 as as SCONJ iajs-1919 239 63 m m PROPN iajs-1919 239 64 is be AUX iajs-1919 239 65 strongly strongly ADV iajs-1919 239 66 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 239 67 , , PUNCT iajs-1919 239 68 hence hence ADV iajs-1919 239 69 𝐼𝑚𝜓 𝐼𝑚𝜓 PROPN iajs-1919 239 70 𝜓 𝜓 PROPN iajs-1919 239 71 𝑀 𝑀 PROPN iajs-1919 239 72 𝑖 𝑖 PROPN iajs-1919 239 73 𝜓 𝜓 PROPN iajs-1919 239 74 𝑀 𝑀 PROPN iajs-1919 239 75 ℎ ℎ ADP iajs-1919 239 76 𝑀 𝑀 PROPN iajs-1919 239 77 0 0 NUM iajs-1919 239 78 , , PUNCT iajs-1919 239 79 thus thus ADV iajs-1919 239 80 𝜓 𝜓 PROPN iajs-1919 239 81 0 0 NUM iajs-1919 239 82 . . PUNCT iajs-1919 240 1 ∎ ∎ PROPN iajs-1919 240 2 lemma lemma PROPN iajs-1919 240 3 24 24 NUM iajs-1919 240 4 . . PUNCT iajs-1919 241 1 for for ADP iajs-1919 241 2 a a DET iajs-1919 241 3 module module NOUN iajs-1919 241 4 m m NOUN iajs-1919 241 5 , , PUNCT iajs-1919 241 6 if if SCONJ iajs-1919 241 7 𝑁 𝑁 PROPN iajs-1919 241 8 ⊴ ⊴ ADP iajs-1919 241 9 𝐾 𝐾 PROPN iajs-1919 241 10 𝑀 𝑀 PROPN iajs-1919 241 11 for for ADP iajs-1919 241 12 𝑖 𝑖 PRON iajs-1919 241 13 ∈∧ ∈∧ NOUN iajs-1919 241 14 1,2 1,2 NUM iajs-1919 241 15 , , PUNCT iajs-1919 241 16 … … PUNCT iajs-1919 241 17 , , PUNCT iajs-1919 241 18 𝑛 𝑛 PROPN iajs-1919 241 19 , , PUNCT iajs-1919 241 20 then then ADV iajs-1919 241 21 ⋂ ⋂ PROPN iajs-1919 241 22 𝑁 𝑁 PROPN iajs-1919 241 23 ⊴ ⊴ ADP iajs-1919 241 24 ⋂ ⋂ PROPN iajs-1919 241 25 𝐾 𝐾 PROPN iajs-1919 241 26 . . PUNCT iajs-1919 242 1 proof proof NOUN iajs-1919 242 2 . . PUNCT iajs-1919 243 1 consider consider VERB iajs-1919 243 2 the the DET iajs-1919 243 3 case case NOUN iajs-1919 243 4 when when SCONJ iajs-1919 243 5 the the DET iajs-1919 243 6 index index NOUN iajs-1919 243 7 set set VERB iajs-1919 243 8 ∧ ∧ PROPN iajs-1919 243 9 1,2 1,2 NUM iajs-1919 243 10 . . PUNCT iajs-1919 244 1 let let VERB iajs-1919 244 2 𝑋 𝑋 NOUN iajs-1919 244 3 ≪ ≪ PUNCT iajs-1919 244 4 𝐾 𝐾 PROPN iajs-1919 244 5 ∩ ∩ ADJ iajs-1919 244 6 𝐾 𝐾 NOUN iajs-1919 244 7 with with ADP iajs-1919 244 8 𝑁 𝑁 PROPN iajs-1919 244 9 ∩ ∩ NOUN iajs-1919 244 10 𝑁 𝑁 PROPN iajs-1919 244 11 ∩ ∩ ADJ iajs-1919 244 12 𝑋 𝑋 NOUN iajs-1919 244 13 0 0 NUM iajs-1919 244 14 , , PUNCT iajs-1919 244 15 then then ADV iajs-1919 244 16 𝑁 𝑁 PROPN iajs-1919 244 17 ∩ ∩ ADJ iajs-1919 244 18 𝑁 𝑁 PROPN iajs-1919 244 19 ∩ ∩ ADJ iajs-1919 244 20 𝑋 𝑋 NOUN iajs-1919 244 21 0 0 NUM iajs-1919 244 22 . . PUNCT iajs-1919 245 1 since since SCONJ iajs-1919 245 2 𝑋 𝑋 NOUN iajs-1919 245 3 ≪ ≪ PUNCT iajs-1919 245 4 𝐾 𝐾 PROPN iajs-1919 245 5 ∩ ∩ ADJ iajs-1919 245 6 𝐾 𝐾 PROPN iajs-1919 245 7 ⊆ ⊆ NUM iajs-1919 245 8 𝐾 𝐾 PROPN iajs-1919 245 9 , , PUNCT iajs-1919 245 10 then then ADV iajs-1919 245 11 𝑋 𝑋 NOUN iajs-1919 245 12 ≪ ≪ PUNCT iajs-1919 245 13 𝐾 𝐾 PROPN iajs-1919 245 14 and and CCONJ iajs-1919 245 15 hence hence ADV iajs-1919 245 16 𝑁 𝑁 PROPN iajs-1919 245 17 ∩ ∩ ADJ iajs-1919 245 18 𝑋 𝑋 NOUN iajs-1919 245 19 ≪ ≪ PUNCT iajs-1919 245 20 𝐾 𝐾 PROPN iajs-1919 245 21 implies imply VERB iajs-1919 245 22 𝑁 𝑁 PROPN iajs-1919 245 23 ∩ ∩ ADJ iajs-1919 245 24 𝑋 𝑋 NOUN iajs-1919 245 25 0 0 NUM iajs-1919 245 26 , , PUNCT iajs-1919 245 27 as as ADP iajs-1919 245 28 𝑁 𝑁 PROPN iajs-1919 245 29 ⊴ ⊴ ADP iajs-1919 245 30 𝐾 𝐾 PROPN iajs-1919 245 31 . . PUNCT iajs-1919 246 1 also also ADV iajs-1919 246 2 , , PUNCT iajs-1919 246 3 𝑋 𝑋 PROPN iajs-1919 246 4 ≪ ≪ PUNCT iajs-1919 246 5 𝐾 𝐾 PROPN iajs-1919 246 6 and and CCONJ iajs-1919 246 7 𝑁 𝑁 PROPN iajs-1919 246 8 ⊴ ⊴ NOUN iajs-1919 246 9 𝐾 𝐾 PROPN iajs-1919 246 10 , , PUNCT iajs-1919 246 11 hence hence ADV iajs-1919 246 12 𝑋 𝑋 PROPN iajs-1919 246 13 0 0 NUM iajs-1919 246 14 . . PUNCT iajs-1919 247 1 thus thus ADV iajs-1919 247 2 𝑁 𝑁 ADJ iajs-1919 247 3 ∩ ∩ NOUN iajs-1919 247 4 𝑁 𝑁 PROPN iajs-1919 247 5 ⊴ ⊴ ADJ iajs-1919 247 6 𝐾 𝐾 PROPN iajs-1919 247 7 ∩ ∩ ADJ iajs-1919 247 8 𝐾 𝐾 PROPN iajs-1919 247 9 .∎ .∎ PROPN iajs-1919 247 10 theorem theorem NOUN iajs-1919 247 11 25 25 NUM iajs-1919 247 12 . . PUNCT iajs-1919 248 1 let let VERB iajs-1919 248 2 𝑀 𝑀 PROPN iajs-1919 248 3 𝑀 𝑀 PROPN iajs-1919 248 4 ⨁𝑀 ⨁𝑀 NOUN iajs-1919 248 5 be be AUX iajs-1919 248 6 an an DET iajs-1919 248 7 r r NOUN iajs-1919 248 8 - - PUNCT iajs-1919 248 9 module module NOUN iajs-1919 248 10 . . PUNCT iajs-1919 249 1 then then ADV iajs-1919 249 2 m m PROPN iajs-1919 249 3 is be AUX iajs-1919 249 4 strongly strongly ADV iajs-1919 249 5 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 249 6 if if SCONJ iajs-1919 249 7 and and CCONJ iajs-1919 249 8 only only ADV iajs-1919 249 9 if if SCONJ iajs-1919 249 10 𝑀 𝑀 PROPN iajs-1919 249 11 is be AUX iajs-1919 249 12 strongly strongly ADV iajs-1919 249 13 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 249 14 relative relative NOUN iajs-1919 249 15 to to ADP iajs-1919 249 16 𝑀 𝑀 PROPN iajs-1919 249 17 , , PUNCT iajs-1919 249 18 for for ADP iajs-1919 249 19 𝑖 𝑖 SYM iajs-1919 249 20 , , PUNCT iajs-1919 249 21 𝑗 𝑗 PROPN iajs-1919 249 22 ∈ ∈ NOUN iajs-1919 249 23 1,2 1,2 NUM iajs-1919 249 24 . . PUNCT iajs-1919 250 1 proof proof NOUN iajs-1919 250 2 . . PUNCT iajs-1919 251 1 assume assume VERB iajs-1919 251 2 𝑀 𝑀 PROPN iajs-1919 251 3 𝑀 𝑀 PROPN iajs-1919 251 4 ⨁𝑀 ⨁𝑀 NOUN iajs-1919 251 5 a a DET iajs-1919 251 6 strongly strongly ADV iajs-1919 251 7 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 251 8 module module NOUN iajs-1919 251 9 . . PUNCT iajs-1919 252 1 by by ADP iajs-1919 252 2 proposition proposition NOUN iajs-1919 252 3 21 21 NUM iajs-1919 252 4 , , PUNCT iajs-1919 252 5 𝑀 𝑀 PROPN iajs-1919 252 6 is be AUX iajs-1919 252 7 strongly strongly ADV iajs-1919 252 8 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 252 9 , , PUNCT iajs-1919 252 10 for for ADP iajs-1919 252 11 𝑖 𝑖 DET iajs-1919 252 12 ∈ ∈ PROPN iajs-1919 252 13 1,2 1,2 NUM iajs-1919 252 14 . . PUNCT iajs-1919 253 1 hence hence ADV iajs-1919 253 2 𝑀 𝑀 PROPN iajs-1919 253 3 is be AUX iajs-1919 253 4 strongly strongly ADV iajs-1919 253 5 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 253 6 relative relative NOUN iajs-1919 253 7 to to ADP iajs-1919 253 8 𝑀 𝑀 PROPN iajs-1919 253 9 , , PUNCT iajs-1919 253 10 for for ADP iajs-1919 253 11 𝑖 𝑖 DET iajs-1919 253 12 ∈ ∈ PROPN iajs-1919 253 13 1,2 1,2 NUM iajs-1919 253 14 . . PUNCT iajs-1919 254 1 now now ADV iajs-1919 254 2 , , PUNCT iajs-1919 254 3 let let VERB iajs-1919 254 4 𝜑 𝜑 PRON iajs-1919 254 5 ∈ ∈ PROPN iajs-1919 254 6 𝐻𝑜𝑚 𝐻𝑜𝑚 PROPN iajs-1919 254 7 𝑀 𝑀 PROPN iajs-1919 254 8 , , PUNCT iajs-1919 254 9 𝑀 𝑀 PROPN iajs-1919 254 10 such such ADJ iajs-1919 254 11 that that SCONJ iajs-1919 254 12 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 254 13 ⊴ ⊴ ADP iajs-1919 254 14 𝑀 𝑀 PROPN iajs-1919 254 15 . . PUNCT iajs-1919 255 1 consider consider VERB iajs-1919 255 2 𝜓 𝜓 PRON iajs-1919 255 3 𝑖 𝑖 X iajs-1919 255 4 ∘ ∘ NOUN iajs-1919 255 5 𝜑 𝜑 X iajs-1919 255 6 ∘ ∘ X iajs-1919 255 7 𝜌 𝜌 ADP iajs-1919 255 8 ∈ ∈ PROPN iajs-1919 255 9 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 255 10 𝑀 𝑀 PROPN iajs-1919 255 11 , , PUNCT iajs-1919 255 12 where where SCONJ iajs-1919 255 13 𝜌 𝜌 PRON iajs-1919 255 14 is be AUX iajs-1919 255 15         SPACE iajs-1919 255 16 174 174 NUM iajs-1919 255 17     SPACE iajs-1919 255 18 ibn ibn PROPN iajs-1919 255 19 al al PROPN iajs-1919 255 20 - - PUNCT iajs-1919 255 21 haitham haitham PROPN iajs-1919 255 22 jour.for jour.for PROPN iajs-1919 255 23 pure&appl.sci pure&appl.sci PROPN iajs-1919 255 24 . . PUNCT iajs-1919 256 1 ihjpas ihjpas PROPN iajs-1919 256 2 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 256 3 vol vol NOUN iajs-1919 256 4 . . PUNCT iajs-1919 256 5 32 32 NUM iajs-1919 256 6 ( ( PUNCT iajs-1919 256 7 1 1 NUM iajs-1919 256 8 ) ) PUNCT iajs-1919 256 9 2019 2019 NUM iajs-1919 256 10 the the DET iajs-1919 256 11 canonical canonical ADJ iajs-1919 256 12 projection projection NOUN iajs-1919 256 13 map map NOUN iajs-1919 256 14 onto onto ADP iajs-1919 256 15 𝑀 𝑀 PROPN iajs-1919 256 16 , , PUNCT iajs-1919 256 17 𝑖 𝑖 PROPN iajs-1919 256 18 : : PUNCT iajs-1919 256 19 𝑀 𝑀 PROPN iajs-1919 256 20 ⟶ ⟶ NOUN iajs-1919 256 21 𝑀 𝑀 PROPN iajs-1919 256 22 is be AUX iajs-1919 256 23 the the DET iajs-1919 256 24 inclusion inclusion NOUN iajs-1919 256 25 map map NOUN iajs-1919 256 26 . . PUNCT iajs-1919 257 1 clearly clearly ADV iajs-1919 257 2 , , PUNCT iajs-1919 257 3 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 PROPN iajs-1919 257 4 𝑘𝑒𝑟𝜑⨁𝑀 𝑘𝑒𝑟𝜑⨁𝑀 X iajs-1919 257 5 , , PUNCT iajs-1919 257 6 so so ADV iajs-1919 257 7 𝑘𝑒𝑟𝜓 𝑘𝑒𝑟𝜓 VERB iajs-1919 257 8 𝑘𝑒𝑟𝜑⨁𝑀 𝑘𝑒𝑟𝜑⨁𝑀 X iajs-1919 257 9 ⊴ ⊴ ADP iajs-1919 257 10 𝑀 𝑀 PROPN iajs-1919 257 11 ⨁𝑀 ⨁𝑀 PROPN iajs-1919 257 12 𝑀 𝑀 PROPN iajs-1919 257 13 , , PUNCT iajs-1919 257 14 hence hence ADV iajs-1919 257 15 𝜓 𝜓 PROPN iajs-1919 257 16 0 0 NUM iajs-1919 257 17 ( ( PUNCT iajs-1919 257 18 since since SCONJ iajs-1919 257 19 m m PROPN iajs-1919 257 20 is be AUX iajs-1919 257 21 strongly strongly ADV iajs-1919 257 22 𝒦nonsigular 𝒦nonsigular ADJ iajs-1919 257 23 ) ) PUNCT iajs-1919 257 24 . . PUNCT iajs-1919 258 1 thus thus ADV iajs-1919 258 2 , , PUNCT iajs-1919 258 3 𝜑 𝜑 PROPN iajs-1919 258 4 0 0 PUNCT iajs-1919 259 1 and and CCONJ iajs-1919 259 2 so so ADV iajs-1919 259 3 𝑀 𝑀 PROPN iajs-1919 259 4 is be