id sid tid token lemma pos cana-3155 1 1 communications communication NOUN cana-3155 1 2 on on ADP cana-3155 1 3 applied apply VERB cana-3155 1 4 nonlinear nonlinear ADJ cana-3155 1 5 analysis analysis NOUN cana-3155 1 6 issn issn NOUN cana-3155 1 7 : : PUNCT cana-3155 1 8 1074 1074 NUM cana-3155 1 9 - - PUNCT cana-3155 1 10 133x 133x NUM cana-3155 1 11 vol vol NOUN cana-3155 1 12 32 32 NUM cana-3155 1 13 no no NOUN cana-3155 1 14 . . PUNCT cana-3155 2 1 5s 5s NUM cana-3155 2 2 ( ( PUNCT cana-3155 2 3 2025 2025 NUM cana-3155 