AUX iajs-1919 259 5 strongly strongly ADV iajs-1919 259 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 259 7 relative relative NOUN iajs-1919 259 8 to to ADP iajs-1919 259 9 𝑀 𝑀 PROPN iajs-1919 259 10 . . PUNCT iajs-1919 260 1 𝑀 𝑀 PROPN iajs-1919 260 2 is be AUX iajs-1919 260 3 strongly strongly ADV iajs-1919 260 4 𝒦nonsigular 𝒦nonsigular ADJ iajs-1919 260 5 relative relative NOUN iajs-1919 260 6 to to ADP iajs-1919 260 7 𝑀 𝑀 PROPN iajs-1919 260 8 , , PUNCT iajs-1919 260 9 similarly similarly ADV iajs-1919 260 10 . . PUNCT iajs-1919 261 1 conversely conversely ADV iajs-1919 261 2 , , PUNCT iajs-1919 261 3 if if SCONJ iajs-1919 261 4 𝑓 𝑓 PRON iajs-1919 261 5 ∈ ∈ PROPN iajs-1919 261 6 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 261 7 𝑀 𝑀 PROPN iajs-1919 261 8 such such ADJ iajs-1919 261 9 that that DET iajs-1919 261 10 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 261 11 ⊴ ⊴ PROPN iajs-1919 261 12 𝑀 𝑀 PROPN iajs-1919 261 13 , , PUNCT iajs-1919 261 14 so so SCONJ iajs-1919 261 15 we we PRON iajs-1919 261 16 have have VERB iajs-1919 261 17 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 261 18 ∩ ∩ PROPN iajs-1919 261 19 𝑀 𝑀 PROPN iajs-1919 261 20 ⊴ ⊴ ADP iajs-1919 261 21 𝑀 𝑀 PROPN iajs-1919 261 22 , , PUNCT iajs-1919 261 23 by by ADP iajs-1919 261 24 lemma lemma PROPN iajs-1919 261 25 24 24 NUM iajs-1919 261 26 . . PUNCT iajs-1919 262 1 consider consider VERB iajs-1919 262 2 𝑓| 𝑓| PROPN iajs-1919 262 3 : : PUNCT iajs-1919 262 4 𝑀 𝑀 PROPN iajs-1919 262 5 → → SYM iajs-1919 262 6 𝑀 𝑀 PROPN iajs-1919 262 7 which which PRON iajs-1919 262 8 defined define VERB iajs-1919 262 9 by by ADP iajs-1919 262 10 𝑓| 𝑓| PROPN iajs-1919 262 11 𝑥 𝑥 PROPN iajs-1919 262 12 𝑓 𝑓 ADV iajs-1919 262 13 𝑥 𝑥 NOUN iajs-1919 262 14 0 0 NUM iajs-1919 262 15 for for ADP iajs-1919 262 16 all all PRON iajs-1919 262 17 𝑥 𝑥 DET iajs-1919 262 18 ∈ ∈ NOUN iajs-1919 262 19 𝑀. 𝑀. NOUN iajs-1919 262 20 we we PRON iajs-1919 262 21 have have VERB iajs-1919 262 22 𝑘𝑒𝑟 𝑘𝑒𝑟 PROPN iajs-1919 262 23 𝑓| 𝑓| PROPN iajs-1919 262 24 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 262 25 ∩ ∩ X iajs-1919 262 26 𝑀 𝑀 PROPN iajs-1919 262 27 as as SCONJ iajs-1919 262 28 follows follow VERB iajs-1919 262 29 : : PUNCT iajs-1919 262 30 if if SCONJ iajs-1919 262 31 𝑎 𝑎 PROPN iajs-1919 262 32 ∈ ∈ PROPN iajs-1919 262 33 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 262 34 ∩ ∩ PROPN iajs-1919 262 35 𝑀 𝑀 PROPN iajs-1919 262 36 then then ADV iajs-1919 262 37 0 0 NUM iajs-1919 263 1 𝑓 𝑓 PRON iajs-1919 263 2 𝑎 𝑎 X iajs-1919 263 3 𝑓 𝑓 PRON iajs-1919 263 4 𝑎 𝑎 ADJ iajs-1919 263 5 0 0 NUM iajs-1919 263 6 𝑓| 𝑓| PROPN iajs-1919 263 7 𝑎 𝑎 NOUN iajs-1919 263 8 and and CCONJ iajs-1919 263 9 𝑎 𝑎 PROPN iajs-1919 263 10 ∈ ∈ PROPN iajs-1919 263 11 𝑀 𝑀 PROPN iajs-1919 263 12 , , PUNCT iajs-1919 263 13 thus thus ADV iajs-1919 263 14 𝑎 𝑎 X iajs-1919 263 15 ∈ ∈ PROPN iajs-1919 263 16 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 263 17 𝑓| 𝑓| PROPN iajs-1919 263 18 . . PUNCT iajs-1919 264 1 now now ADV iajs-1919 264 2 , , PUNCT iajs-1919 264 3 if if SCONJ iajs-1919 264 4 𝑥 𝑥 DET iajs-1919 264 5 ∈ ∈ PROPN iajs-1919 264 6 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 264 7 𝑓| 𝑓| PROPN iajs-1919 264 8 then then ADV iajs-1919 264 9 0 0 NUM iajs-1919 264 10 𝑓| 𝑓| PROPN iajs-1919 264 11 𝑥 𝑥 PROPN iajs-1919 264 12 𝑓 𝑓 PRON iajs-1919 264 13 𝑥 𝑥 NOUN iajs-1919 264 14 0 0 NUM iajs-1919 265 1 𝑓 𝑓 PRON iajs-1919 265 2 𝑥 𝑥 X iajs-1919 265 3 , , PUNCT iajs-1919 265 4 so so ADV iajs-1919 265 5 𝑥 𝑥 DET iajs-1919 265 6 ∈ ∈ PROPN iajs-1919 265 7 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 265 8 ∩ ∩ PROPN iajs-1919 265 9 𝑀 𝑀 PROPN iajs-1919 265 10 . . PUNCT iajs-1919 266 1 consider consider VERB iajs-1919 266 2 𝑔 𝑔 PRON iajs-1919 266 3 𝜌 𝜌 ADP iajs-1919 266 4 ∘ ∘ X iajs-1919 266 5 𝑓| 𝑓| PROPN iajs-1919 266 6 , , PUNCT iajs-1919 266 7 where where SCONJ iajs-1919 266 8 𝜌 𝜌 PRON iajs-1919 266 9 is be AUX iajs-1919 266 10 the the DET iajs-1919 266 11 canonical canonical ADJ iajs-1919 266 12 projection projection NOUN iajs-1919 266 13 map map NOUN iajs-1919 266 14 onto onto ADP iajs-1919 266 15 𝑀 𝑀 PROPN iajs-1919 266 16 , , PUNCT iajs-1919 266 17 for for ADP iajs-1919 266 18 𝑖 𝑖 DET iajs-1919 266 19 ∈ ∈ PROPN iajs-1919 266 20 1,2 1,2 NUM iajs-1919 266 21 . . PUNCT iajs-1919 267 1 to to PART iajs-1919 267 2 prove prove VERB iajs-1919 267 3 that that DET iajs-1919 267 4 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 267 5 𝑓| 𝑓| PROPN iajs-1919 267 6 ⋂ ⋂ PROPN iajs-1919 267 7 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 267 8 . . PUNCT iajs-1919 268 1 if if SCONJ iajs-1919 268 2 𝑥 𝑥 PRON iajs-1919 268 3 ∈ ∈ PROPN iajs-1919 268 4 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 268 5 𝑓| 𝑓| PROPN iajs-1919 268 6 , , PUNCT iajs-1919 268 7 0 0 NUM iajs-1919 268 8 𝑓| 𝑓| PROPN iajs-1919 268 9 𝑥 𝑥 X iajs-1919 268 10 , , PUNCT iajs-1919 268 11 so so ADV iajs-1919 268 12 𝑔 𝑔 PROPN iajs-1919 268 13 𝑥 𝑥 X iajs-1919 268 14 𝜌 𝜌 X iajs-1919 268 15 ∘ ∘ ADJ iajs-1919 268 16 𝑓| 𝑓| PROPN iajs-1919 268 17 𝑥 𝑥 X iajs-1919 268 18 𝜌 𝜌 X iajs-1919 268 19 𝑓| 𝑓| PROPN iajs-1919 268 20 𝑥 𝑥 X iajs-1919 268 21 𝜌 𝜌 ADP iajs-1919 268 22 0 0 NUM iajs-1919 268 23 0 0 NUM iajs-1919 268 24 , , PUNCT iajs-1919 268 25 this this PRON iajs-1919 268 26 implies imply VERB iajs-1919 268 27 𝑥 𝑥 DET iajs-1919 268 28 ∈ ∈ PROPN iajs-1919 268 29 ⋂ ⋂ PROPN iajs-1919 268 30 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 268 31 . . PUNCT iajs-1919 269 1 now now ADV iajs-1919 269 2 , , PUNCT iajs-1919 269 3 if if SCONJ iajs-1919 269 4 𝑥 𝑥 PROPN iajs-1919 269 5 ∈ ∈ PROPN iajs-1919 269 6 ⋂ ⋂ PROPN iajs-1919 269 7 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 269 8 , , PUNCT iajs-1919 269 9 so so ADV iajs-1919 269 10 𝑔 𝑔 PROPN iajs-1919 269 11 𝑥 𝑥 PART iajs-1919 269 12 0 0 PUNCT iajs-1919 269 13 ⇒ ⇒ NOUN iajs-1919 269 14 𝜌 𝜌 ADP iajs-1919 269 15 𝑓| 𝑓| PROPN iajs-1919 269 16 𝑥 𝑥 PART iajs-1919 269 17 0 0 PUNCT iajs-1919 269 18 ⇒ ⇒ NOUN iajs-1919 269 19 𝑓| 𝑓| PROPN iajs-1919 269 20 𝑥 𝑥 X iajs-1919 269 21 ∈ ∈ PROPN iajs-1919 269 22 ⋂ ⋂ PROPN iajs-1919 269 23 𝑘𝑒𝑟𝜌 𝑘𝑒𝑟𝜌 NOUN iajs-1919 269 24 𝑀 𝑀 PROPN iajs-1919 269 25 ∩ ∩ NOUN iajs-1919 269 26 𝑀 𝑀 PROPN iajs-1919 269 27 0 0 NUM iajs-1919 269 28 ⇒ ⇒ VERB iajs-1919 269 29 𝑥 𝑥 X iajs-1919 269 30 ∈ ∈ PROPN iajs-1919 269 31 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 269 32 𝑓| 𝑓| PROPN iajs-1919 269 33 for for ADP iajs-1919 269 34 𝑖 𝑖 DET iajs-1919 269 35 ∈ ∈ PROPN iajs-1919 269 36 1,2 1,2 NUM iajs-1919 269 37 . . PUNCT iajs-1919 270 1 so so ADV iajs-1919 270 2 ⋂ ⋂ PROPN iajs-1919 270 3 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 270 4 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-1919 270 5 𝑓| 𝑓| PROPN iajs-1919 270 6 𝑘𝑒𝑟𝑓 𝑘𝑒𝑟𝑓 NOUN iajs-1919 270 7 ∩ ∩ PROPN iajs-1919 270 8 𝑀 𝑀 PROPN iajs-1919 270 9 ⊴ ⊴ ADP iajs-1919 270 10 𝑀 𝑀 PROPN iajs-1919 270 11 , , PUNCT iajs-1919 270 12 hence hence ADV iajs-1919 270 13 by by ADP iajs-1919 270 14 proposition proposition NOUN iajs-1919 270 15 1 1 NUM iajs-1919 270 16 , , PUNCT iajs-1919 270 17 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 270 18 ⊴ ⊴ ADP iajs-1919 270 19 𝑀 𝑀 PROPN iajs-1919 270 20 and and CCONJ iajs-1919 270 21 𝑘𝑒𝑟𝑔 𝑘𝑒𝑟𝑔 NOUN iajs-1919 270 22 ⊴ ⊴ ADP iajs-1919 270 23 𝑀 𝑀 PROPN iajs-1919 270 24 . . PUNCT iajs-1919 271 1 by by ADP iajs-1919 271 2 hypothesis hypothesis NOUN iajs-1919 271 3 , , PUNCT iajs-1919 271 4 𝑔 𝑔 PROPN iajs-1919 271 5 0 0 NUM iajs-1919 271 6 ⇒ ⇒ NOUN iajs-1919 271 7 𝜌 𝜌 ADP iajs-1919 271 8 𝐼𝑚 𝐼𝑚 PROPN iajs-1919 271 9 𝑓| 𝑓| PROPN iajs-1919 271 10 0 0 NUM iajs-1919 271 11 ⇒ ⇒ NOUN iajs-1919 271 12 𝐼𝑚𝑓| 𝐼𝑚𝑓| VERB iajs-1919 271 13 ⊆ ⊆ NUM iajs-1919 271 14 ⋂ ⋂ PROPN iajs-1919 271 15 𝑘𝑒𝑟𝜌 𝑘𝑒𝑟𝜌 NOUN iajs-1919 271 16 0 0 NUM iajs-1919 271 17 for for ADP iajs-1919 271 18 𝑖 𝑖 DET iajs-1919 271 19 ∈ ∈ PROPN iajs-1919 271 20 1,2 1,2 NUM iajs-1919 271 21 , , PUNCT iajs-1919 271 22 implies imply VERB iajs-1919 271 23 𝑓| 𝑓| PROPN iajs-1919 271 24 0 0 NUM iajs-1919 271 25 . . PUNCT iajs-1919 272 1 similarly similarly ADV iajs-1919 272 2 , , PUNCT iajs-1919 272 3 we we PRON iajs-1919 272 4 obtain obtain VERB iajs-1919 272 5 ℎ ℎ ADP iajs-1919 272 6 𝜌 𝜌 ADP iajs-1919 272 7 ∘ ∘ ADJ iajs-1919 272 8 𝑓| 𝑓| PROPN iajs-1919 272 9 0 0 NUM iajs-1919 272 10 for for ADP iajs-1919 272 11 𝑖 𝑖 DET iajs-1919 272 12 ∈ ∈ PROPN iajs-1919 272 13 1,2 1,2 NUM iajs-1919 272 14 , , PUNCT iajs-1919 272 15 and and CCONJ iajs-1919 272 16 hence hence ADV iajs-1919 272 17 𝑓| 𝑓| PROPN iajs-1919 272 18 0 0 NUM iajs-1919 272 19 . . PUNCT iajs-1919 273 1 so so ADV iajs-1919 273 2 𝑓| 𝑓| PROPN iajs-1919 273 3 0 0 NUM iajs-1919 273 4 for for ADP iajs-1919 273 5 𝑖 𝑖 DET iajs-1919 273 6 ∈ ∈ PROPN iajs-1919 273 7 1,2 1,2 NUM iajs-1919 273 8 . . PUNCT iajs-1919 274 1 therefore therefore ADV iajs-1919 274 2 𝑓 𝑓 DET iajs-1919 274 3 0 0 NUM iajs-1919 274 4 , , PUNCT iajs-1919 274 5 and and CCONJ iajs-1919 274 6 𝑀 𝑀 PROPN iajs-1919 274 7 𝑀 𝑀 PROPN iajs-1919 274 8 ⨁𝑀 ⨁𝑀 NOUN iajs-1919 274 9 is be AUX iajs-1919 274 10 strongly strongly ADV iajs-1919 274 11 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 274 12 . . PUNCT iajs-1919 275 1 ∎ ∎ PROPN iajs-1919 275 2 corollary corollary NOUN iajs-1919 275 3 26 26 NUM iajs-1919 275 4 . . PUNCT iajs-1919 276 1 if if SCONJ iajs-1919 276 2 𝑀 𝑀 PROPN iajs-1919 276 3 ⊕ ⊕ PROPN iajs-1919 276 4 𝑀 𝑀 PROPN iajs-1919 276 5 . . PUNCT iajs-1919 277 1 then then ADV iajs-1919 277 2 m m PROPN iajs-1919 277 3 is be AUX iajs-1919 277 4 a a DET iajs-1919 277 5 strongly strongly ADV iajs-1919 277 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 277 7 module module NOUN iajs-1919 277 8 if if SCONJ iajs-1919 277 9 and and CCONJ iajs-1919 277 10 only only ADV iajs-1919 277 11 if if SCONJ iajs-1919 277 12 𝑀 𝑀 PROPN iajs-1919 277 13 is be AUX iajs-1919 277 14 strongly strongly ADV iajs-1919 277 15 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 277 16 relative relative NOUN iajs-1919 277 17 to to ADP iajs-1919 277 18 𝑀 𝑀 PROPN iajs-1919 277 19 , , PUNCT iajs-1919 277 20 for for ADP iajs-1919 277 21 𝑖 𝑖 SYM iajs-1919 277 22 , , PUNCT iajs-1919 277 23 𝑗 𝑗 PROPN iajs-1919 277 24 ∈ ∈ NOUN iajs-1919 277 25 1,2 1,2 NUM iajs-1919 277 26 , , PUNCT iajs-1919 277 27 … … PUNCT iajs-1919 277 28 , , PUNCT iajs-1919 277 29 𝑛 𝑛 PROPN iajs-1919 277 30 . . PUNCT iajs-1919 278 1 proposition proposition NOUN iajs-1919 278 2 27 27 NUM iajs-1919 278 3 . . PUNCT iajs-1919 279 1 let let VERB iajs-1919 279 2 𝑀 𝑀 PROPN iajs-1919 279 3 𝑀 𝑀 PROPN iajs-1919 279 4 𝑀 𝑀 PROPN iajs-1919 279 5 be be VERB iajs-1919 279 6 an an DET iajs-1919 279 7 r r NOUN iajs-1919 279 8 - - PUNCT iajs-1919 279 9 module module NOUN iajs-1919 279 10 , , PUNCT iajs-1919 279 11 where where SCONJ iajs-1919 279 12 𝑀 𝑀 PROPN iajs-1919 279 13 , , PUNCT iajs-1919 279 14 𝑀 𝑀 PROPN iajs-1919 279 15 𝑀. 𝑀. PROPN iajs-1919 279 16 if if SCONJ iajs-1919 279 17 ∩ ∩ NOUN iajs-1919 279 18 is be AUX iajs-1919 279 19 a a DET iajs-1919 279 20 strongly strongly ADV iajs-1919 279 21 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 279 22 r r NOUN iajs-1919 279 23 - - PUNCT iajs-1919 279 24 module module NOUN iajs-1919 279 25 , , PUNCT iajs-1919 279 26 then then ADV iajs-1919 279 27 both both PRON iajs-1919 279 28 of of ADP iajs-1919 279 29 and and CCONJ iajs-1919 279 30 is be AUX iajs-1919 279 31 strongly strongly ADV iajs-1919 279 32 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 279 33 . . PUNCT iajs-1919 280 1 proof proof NOUN iajs-1919 280 2 . . PUNCT iajs-1919 281 1 we we PRON iajs-1919 281 2 have have VERB iajs-1919 281 3 ∩ ∩ NOUN iajs-1919 281 4 ∩ ∩ NOUN iajs-1919 281 5 ∩ ∩ NOUN iajs-1919 281 6 ∩ ∩ NOUN iajs-1919 281 7 , , PUNCT iajs-1919 281 8 also also ADV iajs-1919 281 9 ∩ ∩ ADJ iajs-1919 281 10 ∩ ∩ NOUN iajs-1919 281 11 ∩ ∩ NOUN iajs-1919 281 12 ∩ ∩ NOUN iajs-1919 281 13 ∩ ∩ NOUN iajs-1919 281 14 0 0 NUM iajs-1919 281 15 ∩ ∩ NOUN iajs-1919 281 16 , , PUNCT iajs-1919 281 17 thus thus ADV iajs-1919 281 18 ∩ ∩ NOUN iajs-1919 281 19 ∩ ∩ ADJ iajs-1919 281 20 ⨁ ⨁ PROPN iajs-1919 281 21 ∩ ∩ NOUN iajs-1919 281 22 . . PUNCT iajs-1919 282 1 as as SCONJ iajs-1919 282 2 ∩ ∩ NOUN iajs-1919 282 3 is be AUX iajs-1919 282 4 strongly strongly ADV iajs-1919 282 5 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 282 6 , , PUNCT iajs-1919 282 7 so so ADV iajs-1919 282 8 by by ADP iajs-1919 282 9 proposition proposition NOUN iajs-1919 282 10 3.2 3.2 NUM iajs-1919 282 11 , , PUNCT iajs-1919 282 12 ∩ ∩ NOUN iajs-1919 282 13 is be AUX iajs-1919 282 14 strongly strongly ADV iajs-1919 282 15 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 282 16 for for ADP iajs-1919 282 17 𝑖 𝑖 NOUN iajs-1919 282 18 1,2 1,2 NUM iajs-1919 282 19 . . PUNCT iajs-1919 283 1 but but CCONJ iajs-1919 283 2 , , PUNCT iajs-1919 283 3 we we PRON iajs-1919 283 4 have have VERB iajs-1919 283 5 ∩ ∩ ADJ iajs-1919 283 6 ≅ ≅ PROPN iajs-1919 283 7 and and CCONJ iajs-1919 283 8 ∩ ∩ PROPN iajs-1919 283 9 ≅ ≅ PROPN iajs-1919 283 10 , , PUNCT iajs-1919 283 11 so so ADV iajs-1919 283 12 by by ADP iajs-1919 283 13 proposition proposition NOUN iajs-1919 283 14 16 16 NUM iajs-1919 283 15 , , PUNCT iajs-1919 283 16 and and CCONJ iajs-1919 283 17 are be AUX iajs-1919 283 18 strongly strongly ADV iajs-1919 283 19 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 283 20 . . PUNCT iajs-1919 284 1 ∎ ∎ PROPN iajs-1919 284 2 4 4 NUM iajs-1919 284 3 . . PUNCT iajs-1919 284 4 connections connection NOUN iajs-1919 284 5 to to ADP iajs-1919 284 6 other other ADJ iajs-1919 284 7 topics topic NOUN iajs-1919 284 8 in in ADP iajs-1919 284 9 this this DET iajs-1919 284 10 section section NOUN iajs-1919 284 11 , , PUNCT iajs-1919 284 12 we we PRON iajs-1919 284 13 can can AUX iajs-1919 284 14 prove prove VERB iajs-1919 284 15 some some DET iajs-1919 284 16 relations relation NOUN iajs-1919 284 17 between between ADP iajs-1919 284 18 strongly strongly ADV iajs-1919 284 19 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 284 20 modules module NOUN iajs-1919 284 21 and and CCONJ iajs-1919 284 22 other other ADJ iajs-1919 284 23 classes class NOUN iajs-1919 284 24 of of ADP iajs-1919 284 25 modules module NOUN iajs-1919 284 26 , , PUNCT iajs-1919 284 27 such such ADJ iajs-1919 284 28 examples example NOUN iajs-1919 284 29 , , PUNCT iajs-1919 284 30 semisimple semisimple NOUN iajs-1919 284 31 , , PUNCT iajs-1919 284 32 rickart rickart NOUN iajs-1919 284 33 , , PUNCT iajs-1919 284 34 quasi quasi ADJ iajs-1919 284 35 - - ADJ iajs-1919 284 36 dedekind dedekind ADJ iajs-1919 284 37 and and CCONJ iajs-1919 284 38 prime prime ADJ iajs-1919 284 39 modules module NOUN iajs-1919 284 40 . . PUNCT iajs-1919 285 1 example example NOUN iajs-1919 285 2 28 28 NUM iajs-1919 285 3 . . PUNCT iajs-1919 286 1 every every DET iajs-1919 286 2 module module NOUN iajs-1919 286 3 has have VERB iajs-1919 286 4 no no DET iajs-1919 286 5 nonzero nonzero ADJ iajs-1919 286 6 small small ADJ iajs-1919 286 7 submodule submodule NOUN iajs-1919 286 8 , , PUNCT iajs-1919 286 9 all all DET iajs-1919 286 10 its its PRON iajs-1919 286 11 submodules submodule NOUN iajs-1919 286 12 are be AUX iajs-1919 286 13 s s NOUN iajs-1919 286 14 - - ADJ iajs-1919 286 15 essential essential ADJ iajs-1919 286 16 , , PUNCT iajs-1919 286 17 and and CCONJ iajs-1919 286 18 hence hence ADV iajs-1919 286 19 does do AUX iajs-1919 286 20 not not PART iajs-1919 286 21 strongly strongly ADV iajs-1919 286 22 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 286 23 . . PROPN iajs-1919 286 24 notice notice NOUN iajs-1919 286 25 , , PUNCT iajs-1919 286 26 every every DET iajs-1919 286 27 submodule submodule NOUN iajs-1919 286 28 in in ADP iajs-1919 286 29 𝑍 𝑍 PROPN iajs-1919 286 30 is be AUX iajs-1919 286 31 s s NOUN iajs-1919 286 32 - - ADJ iajs-1919 286 33 essential essential ADJ iajs-1919 286 34 , , PUNCT iajs-1919 286 35 because because SCONJ iajs-1919 286 36 the the DET iajs-1919 286 37 zero zero NUM iajs-1919 286 38 is be AUX iajs-1919 286 39 the the DET iajs-1919 286 40 only only ADJ iajs-1919 286 41 small small ADJ iajs-1919 286 42 submodule submodule NOUN iajs-1919 286 43 of of ADP iajs-1919 286 44 𝑍 𝑍 PROPN iajs-1919 286 45 , , PUNCT iajs-1919 286 46 hence hence ADV iajs-1919 286 47 𝑍 𝑍 NOUN iajs-1919 286 48 is be AUX iajs-1919 286 49 not not PART iajs-1919 286 50 strongly strongly ADV iajs-1919 286 51 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 286 52 . . PUNCT iajs-1919 287 1 in in ADP iajs-1919 287 2 particular particular ADJ iajs-1919 287 3 , , PUNCT iajs-1919 287 4 every every DET iajs-1919 287 5 simple simple ADJ iajs-1919 287 6 ( ( PUNCT iajs-1919 287 7 semisimple semisimple NOUN iajs-1919 287 8 ) ) PUNCT iajs-1919 287 9 module module NOUN iajs-1919 287 10 is be AUX iajs-1919 287 11 not not PART iajs-1919 287 12 strongly strongly ADV iajs-1919 287 13 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 287 14 . . PUNCT iajs-1919 288 1 but but CCONJ iajs-1919 288 2 , , PUNCT iajs-1919 288 3 we we PRON iajs-1919 288 4 know know VERB iajs-1919 288 5 every every DET iajs-1919 288 6 semisimple semisimple NOUN iajs-1919 288 7 module module NOUN iajs-1919 288 8 is be AUX iajs-1919 288 9 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 288 10 . . PUNCT iajs-1919 288 11         SPACE iajs-1919 289 1 175 175 NUM iajs-1919 289 2     SPACE iajs-1919 289 3 ibn ibn PROPN iajs-1919 289 4 al al PROPN iajs-1919 289 5 - - PUNCT iajs-1919 289 6 haitham haitham PROPN iajs-1919 289 7 jour.for jour.for PROPN iajs-1919 289 8 pure&appl.sci pure&appl.sci PROPN iajs-1919 289 9 . . PUNCT iajs-1919 289 10 ihjpas ihjpas PROPN iajs-1919 289 11 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 289 12 vol vol NOUN iajs-1919 289 13 . . PUNCT iajs-1919 290 1 32 32 NUM iajs-1919 291 1 ( ( PUNCT iajs-1919 291 2 1 1 NUM iajs-1919 291 3 ) ) PUNCT iajs-1919 291 4 2019 2019 NUM iajs-1919 291 5 remark remark NOUN iajs-1919 291 6 29 29 NUM iajs-1919 291 7 . . PUNCT iajs-1919 292 1 it it PRON iajs-1919 292 2 is be AUX iajs-1919 292 3 clear clear ADJ iajs-1919 292 4 that that SCONJ iajs-1919 292 5 every every DET iajs-1919 292 6 strongly strongly ADV iajs-1919 292 7 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 292 8 module module NOUN iajs-1919 292 9 is be AUX iajs-1919 292 10 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 292 11 , , PUNCT iajs-1919 292 12 but but CCONJ iajs-1919 292 13 the the DET iajs-1919 292 14 converse converse NOUN iajs-1919 292 15 need need AUX iajs-1919 292 16 not not PART iajs-1919 292 17 be be AUX iajs-1919 292 18 true true ADJ iajs-1919 292 19 , , PUNCT iajs-1919 292 20 in in ADP iajs-1919 292 21 general general ADJ iajs-1919 292 22 , , PUNCT iajs-1919 292 23 a a DET iajs-1919 292 24 semisimple semisimple NOUN iajs-1919 292 25 module module NOUN iajs-1919 292 26 is be AUX iajs-1919 292 27 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 292 28 but but CCONJ iajs-1919 292 29 not not PART iajs-1919 292 30 strongly strongly ADV iajs-1919 292 31 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 292 32 . . PUNCT iajs-1919 293 1 lemma lemma PROPN iajs-1919 293 2 30 30 NUM iajs-1919 293 3 . . PUNCT iajs-1919 294 1 let let VERB iajs-1919 294 2 m m PRON iajs-1919 294 3 be be AUX iajs-1919 294 4 a a DET iajs-1919 294 5 hollow hollow ADJ iajs-1919 294 6 ( ( PUNCT iajs-1919 294 7 not not PART iajs-1919 294 8 simple simple ADJ iajs-1919 294 9 ) ) PUNCT iajs-1919 294 10 module module NOUN iajs-1919 294 11 , , PUNCT iajs-1919 294 12 and and CCONJ iajs-1919 294 13 𝐴 𝐴 PROPN iajs-1919 294 14 𝑀. 𝑀. PROPN iajs-1919 294 15 then then ADV iajs-1919 294 16 𝐴 𝐴 PROPN iajs-1919 294 17 is be AUX iajs-1919 294 18 essential essential ADJ iajs-1919 294 19 if if SCONJ iajs-1919 294 20 and and CCONJ iajs-1919 294 21 only only ADV iajs-1919 294 22 if if SCONJ iajs-1919 294 23 𝐴 𝐴 PROPN iajs-1919 294 24 is be AUX iajs-1919 294 25 s s NOUN iajs-1919 294 26 - - ADJ iajs-1919 294 27 essential essential ADJ iajs-1919 294 28 . . PUNCT iajs-1919 295 1 proof proof NOUN iajs-1919 295 2 . . PUNCT iajs-1919 296 1 ⇒ ⇒ PROPN iajs-1919 296 2 clear clear ADJ iajs-1919 296 3 . . PUNCT iajs-1919 297 1 ⇐ ⇐ PROPN iajs-1919 297 2 assume assume VERB iajs-1919 297 3 0 0 NUM iajs-1919 297 4 𝐴 𝐴 PROPN iajs-1919 297 5 ⊴ ⊴ ADP iajs-1919 297 6 𝑀 𝑀 PROPN iajs-1919 297 7 such such ADJ iajs-1919 297 8 that that SCONJ iajs-1919 297 9 𝐴 𝐴 PROPN iajs-1919 297 10 ∩ ∩ ADJ iajs-1919 297 11 𝐵 𝐵 NOUN iajs-1919 297 12 0 0 NUM iajs-1919 297 13 , , PUNCT iajs-1919 297 14 where where SCONJ iajs-1919 297 15 𝐵 𝐵 PROPN iajs-1919 297 16 𝑀. 𝑀. PROPN iajs-1919 297 17 if if SCONJ iajs-1919 297 18 𝐵 𝐵 PROPN iajs-1919 297 19 𝑀 𝑀 PROPN iajs-1919 297 20 , , PUNCT iajs-1919 297 21 then then ADV iajs-1919 297 22 𝐴 𝐴 PROPN iajs-1919 297 23 0 0 NUM iajs-1919 297 24 , , PUNCT iajs-1919 297 25 a a DET iajs-1919 297 26 contradiction contradiction NOUN iajs-1919 297 27 . . PUNCT iajs-1919 298 1 thus thus ADV iajs-1919 298 2 b b X iajs-1919 298 3 is be AUX iajs-1919 298 4 a a DET iajs-1919 298 5 proper proper ADJ iajs-1919 298 6 in in ADP iajs-1919 298 7 m m PROPN iajs-1919 298 8 , , PUNCT iajs-1919 298 9 hence hence ADV iajs-1919 298 10 𝐵 𝐵 NOUN iajs-1919 298 11 ≪ ≪ PUNCT iajs-1919 298 12 𝑀 𝑀 PROPN iajs-1919 298 13 ( ( PUNCT iajs-1919 298 14 since since SCONJ iajs-1919 298 15 m m PROPN iajs-1919 298 16 is be AUX iajs-1919 298 17 hollow hollow ADJ iajs-1919 298 18 ) ) PUNCT iajs-1919 298 19 , , PUNCT iajs-1919 298 20 and and CCONJ iajs-1919 298 21 so so ADV iajs-1919 298 22 𝐵 𝐵 NOUN iajs-1919 298 23 0 0 NUM iajs-1919 298 24 , , PUNCT iajs-1919 298 25 as as SCONJ iajs-1919 298 26 𝐴 𝐴 PROPN iajs-1919 298 27 ⊴ ⊴ ADP iajs-1919 298 28 𝑀. 𝑀. PROPN iajs-1919 298 29 therfore therfore ADJ iajs-1919 298 30 𝐴 𝐴 PROPN iajs-1919 298 31 ⊴ ⊴ ADP iajs-1919 298 32 𝑀. 𝑀. PROPN iajs-1919 298 33 ∎ ∎ PROPN iajs-1919 298 34 however however ADV iajs-1919 298 35 , , PUNCT iajs-1919 298 36 we we PRON iajs-1919 298 37 consider consider VERB iajs-1919 298 38 the the DET iajs-1919 298 39 following follow VERB iajs-1919 298 40 proposition proposition NOUN iajs-1919 298 41 by by ADP iajs-1919 298 42 lemma lemma PROPN iajs-1919 298 43 30 30 NUM iajs-1919 298 44 . . PUNCT iajs-1919 299 1 proposition proposition NOUN iajs-1919 299 2 31 31 NUM iajs-1919 299 3 . . PUNCT iajs-1919 300 1 let let VERB iajs-1919 300 2 m m PRON iajs-1919 300 3 be be AUX iajs-1919 300 4 a a DET iajs-1919 300 5 hollow hollow ADJ iajs-1919 300 6 ( ( PUNCT iajs-1919 300 7 not not PART iajs-1919 300 8 simple simple ADJ iajs-1919 300 9 ) ) PUNCT iajs-1919 300 10 module module NOUN iajs-1919 300 11 . . PUNCT iajs-1919 301 1 then then ADV iajs-1919 301 2 m m PROPN iajs-1919 301 3 is be AUX iajs-1919 301 4 strongly strongly ADV iajs-1919 301 5 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 301 6 if if SCONJ iajs-1919 301 7 and and CCONJ iajs-1919 301 8 only only ADV iajs-1919 301 9 if if SCONJ iajs-1919 301 10 m m NOUN iajs-1919 301 11 is be AUX iajs-1919 301 12 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 301 13 . . PUNCT iajs-1919 302 1 an an DET iajs-1919 302 2 r r NOUN iajs-1919 302 3 - - PUNCT iajs-1919 302 4 module module NOUN iajs-1919 302 5 m m NOUN iajs-1919 302 6 is be AUX iajs-1919 302 7 said say VERB iajs-1919 302 8 to to PART iajs-1919 302 9 be be AUX iajs-1919 302 10 rickart rickart NOUN iajs-1919 302 11 if if SCONJ iajs-1919 302 12 𝑟 𝑟 NOUN iajs-1919 302 13 𝜑 𝜑 X iajs-1919 302 14 𝐾𝑒𝑟𝜑 𝐾𝑒𝑟𝜑 PROPN iajs-1919 302 15 is be AUX iajs-1919 302 16 a a DET iajs-1919 302 17 direct direct ADJ iajs-1919 302 18 summand summand NOUN iajs-1919 302 19 of of ADP iajs-1919 302 20 m m PROPN iajs-1919 302 21 for for ADP iajs-1919 302 22 each each PRON iajs-1919 302 23 𝜑 𝜑 PRON iajs-1919 302 24 ∈ ∈ PROPN iajs-1919 302 25 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 302 26 𝑀 𝑀 PROPN iajs-1919 303 1 [ [ X iajs-1919 303 2 16 16 NUM iajs-1919 303 3 ] ] PUNCT iajs-1919 303 4 . . PUNCT iajs-1919 304 1 recall recall VERB iajs-1919 304 2 that that SCONJ iajs-1919 304 3 an an DET iajs-1919 304 4 r r NOUN iajs-1919 304 5 - - PUNCT iajs-1919 304 6 module module NOUN iajs-1919 304 7 m m NOUN iajs-1919 304 8 is be AUX iajs-1919 304 9 quasi quasi ADJ iajs-1919 304 10 - - ADJ iajs-1919 304 11 dedekind dedekind ADJ iajs-1919 304 12 if if SCONJ iajs-1919 304 13 , , PUNCT iajs-1919 304 14 for for ADP iajs-1919 304 15 any any PRON iajs-1919 304 16 0 0 NUM iajs-1919 304 17 𝜑 𝜑 NOUN iajs-1919 304 18 ∈ ∈ PROPN iajs-1919 304 19 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 304 20 𝑀 𝑀 PROPN iajs-1919 304 21 , , PUNCT iajs-1919 304 22 is be AUX iajs-1919 304 23 a a DET iajs-1919 304 24 monomorphism monomorphism NOUN iajs-1919 304 25 ( ( PUNCT iajs-1919 304 26 i.e. i.e. X iajs-1919 304 27 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 304 28 0 0 NUM iajs-1919 304 29 ) ) PUNCT iajs-1919 305 1 [ [ X iajs-1919 305 2 7 7 NUM iajs-1919 305 3 ] ] PUNCT iajs-1919 305 4 . . PUNCT iajs-1919 306 1 obviously obviously ADV iajs-1919 306 2 , , PUNCT iajs-1919 306 3 rickart rickart NOUN iajs-1919 306 4 , , PUNCT iajs-1919 306 5 quasi quasi ADJ iajs-1919 306 6 - - ADJ iajs-1919 306 7 dedekind dedekind ADJ iajs-1919 306 8 modules module NOUN iajs-1919 306 9 are be AUX iajs-1919 306 10 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 306 11 . . PUNCT iajs-1919 307 1 note note VERB iajs-1919 307 2 that that SCONJ iajs-1919 307 3 the the DET iajs-1919 307 4 z z NOUN iajs-1919 307 5 - - PUNCT iajs-1919 307 6 module module NOUN iajs-1919 307 7 𝑍 𝑍 NOUN iajs-1919 307 8 is be AUX iajs-1919 307 9 semisimple semisimple NOUN iajs-1919 307 10 , , PUNCT iajs-1919 307 11 so so SCONJ iajs-1919 307 12 it it PRON iajs-1919 307 13 is be AUX iajs-1919 307 14 rickart rickart NOUN iajs-1919 307 15 , , PUNCT iajs-1919 307 16 but but CCONJ iajs-1919 307 17 not not PART iajs-1919 307 18 strongly strongly ADV iajs-1919 307 19 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 307 20 . . PUNCT iajs-1919 308 1 also also ADV iajs-1919 308 2 we we PRON iajs-1919 308 3 know know VERB iajs-1919 308 4 𝑍 𝑍 NOUN iajs-1919 308 5 is be AUX iajs-1919 308 6 quasi quasi ADJ iajs-1919 308 7 - - ADJ iajs-1919 308 8 dedekind dedekind ADJ iajs-1919 308 9 , , PUNCT iajs-1919 308 10 but but CCONJ iajs-1919 308 11 it it PRON iajs-1919 308 12 is be AUX iajs-1919 308 13 not not PART iajs-1919 308 14 strongly strongly ADV iajs-1919 308 15 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 308 16 . . PUNCT iajs-1919 309 1 however however ADV iajs-1919 309 2 , , PUNCT iajs-1919 309 3 we we PRON iajs-1919 309 4 have have VERB iajs-1919 309 5 the the DET iajs-1919 309 6 following follow VERB iajs-1919 309 7 corollary corollary NOUN iajs-1919 309 8 which which PRON iajs-1919 309 9 follows follow VERB iajs-1919 309 10 by by ADP iajs-1919 309 11 proposition proposition NOUN iajs-1919 309 12 4.4 4.4 NUM iajs-1919 309 13 . . PUNCT iajs-1919 310 1 corollary corollary ADJ iajs-1919 310 2 32 32 NUM iajs-1919 310 3 . . PUNCT iajs-1919 311 1 for for ADP iajs-1919 311 2 a a DET iajs-1919 311 3 hollow hollow ADJ iajs-1919 311 4 ( ( PUNCT iajs-1919 311 5 not not PART iajs-1919 311 6 simple simple ADJ iajs-1919 311 7 ) ) PUNCT iajs-1919 311 8 module module NOUN iajs-1919 311 9 m. m. NOUN iajs-1919 311 10 if if SCONJ iajs-1919 311 11 m m NOUN iajs-1919 311 12 is be AUX iajs-1919 311 13 rickart rickart NOUN iajs-1919 311 14 ( ( PUNCT iajs-1919 311 15 or or CCONJ iajs-1919 311 16 quasi quasi ADJ iajs-1919 311 17 - - ADJ iajs-1919 311 18 dedekind dedekind ADJ iajs-1919 311 19 ) ) PUNCT iajs-1919 311 20 , , PUNCT iajs-1919 311 21 then then ADV iajs-1919 311 22 m m PROPN iajs-1919 311 23 is be AUX iajs-1919 311 24 strongly strongly ADV iajs-1919 311 25 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 311 26 . . PUNCT iajs-1919 312 1 lemma lemma PROPN iajs-1919 312 2 33 33 NUM iajs-1919 312 3 . . PUNCT iajs-1919 313 1 let let VERB iajs-1919 313 2 m m PRON iajs-1919 313 3 be be AUX iajs-1919 313 4 an an DET iajs-1919 313 5 r r NOUN iajs-1919 313 6 - - PUNCT iajs-1919 313 7 module module NOUN iajs-1919 313 8 . . PUNCT iajs-1919 314 1 if if SCONJ iajs-1919 314 2 𝑆 𝑆 PROPN iajs-1919 314 3 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 314 4 𝑀 𝑀 PROPN iajs-1919 314 5 is be AUX iajs-1919 314 6 a a DET iajs-1919 314 7 regular regular ADJ iajs-1919 314 8 ring ring NOUN iajs-1919 314 9 , , PUNCT iajs-1919 314 10 then then ADV iajs-1919 314 11 m m VERB iajs-1919 314 12 is be AUX iajs-1919 314 13 rickart rickart NOUN iajs-1919 314 14 . . PUNCT iajs-1919 315 1 proof proof NOUN iajs-1919 315 2 . . PUNCT iajs-1919 316 1 assume assume VERB iajs-1919 316 2 𝜑 𝜑 X iajs-1919 316 3 ∈ ∈ PROPN iajs-1919 316 4 𝑆 𝑆 PROPN iajs-1919 316 5 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 316 6 𝑀 𝑀 PROPN iajs-1919 316 7 . . PUNCT iajs-1919 317 1 since since SCONJ iajs-1919 317 2 𝑆 𝑆 PROPN iajs-1919 317 3 is be AUX iajs-1919 317 4 a a DET iajs-1919 317 5 regular regular ADJ iajs-1919 317 6 ring ring NOUN iajs-1919 317 7 , , PUNCT iajs-1919 317 8 so so ADV iajs-1919 317 9 𝜑 𝜑 PROPN iajs-1919 317 10 a a DET iajs-1919 317 11 regular regular ADJ iajs-1919 317 12 element element NOUN iajs-1919 317 13 , , PUNCT iajs-1919 317 14 thus thus ADV iajs-1919 317 15 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 ADJ iajs-1919 317 16 ⨁ ⨁ PROPN iajs-1919 317 17 𝑀 𝑀 PROPN iajs-1919 317 18 , , PUNCT iajs-1919 317 19 by by ADP iajs-1919 317 20 [ [ X iajs-1919 317 21 17 17 NUM iajs-1919 317 22 , , PUNCT iajs-1919 317 23 cor cor NOUN iajs-1919 317 24 . . PROPN iajs-1919 317 25 3.2 3.2 NUM iajs-1919 317 26 ] ] PUNCT iajs-1919 317 27 . . PUNCT iajs-1919 318 1 hence hence ADV iajs-1919 318 2 m m PROPN iajs-1919 318 3 is be AUX iajs-1919 318 4 a a DET iajs-1919 318 5 rickart rickart NOUN iajs-1919 318 6 module module NOUN iajs-1919 318 7 . . PUNCT iajs-1919 319 1 ∎ ∎ PROPN iajs-1919 319 2 corollary corollary NOUN iajs-1919 319 3 34 34 NUM iajs-1919 319 4 . . PUNCT iajs-1919 320 1 if if SCONJ iajs-1919 320 2 m m NOUN iajs-1919 320 3 is be AUX iajs-1919 320 4 a a DET iajs-1919 320 5 hollow hollow ADJ iajs-1919 320 6 ( ( PUNCT iajs-1919 320 7 not not PART iajs-1919 320 8 simple simple ADJ iajs-1919 320 9 ) ) PUNCT iajs-1919 320 10 r r NOUN iajs-1919 320 11 - - PUNCT iajs-1919 320 12 module module NOUN iajs-1919 320 13 with with ADP iajs-1919 320 14 𝑆 𝑆 PROPN iajs-1919 320 15 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 320 16 𝑀 𝑀 PROPN iajs-1919 320 17 is be AUX iajs-1919 320 18 a a DET iajs-1919 320 19 regular regular ADJ iajs-1919 320 20 ring ring NOUN iajs-1919 320 21 , , PUNCT iajs-1919 320 22 then then ADV iajs-1919 320 23 m m NOUN iajs-1919 320 24 is be AUX iajs-1919 320 25 strongly strongly ADV iajs-1919 320 26 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 320 27 . . PUNCT iajs-1919 321 1 proof proof NOUN iajs-1919 321 2 . . PUNCT iajs-1919 322 1 it it PRON iajs-1919 322 2 follows follow VERB iajs-1919 322 3 directly directly ADV iajs-1919 322 4 by by ADP iajs-1919 322 5 lemma lemma PROPN iajs-1919 322 6 33 33 NUM iajs-1919 322 7 and and CCONJ iajs-1919 322 8 corollary corollary ADJ iajs-1919 322 9 34 34 NUM iajs-1919 322 10 . . PUNCT iajs-1919 323 1 ∎ ∎ PROPN iajs-1919 323 2 lemma lemma PROPN iajs-1919 323 3 35 35 NUM iajs-1919 323 4 . . PUNCT iajs-1919 324 1 if if SCONJ iajs-1919 324 2 m m NOUN iajs-1919 324 3 is be AUX iajs-1919 324 4 a a DET iajs-1919 324 5 uniform uniform ADJ iajs-1919 324 6 module module NOUN iajs-1919 324 7 has have VERB iajs-1919 324 8 nonzero nonzero ADJ iajs-1919 324 9 small small ADJ iajs-1919 324 10 submodule submodule NOUN iajs-1919 324 11 , , PUNCT iajs-1919 324 12 then then ADV iajs-1919 324 13 s s NOUN iajs-1919 324 14 - - ADJ iajs-1919 324 15 essential essential ADJ iajs-1919 324 16 submodule submodule NOUN iajs-1919 324 17 implies imply VERB iajs-1919 324 18 essential essential ADJ iajs-1919 324 19 . . PUNCT iajs-1919 325 1 proof proof NOUN iajs-1919 325 2 . . PUNCT iajs-1919 326 1 assume assume VERB iajs-1919 326 2 𝑋 𝑋 PROPN iajs-1919 326 3 𝑀. 𝑀. PROPN iajs-1919 326 4 put put VERB iajs-1919 326 5 𝑋 𝑋 NOUN iajs-1919 326 6 0 0 NUM iajs-1919 326 7 . . PUNCT iajs-1919 327 1 let let VERB iajs-1919 327 2 n n PRON iajs-1919 327 3 be be AUX iajs-1919 327 4 a a DET iajs-1919 327 5 nonzero nonzero ADJ iajs-1919 327 6 small small ADJ iajs-1919 327 7 submodule submodule NOUN iajs-1919 327 8 of of ADP iajs-1919 327 9 m m PROPN iajs-1919 327 10 , , PUNCT iajs-1919 327 11 then then ADV iajs-1919 327 12 𝑋 𝑋 NOUN iajs-1919 327 13 ∩ ∩ NOUN iajs-1919 327 14 𝑁 𝑁 PROPN iajs-1919 327 15 0 0 NUM iajs-1919 327 16 which which PRON iajs-1919 327 17 implies imply VERB iajs-1919 327 18 𝑋 𝑋 PROPN iajs-1919 327 19 ⋬ ⋬ NOUN iajs-1919 327 20 𝑀. 𝑀. NOUN iajs-1919 327 21 hence hence ADV iajs-1919 327 22 the the DET iajs-1919 327 23 result result NOUN iajs-1919 327 24 is be AUX iajs-1919 327 25 obtained obtain VERB iajs-1919 327 26 . . PUNCT iajs-1919 328 1 ∎ ∎ PROPN iajs-1919 328 2 note note VERB iajs-1919 328 3 that that SCONJ iajs-1919 328 4 z z NOUN iajs-1919 328 5 - - PUNCT iajs-1919 328 6 module module NOUN iajs-1919 328 7 z z NOUN iajs-1919 328 8 is be AUX iajs-1919 328 9 uniform uniform ADJ iajs-1919 328 10 , , PUNCT iajs-1919 328 11 the the DET iajs-1919 328 12 zero zero NUM iajs-1919 328 13 submodule submodule NOUN iajs-1919 328 14 of of ADP iajs-1919 328 15 𝑍 𝑍 PROPN iajs-1919 328 16 is be AUX iajs-1919 328 17 s s NOUN iajs-1919 328 18 - - ADJ iajs-1919 328 19 essential essential ADJ iajs-1919 328 20 but but CCONJ iajs-1919 328 21 not not PART iajs-1919 328 22 essential essential ADJ iajs-1919 328 23 ( ( PUNCT iajs-1919 328 24 in in ADP iajs-1919 328 25 fact fact NOUN iajs-1919 328 26 , , PUNCT iajs-1919 328 27 0 0 NUM iajs-1919 328 28 is be AUX iajs-1919 328 29 the the DET iajs-1919 328 30 only only ADJ iajs-1919 328 31 small small ADJ iajs-1919 328 32 submodule submodule NOUN iajs-1919 328 33 of of ADP iajs-1919 328 34 𝑍 𝑍 PROPN iajs-1919 328 35 ) ) PUNCT iajs-1919 328 36 . . PUNCT iajs-1919 329 1 however however ADV iajs-1919 329 2 , , PUNCT iajs-1919 329 3 we we PRON iajs-1919 329 4 have have VERB iajs-1919 329 5 the the DET iajs-1919 329 6 following following NOUN iajs-1919 329 7 . . PUNCT iajs-1919 329 8         SPACE iajs-1919 330 1 176 176 NUM iajs-1919 330 2     SPACE iajs-1919 330 3 ibn ibn PROPN iajs-1919 330 4 al al PROPN iajs-1919 330 5 - - PUNCT iajs-1919 330 6 haitham haitham PROPN iajs-1919 330 7 jour.for jour.for PROPN iajs-1919 330 8 pure&appl.sci pure&appl.sci PROPN iajs-1919 330 9 . . PUNCT iajs-1919 330 10 ihjpas ihjpas PROPN iajs-1919 330 11 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 330 12 vol vol NOUN iajs-1919 330 13 . . PUNCT iajs-1919 331 1 32 32 NUM iajs-1919 332 1 ( ( PUNCT iajs-1919 332 2 1 1 NUM iajs-1919 332 3 ) ) PUNCT iajs-1919 332 4 2019 2019 NUM iajs-1919 332 5 proposition proposition NOUN iajs-1919 332 6 36 36 NUM iajs-1919 332 7 . . PUNCT iajs-1919 333 1 let let VERB iajs-1919 333 2 m m PRON iajs-1919 333 3 be be AUX iajs-1919 333 4 a a DET iajs-1919 333 5 uniform uniform ADJ iajs-1919 333 6 module module NOUN iajs-1919 333 7 has have VERB iajs-1919 333 8 nonzero nonzero ADJ iajs-1919 333 9 small small ADJ iajs-1919 333 10 submodule submodule NOUN iajs-1919 333 11 . . PUNCT iajs-1919 334 1 then then ADV iajs-1919 334 2 m m PROPN iajs-1919 334 3 is be AUX iajs-1919 334 4 strongly strongly ADV iajs-1919 334 5 𝒦nonsigular 𝒦nonsigular ADJ iajs-1919 334 6 if if SCONJ iajs-1919 335 1 and and CCONJ iajs-1919 335 2 only only ADV iajs-1919 335 3 if if SCONJ iajs-1919 335 4 m m NOUN iajs-1919 335 5 is be AUX iajs-1919 335 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 335 7 . . PUNCT iajs-1919 336 1 proof proof NOUN iajs-1919 336 2 . . PUNCT iajs-1919 337 1 it it PRON iajs-1919 337 2 follows follow VERB iajs-1919 337 3 by by ADP iajs-1919 337 4 lemma lemma PROPN iajs-1919 337 5 35 35 NUM iajs-1919 337 6 . . PUNCT iajs-1919 338 1 ∎ ∎ PROPN iajs-1919 338 2 recall recall VERB iajs-1919 338 3 [ [ X iajs-1919 338 4 18 18 NUM iajs-1919 338 5 ] ] PUNCT iajs-1919 338 6 , , PUNCT iajs-1919 338 7 a a DET iajs-1919 338 8 module module NOUN iajs-1919 338 9 m m VERB iajs-1919 338 10 is be AUX iajs-1919 338 11 called call VERB iajs-1919 338 12 prime prime ADJ iajs-1919 338 13 if if SCONJ iajs-1919 338 14 for for ADP iajs-1919 338 15 all all DET iajs-1919 338 16 nonzero nonzero PROPN iajs-1919 338 17 submodule submodule PROPN iajs-1919 338 18 n n PROPN iajs-1919 338 19 of of ADP iajs-1919 338 20 m m PROPN iajs-1919 338 21 , , PUNCT iajs-1919 338 22 𝑟 𝑟 X iajs-1919 338 23 𝑁 𝑁 PROPN iajs-1919 338 24 𝑟 𝑟 PRON iajs-1919 338 25 𝑀 𝑀 PROPN iajs-1919 338 26 . . PUNCT iajs-1919 339 1 mijbass mijbass VERB iajs-1919 339 2 in in ADP iajs-1919 339 3 [ [ X iajs-1919 339 4 7 7 NUM iajs-1919 339 5 , , PUNCT iajs-1919 339 6 th th X iajs-1919 339 7 . . PUNCT iajs-1919 339 8 2.3.14 2.3.14 NUM iajs-1919 339 9 ] ] PUNCT iajs-1919 339 10 , , PUNCT iajs-1919 339 11 presented present VERB iajs-1919 339 12 the the DET iajs-1919 339 13 following follow VERB iajs-1919 339 14 theorem theorem PROPN iajs-1919 339 15 . . PUNCT iajs-1919 339 16 theorem theorem VERB iajs-1919 339 17 37 37 NUM iajs-1919 339 18 . . PUNCT iajs-1919 340 1 a a DET iajs-1919 340 2 module module NOUN iajs-1919 340 3 𝑀 𝑀 PROPN iajs-1919 340 4 is be AUX iajs-1919 340 5 uniform uniform ADJ iajs-1919 340 6 quasi quasi ADJ iajs-1919 340 7 - - NOUN iajs-1919 340 8 dedekind dedekind ADJ iajs-1919 340 9 if if SCONJ iajs-1919 340 10 and and CCONJ iajs-1919 340 11 only only ADV iajs-1919 340 12 if if SCONJ iajs-1919 340 13 it it PRON iajs-1919 340 14 is be AUX iajs-1919 340 15 uniform uniform ADJ iajs-1919 340 16 prime prime ADJ iajs-1919 340 17 . . PUNCT iajs-1919 341 1 proposition proposition NOUN iajs-1919 341 2 38 38 NUM iajs-1919 341 3 . . PUNCT iajs-1919 342 1 let let VERB iajs-1919 342 2 𝑀 𝑀 PRON iajs-1919 342 3 be be AUX iajs-1919 342 4 a a DET iajs-1919 342 5 uniform uniform ADJ iajs-1919 342 6 r r NOUN iajs-1919 342 7 - - PUNCT iajs-1919 342 8 module module NOUN iajs-1919 342 9 has have VERB iajs-1919 342 10 nonzero nonzero ADJ iajs-1919 342 11 small small ADJ iajs-1919 342 12 submodule submodule NOUN iajs-1919 342 13 . . PUNCT iajs-1919 343 1 then then ADV iajs-1919 343 2 the the DET iajs-1919 343 3 following follow VERB iajs-1919 343 4 asseretions asseretion NOUN iajs-1919 343 5 are be AUX iajs-1919 343 6 equivalent equivalent ADJ iajs-1919 343 7 . . PUNCT iajs-1919 344 1 𝑖 𝑖 PRON iajs-1919 344 2 𝑀 𝑀 PROPN iajs-1919 344 3 is be AUX iajs-1919 344 4 rickart rickart NOUN iajs-1919 344 5 . . PUNCT iajs-1919 345 1 𝑖𝑖 𝑖𝑖 PROPN iajs-1919 345 2 𝑀 𝑀 PROPN iajs-1919 345 3 is be AUX iajs-1919 346 1 𝒦-nonsigular 𝒦-nonsigular PROPN iajs-1919 346 2 . . PUNCT iajs-1919 346 3 𝑖𝑖𝑖 𝑖𝑖𝑖 PROPN iajs-1919 346 4 𝑀 𝑀 PROPN iajs-1919 346 5 is be AUX iajs-1919 346 6 strongly strongly ADV iajs-1919 346 7 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 346 8 . . PUNCT iajs-1919 347 1 𝑖𝑣 𝑖𝑣 PROPN iajs-1919 347 2 𝑀 𝑀 PROPN iajs-1919 347 3 is be AUX iajs-1919 347 4 quasi quasi ADJ iajs-1919 347 5 - - ADJ iajs-1919 347 6 dedekind dedekind ADJ iajs-1919 347 7 . . PUNCT iajs-1919 348 1 𝑣 𝑣 PRON iajs-1919 348 2 𝑀 𝑀 PROPN iajs-1919 348 3 is be AUX iajs-1919 348 4 prime prime ADJ iajs-1919 348 5 . . PUNCT iajs-1919 349 1 𝑣𝑖 𝑣𝑖 VERB iajs-1919 349 2 for for ADP iajs-1919 349 3 𝑁 𝑁 PROPN iajs-1919 349 4 ⊴ ⊴ ADP iajs-1919 349 5 𝑀 𝑀 PROPN iajs-1919 349 6 , , PUNCT iajs-1919 349 7 𝑟 𝑟 X iajs-1919 349 8 𝑁 𝑁 PROPN iajs-1919 349 9 𝑟 𝑟 PRON iajs-1919 349 10 𝑀 𝑀 PROPN iajs-1919 349 11 . . PUNCT iajs-1919 350 1 proof proof NOUN iajs-1919 350 2 . . PUNCT iajs-1919 351 1 𝑖 𝑖 PUNCT iajs-1919 351 2 ⇒ ⇒ NOUN iajs-1919 351 3 𝑖𝑣 𝑖𝑣 VERB iajs-1919 351 4 since since SCONJ iajs-1919 351 5 𝑀 𝑀 PROPN iajs-1919 351 6 is be AUX iajs-1919 351 7 a a DET iajs-1919 351 8 uniform uniform ADJ iajs-1919 351 9 r r NOUN iajs-1919 351 10 - - PUNCT iajs-1919 351 11 module module NOUN iajs-1919 351 12 , , PUNCT iajs-1919 351 13 then then ADV iajs-1919 351 14 𝑀 𝑀 PROPN iajs-1919 351 15 is be AUX iajs-1919 351 16 indecomposable indecomposable ADJ iajs-1919 351 17 . . PUNCT iajs-1919 352 1 let let VERB iajs-1919 352 2 𝜑 𝜑 PRON iajs-1919 352 3 ∈ ∈ PROPN iajs-1919 352 4 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 352 5 𝑀 𝑀 PROPN iajs-1919 352 6 with with ADP iajs-1919 352 7 𝜑 𝜑 PROPN iajs-1919 352 8 0 0 NUM iajs-1919 352 9 , , PUNCT iajs-1919 352 10 then then ADV iajs-1919 352 11 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 352 12 ⨁ ⨁ PROPN iajs-1919 352 13 𝑀 𝑀 PROPN iajs-1919 352 14 , , PUNCT iajs-1919 352 15 as as SCONJ iajs-1919 352 16 𝑀 𝑀 PROPN iajs-1919 352 17 is be AUX iajs-1919 352 18 rickart rickart NOUN iajs-1919 352 19 . . PUNCT iajs-1919 353 1 so so ADV iajs-1919 353 2 , , PUNCT iajs-1919 353 3 either either CCONJ iajs-1919 353 4 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 353 5 𝑀 𝑀 PROPN iajs-1919 353 6 or or CCONJ iajs-1919 353 7 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 NOUN iajs-1919 353 8 0 0 NUM iajs-1919 353 9 . . PUNCT iajs-1919 354 1 if if SCONJ iajs-1919 354 2 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 354 3 𝑀 𝑀 PROPN iajs-1919 354 4 then then ADV iajs-1919 354 5 𝜑 𝜑 PROPN iajs-1919 354 6 0 0 NUM iajs-1919 354 7 , , PUNCT iajs-1919 354 8 a a DET iajs-1919 354 9 contradiction contradiction NOUN iajs-1919 354 10 . . PUNCT iajs-1919 355 1 hence hence ADV iajs-1919 355 2 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 355 3 0 0 NUM iajs-1919 355 4 , , PUNCT iajs-1919 355 5 implies imply VERB iajs-1919 355 6 𝑀 𝑀 PROPN iajs-1919 355 7 is be AUX iajs-1919 355 8 quasi quasi ADJ iajs-1919 355 9 - - ADJ iajs-1919 355 10 dedekind dedekind ADJ iajs-1919 355 11 . . PUNCT iajs-1919 356 1 𝑖𝑣 𝑖𝑣 VERB iajs-1919 356 2 ⇒ ⇒ NOUN iajs-1919 357 1 𝑖 𝑖 PUNCT iajs-1919 357 2 let let VERB iajs-1919 357 3 𝜑 𝜑 X iajs-1919 357 4 ∈ ∈ PROPN iajs-1919 357 5 𝐸𝑛𝑑 𝐸𝑛𝑑 PROPN iajs-1919 357 6 𝑀 𝑀 PROPN iajs-1919 357 7 . . PUNCT iajs-1919 358 1 if if SCONJ iajs-1919 358 2 𝜑 𝜑 PROPN iajs-1919 358 3 0 0 NUM iajs-1919 358 4 , , PUNCT iajs-1919 358 5 then then ADV iajs-1919 358 6 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 PROPN iajs-1919 358 7 𝑀 𝑀 PROPN iajs-1919 358 8 ⨁ ⨁ PROPN iajs-1919 358 9 𝑀. 𝑀. PROPN iajs-1919 358 10 assume assume VERB iajs-1919 358 11 that that SCONJ iajs-1919 358 12 𝜑 𝜑 PROPN iajs-1919 358 13 0 0 NUM iajs-1919 358 14 , , PUNCT iajs-1919 358 15 but but CCONJ iajs-1919 358 16 𝑀 𝑀 PROPN iajs-1919 358 17 is be AUX iajs-1919 358 18 a a DET iajs-1919 358 19 quasi quasi ADJ iajs-1919 358 20 - - ADJ iajs-1919 358 21 dedekind dedekind ADJ iajs-1919 358 22 module module NOUN iajs-1919 358 23 , , PUNCT iajs-1919 358 24 so so CCONJ iajs-1919 358 25 𝑘𝑒𝑟𝜑 𝑘𝑒𝑟𝜑 ADJ iajs-1919 358 26 0 0 NUM iajs-1919 358 27 ⨁ ⨁ PROPN iajs-1919 358 28 𝑀. 𝑀. PROPN iajs-1919 358 29 thus thus ADV iajs-1919 358 30 𝑀 𝑀 PROPN iajs-1919 358 31 is be AUX iajs-1919 358 32 rickart rickart NOUN iajs-1919 358 33 . . PUNCT iajs-1919 359 1 𝑖𝑖 𝑖𝑖 PUNCT iajs-1919 360 1 ⇔ ⇔ PROPN iajs-1919 360 2 𝑖𝑖𝑖 𝑖𝑖𝑖 NOUN iajs-1919 360 3 it it PRON iajs-1919 360 4 follows follow VERB iajs-1919 360 5 by by ADP iajs-1919 360 6 proposition proposition NOUN iajs-1919 360 7 36 36 NUM iajs-1919 360 8 . . PUNCT iajs-1919 360 9 𝑖𝑖 𝑖𝑖 PUNCT iajs-1919 361 1 ⇔ ⇔ PROPN iajs-1919 361 2 𝑖𝑣 𝑖𝑣 ADV iajs-1919 361 3 since since SCONJ iajs-1919 361 4 𝑀 𝑀 PROPN iajs-1919 361 5 is be AUX iajs-1919 361 6 a a DET iajs-1919 361 7 uniform uniform ADJ iajs-1919 361 8 module module NOUN iajs-1919 361 9 , , PUNCT iajs-1919 361 10 the the DET iajs-1919 361 11 result result NOUN iajs-1919 361 12 is be AUX iajs-1919 361 13 follow follow VERB iajs-1919 361 14 . . PUNCT iajs-1919 362 1 𝑖𝑣 𝑖𝑣 ADP iajs-1919 362 2 ⇔ ⇔ PROPN iajs-1919 363 1 𝑣 𝑣 ADP iajs-1919 363 2 it it PRON iajs-1919 363 3 follows follow VERB iajs-1919 363 4 by by ADP iajs-1919 363 5 theorem theorem NOUN iajs-1919 363 6 37 37 NUM iajs-1919 363 7 . . PUNCT iajs-1919 364 1 𝑣 𝑣 ADP iajs-1919 364 2 ⇔ ⇔ PROPN iajs-1919 364 3 𝑣𝑖 𝑣𝑖 ADP iajs-1919 364 4 since since SCONJ iajs-1919 364 5 𝑀 𝑀 PROPN iajs-1919 364 6 is be AUX iajs-1919 364 7 uniform uniform ADJ iajs-1919 364 8 has have VERB iajs-1919 364 9 nonzero nonzero PROPN iajs-1919 364 10 small small ADJ iajs-1919 364 11 submodule submodule NOUN iajs-1919 364 12 , , PUNCT iajs-1919 364 13 then then ADV iajs-1919 364 14 all all DET iajs-1919 364 15 its its PRON iajs-1919 364 16 nonzero nonzero NOUN iajs-1919 364 17 submodules submodule NOUN iajs-1919 364 18 are be AUX iajs-1919 364 19 s s NOUN iajs-1919 364 20 - - ADJ iajs-1919 364 21 essential essential ADJ iajs-1919 364 22 , , PUNCT iajs-1919 364 23 so so CCONJ iajs-1919 364 24 the the DET iajs-1919 364 25 result result NOUN iajs-1919 364 26 is be AUX iajs-1919 364 27 obtained obtain VERB iajs-1919 364 28 . . PUNCT iajs-1919 365 1 ∎ ∎ PROPN iajs-1919 365 2 5 5 NUM iajs-1919 365 3 . . X iajs-1919 365 4 conclusion conclusion VERB iajs-1919 365 5 the the DET iajs-1919 365 6 most most ADV iajs-1919 365 7 important important ADJ iajs-1919 365 8 results result NOUN iajs-1919 365 9 of of ADP iajs-1919 365 10 the the DET iajs-1919 365 11 article article NOUN iajs-1919 365 12 are be AUX iajs-1919 365 13 : : PUNCT iajs-1919 365 14 ( ( PUNCT iajs-1919 365 15 1 1 X iajs-1919 365 16 ) ) PUNCT iajs-1919 365 17 let let VERB iajs-1919 365 18 m m PRON iajs-1919 365 19 be be AUX iajs-1919 365 20 a a DET iajs-1919 365 21 faithful faithful ADJ iajs-1919 365 22 multiplication multiplication NOUN iajs-1919 365 23 r r NOUN iajs-1919 365 24 - - NOUN iajs-1919 365 25 module module NOUN iajs-1919 365 26 . . PUNCT iajs-1919 366 1 if if SCONJ iajs-1919 366 2 m m NOUN iajs-1919 366 3 is be AUX iajs-1919 366 4 a a DET iajs-1919 366 5 strongly strongly ADV iajs-1919 366 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 366 7 r r NOUN iajs-1919 366 8 - - PUNCT iajs-1919 366 9 module module NOUN iajs-1919 366 10 , , PUNCT iajs-1919 366 11 then then ADV iajs-1919 366 12 r r NOUN iajs-1919 366 13 is be AUX iajs-1919 366 14 strongly strongly ADV iajs-1919 366 15 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 366 16 . . PUNCT iajs-1919 367 1 the the DET iajs-1919 367 2 converse converse NOUN iajs-1919 367 3 holds hold VERB iajs-1919 367 4 , , PUNCT iajs-1919 367 5 whenever whenever SCONJ iajs-1919 367 6 m m VERB iajs-1919 367 7 is be AUX iajs-1919 367 8 finitely finitely ADV iajs-1919 367 9 generated generate VERB iajs-1919 367 10 . . PUNCT iajs-1919 368 1 ( ( PUNCT iajs-1919 368 2 2 2 X iajs-1919 368 3 ) ) PUNCT iajs-1919 368 4 a a DET iajs-1919 368 5 direct direct ADJ iajs-1919 368 6 summand summand NOUN iajs-1919 368 7 of of ADP iajs-1919 368 8 a a DET iajs-1919 368 9 strongly strongly ADV iajs-1919 368 10 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 368 11 module module NOUN iajs-1919 368 12 is be AUX iajs-1919 368 13 strongly strongly ADV iajs-1919 368 14 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 368 15 . . PUNCT iajs-1919 369 1 ( ( PUNCT iajs-1919 369 2 3 3 X iajs-1919 369 3 ) ) PUNCT iajs-1919 369 4 if if SCONJ iajs-1919 369 5 𝑀 𝑀 PROPN iajs-1919 369 6 ⊕ ⊕ PROPN iajs-1919 369 7 𝑀 𝑀 PROPN iajs-1919 369 8 . . PUNCT iajs-1919 370 1 then then ADV iajs-1919 370 2 m m PROPN iajs-1919 370 3 is be AUX iajs-1919 370 4 a a DET iajs-1919 370 5 strongly strongly ADV iajs-1919 370 6 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 370 7 module module NOUN iajs-1919 370 8 if if SCONJ iajs-1919 370 9 and and CCONJ iajs-1919 370 10 only only ADV iajs-1919 370 11 if if SCONJ iajs-1919 370 12 𝑀 𝑀 PROPN iajs-1919 370 13 is be AUX iajs-1919 370 14 strongly strongly ADV iajs-1919 370 15 𝒦-nonsingular 𝒦-nonsingular ADJ iajs-1919 370 16 relative relative ADJ iajs-1919 370 17 to to ADP iajs-1919 370 18 𝑀 𝑀 PROPN iajs-1919 370 19 , , PUNCT iajs-1919 370 20 for for ADP iajs-1919 370 21 𝑖 𝑖 SYM iajs-1919 370 22 , , PUNCT iajs-1919 370 23 𝑗 𝑗 PROPN iajs-1919 370 24 ∈ ∈ NOUN iajs-1919 370 25 1,2 1,2 NUM iajs-1919 370 26 , , PUNCT iajs-1919 370 27 … … PUNCT iajs-1919 370 28 , , PUNCT iajs-1919 370 29 𝑛 𝑛 PROPN iajs-1919 370 30 . . PUNCT iajs-1919 371 1 references reference NOUN iajs-1919 371 2 1 1 NUM iajs-1919 371 3 . . PUNCT iajs-1919 371 4 leonard leonard PROPN iajs-1919 371 5 , , PUNCT iajs-1919 371 6 w.w w.w PROPN iajs-1919 371 7 . . PROPN iajs-1919 371 8 small small ADJ iajs-1919 371 9 modules module NOUN iajs-1919 371 10 . . PUNCT iajs-1919 372 1 pro pro X iajs-1919 372 2 . . PROPN iajs-1919 372 3 amer amer PROPN iajs-1919 372 4 . . PUNCT iajs-1919 372 5 math math PROPN iajs-1919 372 6 . . PUNCT iajs-1919 373 1 soc soc PROPN iajs-1919 373 2 . . PUNCT iajs-1919 374 1 1966 1966 NUM iajs-1919 374 2 , , PUNCT iajs-1919 374 3 2 2 NUM iajs-1919 374 4 , , PUNCT iajs-1919 374 5 527 527 NUM iajs-1919 374 6 - - SYM iajs-1919 374 7 531 531 NUM iajs-1919 374 8 . . PUNCT iajs-1919 375 1 2 2 NUM iajs-1919 375 2 . . X iajs-1919 375 3 fluery fluery NOUN iajs-1919 375 4 , , PUNCT iajs-1919 375 5 p. p. NOUN iajs-1919 375 6 hollow hollow ADJ iajs-1919 375 7 modules module NOUN iajs-1919 375 8 and and CCONJ iajs-1919 375 9 local local ADJ iajs-1919 375 10 endomorphism endomorphism NOUN iajs-1919 375 11 rings ring NOUN iajs-1919 375 12 . . PUNCT iajs-1919 376 1 pacific pacific PROPN iajs-1919 376 2 j. j. PROPN iajs-1919 376 3 math math PROPN iajs-1919 376 4 . . PUNCT iajs-1919 377 1 1974 1974 NUM iajs-1919 377 2 , , PUNCT iajs-1919 377 3 4 4 NUM iajs-1919 377 4 , , PUNCT iajs-1919 377 5 379385 379385 NUM iajs-1919 377 6 . . PUNCT iajs-1919 377 7         SPACE iajs-1919 378 1 177 177 NUM iajs-1919 378 2     SPACE iajs-1919 378 3 ibn ibn PROPN iajs-1919 378 4 al al PROPN iajs-1919 378 5 - - PUNCT iajs-1919 378 6 haitham haitham PROPN iajs-1919 378 7 jour.for jour.for PROPN iajs-1919 378 8 pure&appl.sci pure&appl.sci PROPN iajs-1919 378 9 . . PUNCT iajs-1919 378 10 ihjpas ihjpas PROPN iajs-1919 378 11 https://doi.org/10.30526/32.1.1919 https://doi.org/10.30526/32.1.1919 X iajs-1919 378 12 vol vol NOUN iajs-1919 378 13 . . PUNCT iajs-1919 378 14 32 32 NUM iajs-1919 378 15 ( ( PUNCT iajs-1919 378 16 1 1 NUM iajs-1919 378 17 ) ) PUNCT iajs-1919 378 18 2019 2019 NUM iajs-1919 378 19 3 3 NUM iajs-1919 378 20 . . PUNCT iajs-1919 379 1 goodearl goodearl PROPN iajs-1919 379 2 , , PUNCT iajs-1919 379 3 k.r k.r PROPN iajs-1919 379 4 . . PROPN iajs-1919 379 5 ring ring PROPN iajs-1919 379 6 theory theory NOUN iajs-1919 379 7 , , PUNCT iajs-1919 379 8 nonsingular nonsingular ADJ iajs-1919 379 9 rings ring NOUN iajs-1919 379 10 and and CCONJ iajs-1919 379 11 modules module NOUN iajs-1919 379 12 . . PUNCT iajs-1919 380 1 marcel marcel PROPN iajs-1919 380 2 dekker dekker PROPN iajs-1919 380 3 . . PUNCT iajs-1919 381 1 newyork newyork PROPN iajs-1919 381 2 and and CCONJ iajs-1919 381 3 basel basel PROPN iajs-1919 381 4 . . PUNCT iajs-1919 381 5 1976 1976 NUM iajs-1919 381 6 . . PUNCT iajs-1919 382 1 4 4 X iajs-1919 382 2 . . X iajs-1919 382 3 zhou zhou PROPN iajs-1919 382 4 , , PUNCT iajs-1919 382 5 d.x d.x PROPN iajs-1919 382 6 . . PROPN iajs-1919 382 7 ; ; PUNCT iajs-1919 382 8 zhang zhang PROPN iajs-1919 382 9 , , PUNCT iajs-1919 382 10 x.r x.r PROPN iajs-1919 382 11 . . PUNCT iajs-1919 382 12 small small ADJ iajs-1919 382 13 - - PUNCT iajs-1919 382 14 essential essential ADJ iajs-1919 382 15 submodules submodule NOUN iajs-1919 382 16 and and CCONJ iajs-1919 382 17 morita morita PROPN iajs-1919 382 18 duality duality PROPN iajs-1919 382 19 . . PUNCT iajs-1919 383 1 southeast southeast ADJ iajs-1919 383 2 asian asian ADJ iajs-1919 383 3 bulletin bulletin NOUN iajs-1919 383 4 of of ADP iajs-1919 383 5 math math NOUN iajs-1919 383 6 . . PUNCT iajs-1919 384 1 2011 2011 NUM iajs-1919 384 2 , , PUNCT iajs-1919 384 3 35 35 NUM iajs-1919 384 4 , , PUNCT iajs-1919 384 5 6 6 NUM iajs-1919 384 6 , , PUNCT iajs-1919 384 7 1051 1051 NUM iajs-1919 384 8 - - SYM iajs-1919 384 9 1062 1062 NUM iajs-1919 384 10 . . PUNCT iajs-1919 385 1 5 5 X iajs-1919 385 2 . . X iajs-1919 385 3 roman roman PROPN iajs-1919 385 4 , , PUNCT iajs-1919 385 5 c.s c.s PROPN iajs-1919 385 6 . . PROPN iajs-1919 385 7 baer baer PROPN iajs-1919 385 8 and and CCONJ iajs-1919 385 9 quasi quasi PROPN iajs-1919 385 10 - - ADJ iajs-1919 385 11 baer baer ADJ iajs-1919 385 12 modules module NOUN iajs-1919 385 13 . . PUNCT iajs-1919 386 1 doctoral doctoral ADJ iajs-1919 386 2 dissertation dissertation NOUN iajs-1919 386 3 . . PUNCT iajs-1919 387 1 the the DET iajs-1919 387 2 ohio ohio PROPN iajs-1919 387 3 state state PROPN iajs-1919 387 4 univ univ PROPN iajs-1919 387 5 . . PUNCT iajs-1919 387 6 2004 2004 NUM iajs-1919 387 7 . . PUNCT iajs-1919 388 1 6 6 X iajs-1919 388 2 . . X iajs-1919 389 1 rizvi rizvi PROPN iajs-1919 389 2 , , PUNCT iajs-1919 389 3 s.t s.t PROPN iajs-1919 389 4 ; ; PUNCT iajs-1919 389 5 roman roman PROPN iajs-1919 389 6 , , PUNCT iajs-1919 389 7 c.s c.s PROPN iajs-1919 389 8 . . PROPN iajs-1919 389 9 on on ADP iajs-1919 389 10 𝒦-nonsigular 𝒦-nonsigular ADJ iajs-1919 389 11 modules module NOUN iajs-1919 389 12 and and CCONJ iajs-1919 389 13 applications application NOUN iajs-1919 389 14 . . PUNCT iajs-1919 390 1 comm comm NOUN iajs-1919 390 2 . . PUNCT iajs-1919 391 1 in in ADP iajs-1919 391 2 algebra algebra PROPN iajs-1919 391 3 . . PUNCT iajs-1919 392 1 2007 2007 NUM iajs-1919 392 2 , , PUNCT iajs-1919 392 3 3 3 NUM iajs-1919 392 4 , , PUNCT iajs-1919 392 5 2960 2960 NUM iajs-1919 392 6 - - SYM iajs-1919 392 7 2982 2982 NUM iajs-1919 392 8 . . PUNCT iajs-1919 393 1 7 7 X iajs-1919 393 2 . . X iajs-1919 393 3 mijbass mijbass PROPN iajs-1919 393 4 , , PUNCT iajs-1919 393 5 a.s a.s PROPN iajs-1919 393 6 . . PROPN iajs-1919 393 7 quasi quasi ADJ iajs-1919 393 8 - - ADJ iajs-1919 393 9 dedekind dedekind ADJ iajs-1919 393 10 modules module NOUN iajs-1919 393 11 . . PUNCT iajs-1919 394 1 ph.d ph.d PROPN iajs-1919 394 2 . . PUNCT iajs-1919 395 1 thesis thesis PROPN iajs-1919 395 2 . . PUNCT iajs-1919 396 1 college college NOUN iajs-1919 396 2 of of ADP iajs-1919 396 3 science science NOUN iajs-1919 396 4 . . PUNCT iajs-1919 397 1 university university NOUN iajs-1919 397 2 of of ADP iajs-1919 397 3 baghdad baghdad PROPN iajs-1919 397 4 . . PUNCT iajs-1919 398 1 iraq iraq PROPN iajs-1919 398 2 . . PUNCT iajs-1919 399 1 1997 1997 NUM iajs-1919 399 2 . . PUNCT iajs-1919 400 1 8 8 X iajs-1919 400 2 . . X iajs-1919 401 1 kasch kasch PROPN iajs-1919 401 2 , , PUNCT iajs-1919 401 3 f. f. PROPN iajs-1919 401 4 modules module NOUN iajs-1919 401 5 and and CCONJ iajs-1919 401 6 rings ring NOUN iajs-1919 401 7 . . PUNCT iajs-1919 402 1 academic academic ADJ iajs-1919 402 2 press press NOUN iajs-1919 402 3 . . PUNCT iajs-1919 403 1 new new PROPN iajs-1919 403 2 york york PROPN iajs-1919 403 3 . . PUNCT iajs-1919 404 1 1982 1982 NUM iajs-1919 404 2 . . PUNCT iajs-1919 405 1 9 9 NUM iajs-1919 405 2 . . NOUN iajs-1919 405 3 wisbauer wisbauer NOUN iajs-1919 405 4 , , PUNCT iajs-1919 405 5 r. r. PROPN iajs-1919 405 6 foundations foundation NOUN iajs-1919 405 7 of of ADP iajs-1919 405 8 module module NOUN iajs-1919 405 9 and and CCONJ iajs-1919 405 10 ring ring NOUN iajs-1919 405 11 theory theory NOUN iajs-1919 405 12 , , PUNCT iajs-1919 405 13 reading reading NOUN iajs-1919 405 14 . . PUNCT iajs-1919 406 1 gordon gordon PROPN iajs-1919 406 2 and and CCONJ iajs-1919 406 3 breach breach VERB iajs-1919 406 4 science science NOUN iajs-1919 406 5 pub pub NOUN iajs-1919 406 6 . . PUNCT iajs-1919 406 7 1991 1991 NUM iajs-1919 406 8 . . PUNCT iajs-1919 407 1 10 10 NUM iajs-1919 407 2 . . X iajs-1919 407 3 clark clark PROPN iajs-1919 407 4 , , PUNCT iajs-1919 407 5 j. j. PROPN iajs-1919 407 6 ; ; PUNCT iajs-1919 407 7 lomp lomp PROPN iajs-1919 407 8 c. c. PROPN iajs-1919 407 9 ; ; PUNCT iajs-1919 407 10 vanaja vanaja PROPN iajs-1919 407 11 , , PUNCT iajs-1919 407 12 n. n. NOUN iajs-1919 407 13 ; ; PUNCT iajs-1919 407 14 wisbauer wisbauer NOUN iajs-1919 407 15 , , PUNCT iajs-1919 407 16 r. r. NOUN iajs-1919 407 17 lifting lifting NOUN iajs-1919 407 18 modules module NOUN iajs-1919 407 19 . . PUNCT iajs-1919 408 1 supplements supplement NOUN iajs-1919 408 2 and and CCONJ iajs-1919 408 3 projectivity projectivity NOUN iajs-1919 408 4 in in ADP iajs-1919 408 5 module module NOUN iajs-1919 408 6 theory theory NOUN iajs-1919 408 7 . . PUNCT iajs-1919 409 1 frontiers frontier NOUN iajs-1919 409 2 in in ADP iajs-1919 409 3 mathematics mathematic NOUN iajs-1919 409 4 . . PUNCT iajs-1919 410 1 birkhauser birkhauser PROPN iajs-1919 410 2 . . PUNCT iajs-1919 411 1 basel basel PROPN iajs-1919 411 2 . . PUNCT iajs-1919 412 1 2006 2006 NUM iajs-1919 412 2 . . PUNCT iajs-1919 413 1 11 11 NUM iajs-1919 413 2 . . PUNCT iajs-1919 414 1 zelmanowitz zelmanowitz PROPN iajs-1919 414 2 , , PUNCT iajs-1919 414 3 j.m j.m PROPN iajs-1919 414 4 . . PROPN iajs-1919 414 5 representation representation NOUN iajs-1919 414 6 of of ADP iajs-1919 414 7 rings ring NOUN iajs-1919 414 8 with with ADP iajs-1919 414 9 faithful faithful ADJ iajs-1919 414 10 polyform polyform NOUN iajs-1919 414 11 modules module NOUN iajs-1919 414 12 . . PUNCT iajs-1919 415 1 comm comm NOUN iajs-1919 415 2 . . PUNCT iajs-1919 416 1 in in ADP iajs-1919 416 2 algebra algebra PROPN iajs-1919 416 3 . . PUNCT iajs-1919 417 1 1986 1986 NUM iajs-1919 417 2 , , PUNCT iajs-1919 417 3 14 14 NUM iajs-1919 417 4 , , PUNCT iajs-1919 417 5 6 6 NUM iajs-1919 417 6 , , PUNCT iajs-1919 417 7 1141 1141 NUM iajs-1919 417 8 - - SYM iajs-1919 417 9 1169 1169 NUM iajs-1919 417 10 . . PUNCT iajs-1919 418 1 12 12 NUM iajs-1919 418 2 . . PUNCT iajs-1919 419 1 branard branard PROPN iajs-1919 419 2 , , PUNCT iajs-1919 419 3 b. b. PROPN iajs-1919 419 4 multiplication multiplication NOUN iajs-1919 419 5 modules module NOUN iajs-1919 419 6 . . PUNCT iajs-1919 420 1 journal journal NOUN iajs-1919 420 2 of of ADP iajs-1919 420 3 algebra algebra PROPN iajs-1919 420 4 . . PUNCT iajs-1919 421 1 1981 1981 NUM iajs-1919 421 2 , , PUNCT iajs-1919 421 3 3 3 NUM iajs-1919 421 4 , , PUNCT iajs-1919 421 5 170 170 NUM iajs-1919 421 6 - - SYM iajs-1919 421 7 178 178 NUM iajs-1919 421 8 . . PUNCT iajs-1919 422 1 13 13 NUM iajs-1919 422 2 . . X iajs-1919 423 1 shihab shihab PROPN iajs-1919 423 2 , , PUNCT iajs-1919 423 3 b.n b.n PROPN iajs-1919 423 4 . . PROPN iajs-1919 423 5 scalar scalar ADJ iajs-1919 423 6 reflexive reflexive ADJ iajs-1919 423 7 modules module NOUN iajs-1919 423 8 . . PUNCT iajs-1919 424 1 ph.d ph.d PROPN iajs-1919 424 2 . . PUNCT iajs-1919 425 1 thesis thesis NOUN iajs-1919 425 2 . . PUNCT iajs-1919 426 1 university university NOUN iajs-1919 426 2 of of ADP iajs-1919 426 3 baghdad baghdad PROPN iajs-1919 426 4 . . PUNCT iajs-1919 427 1 iraq iraq PROPN iajs-1919 427 2 . . PUNCT iajs-1919 428 1 2004 2004 NUM iajs-1919 428 2 . . PUNCT iajs-1919 429 1 14 14 NUM iajs-1919 429 2 . . PUNCT iajs-1919 430 1 naoum naoum PROPN iajs-1919 430 2 , , PUNCT iajs-1919 430 3 a.g a.g PROPN iajs-1919 430 4 . . PROPN iajs-1919 430 5 on on ADP iajs-1919 430 6 the the DET iajs-1919 430 7 ring ring NOUN iajs-1919 430 8 of of ADP iajs-1919 430 9 endomorphisms endomorphism NOUN iajs-1919 430 10 of of ADP iajs-1919 430 11 finitely finitely ADV iajs-1919 430 12 generated generate VERB iajs-1919 430 13 multiplication multiplication NOUN iajs-1919 430 14 modules module NOUN iajs-1919 430 15 . . PUNCT iajs-1919 431 1 periodica periodica PROPN iajs-1919 431 2 math math PROPN iajs-1919 431 3 . . PUNCT iajs-1919 432 1 hungarica hungarica PROPN iajs-1919 432 2 . . PUNCT iajs-1919 433 1 1990 1990 NUM iajs-1919 433 2 , , PUNCT iajs-1919 433 3 21 21 NUM iajs-1919 433 4 , , PUNCT iajs-1919 433 5 3 3 NUM iajs-1919 433 6 , , PUNCT iajs-1919 433 7 249 249 NUM iajs-1919 433 8 - - SYM iajs-1919 433 9 255 255 NUM iajs-1919 433 10 . . PUNCT iajs-1919 434 1 15 15 NUM iajs-1919 434 2 . . PUNCT iajs-1919 434 3 mohamed mohamed PROPN iajs-1919 434 4 - - PUNCT iajs-1919 434 5 ali ali PROPN iajs-1919 434 6 , , PUNCT iajs-1919 434 7 e.a e.a PROPN iajs-1919 434 8 . . PROPN iajs-1919 435 1 on on ADP iajs-1919 435 2 ikeda ikeda PROPN iajs-1919 435 3 - - PUNCT iajs-1919 435 4 nakayama nakayama PROPN iajs-1919 435 5 modules module NOUN iajs-1919 435 6 . . PUNCT iajs-1919 436 1 ph.d ph.d PROPN iajs-1919 436 2 . . PUNCT iajs-1919 437 1 thesis thesis NOUN iajs-1919 437 2 . . PUNCT iajs-1919 438 1 university university NOUN iajs-1919 438 2 of of ADP iajs-1919 438 3 baghdad baghdad PROPN iajs-1919 438 4 . . PUNCT iajs-1919 439 1 iraq iraq PROPN iajs-1919 439 2 . . PUNCT iajs-1919 440 1 2006 2006 NUM iajs-1919 440 2 . . PUNCT iajs-1919 441 1 16 16 NUM iajs-1919 441 2 . . PUNCT iajs-1919 442 1 lee lee PROPN iajs-1919 442 2 , , PUNCT iajs-1919 442 3 g. g. PROPN iajs-1919 442 4 ; ; PUNCT iajs-1919 442 5 rizvi rizvi PROPN iajs-1919 442 6 , , PUNCT iajs-1919 442 7 s.t s.t PROPN iajs-1919 442 8 . . PROPN iajs-1919 442 9 ; ; PUNCT iajs-1919 442 10 roman roman PROPN iajs-1919 442 11 , , PUNCT iajs-1919 442 12 c.s c.s PROPN iajs-1919 442 13 . . PROPN iajs-1919 442 14 rickart rickart PROPN iajs-1919 442 15 modules module NOUN iajs-1919 442 16 . . PUNCT iajs-1919 443 1 commutative commutative ADJ iajs-1919 443 2 in in ADP iajs-1919 443 3 algebra algebra PROPN iajs-1919 443 4 . . PUNCT iajs-1919 444 1 2010 2010 NUM iajs-1919 444 2 , , PUNCT iajs-1919 444 3 4 4 NUM iajs-1919 444 4 , , PUNCT iajs-1919 444 5 4005 4005 NUM iajs-1919 444 6 - - SYM iajs-1919 444 7 4027 4027 NUM iajs-1919 444 8 . . PUNCT iajs-1919 445 1 17 17 NUM iajs-1919 445 2 . . PUNCT iajs-1919 445 3 ware ware PROPN iajs-1919 445 4 , , PUNCT iajs-1919 445 5 r. r. PROPN iajs-1919 445 6 endomorphism endomorphism PROPN iajs-1919 445 7 rings ring NOUN iajs-1919 445 8 of of ADP iajs-1919 445 9 projective projective ADJ iajs-1919 445 10 modules module NOUN iajs-1919 445 11 . . PUNCT iajs-1919 446 1 trans trans PROPN iajs-1919 446 2 . . PUNCT iajs-1919 447 1 amer amer PROPN iajs-1919 447 2 . . PUNCT iajs-1919 447 3 math math PROPN iajs-1919 447 4 . . PUNCT iajs-1919 448 1 soc soc PROPN iajs-1919 448 2 . . PUNCT iajs-1919 449 1 1971 1971 NUM iajs-1919 449 2 , , PUNCT iajs-1919 449 3 233256 233256 NUM iajs-1919 449 4 . . PUNCT iajs-1919 450 1 18 18 NUM iajs-1919 450 2 . . PUNCT iajs-1919 450 3 desale desale NOUN iajs-1919 450 4 , , PUNCT iajs-1919 450 5 g. g. PROPN iajs-1919 450 6 ; ; PUNCT iajs-1919 450 7 nicholson nicholson PROPN iajs-1919 450 8 , , PUNCT iajs-1919 450 9 w.k w.k PROPN iajs-1919 450 10 . . PROPN iajs-1919 450 11 endoprimitive endoprimitive PROPN iajs-1919 450 12 rings ring NOUN iajs-1919 450 13 . . PUNCT iajs-1919 451 1 j. j. PROPN iajs-1919 451 2 algebra algebra PROPN iajs-1919 451 3 . . PUNCT iajs-1919 452 1 1981 1981 NUM iajs-1919 452 2 , , PUNCT iajs-1919 452 3 548 548 NUM iajs-1919 452 4 - - SYM iajs-1919 452 5 560 560 NUM iajs-1919 452 6 . . NOUN iajs-1919 452 7 19 19 NUM iajs-1919 452 8 . . PUNCT iajs-1919 452 9 hadi hadi PROPN iajs-1919 452 10 , , PUNCT iajs-1919 452 11 i. i. PROPN iajs-1919 452 12 m m PROPN iajs-1919 452 13 - - PROPN iajs-1919 452 14 a a NOUN iajs-1919 452 15 and and CCONJ iajs-1919 452 16 ghawi ghawi VERB iajs-1919 452 17 , , PUNCT iajs-1919 452 18 th.y th.y NOUN iajs-1919 452 19 . . PUNCT iajs-1919 453 1 essentially essentially ADV iajs-1919 453 2 quasi quasi ADJ iajs-1919 453 3 - - ADJ iajs-1919 453 4 invertible invertible ADJ iajs-1919 453 5 submodules submodule NOUN iajs-1919 453 6 and and CCONJ iajs-1919 453 7 essentially essentially ADV iajs-1919 453 8 quasi quasi ADJ iajs-1919 453 9 - - ADJ iajs-1919 453 10 dedekind dedekind ADJ iajs-1919 453 11 modules module NOUN iajs-1919 453 12 . . PUNCT iajs-1919 454 1 ibn ibn PROPN iajs-1919 454 2 al al PROPN iajs-1919 454 3 - - PUNCT iajs-1919 454 4 haitham haitham PROPN iajs-1919 454 5 j. j. PROPN iajs-1919 454 6 for for ADP iajs-1919 454 7 pure pure PROPN iajs-1919 454 8 & & CCONJ iajs-1919 454 9 appl appl PROPN iajs-1919 454 10 . . PUNCT iajs-1919 455 1 sci sci PROPN iajs-1919 455 2 . . PROPN iajs-1919 455 3 2011 2011 NUM iajs-1919 455 4 , , PUNCT iajs-1919 455 5 24 24 NUM iajs-1919 455 6 , , PUNCT iajs-1919 455 7 3 3 NUM iajs-1919 455 8 , , PUNCT iajs-1919 455 9 102 102 NUM iajs-1919 455 10 - - SYM iajs-1919 455 11 113 113 NUM iajs-1919 455 12 . . PUNCT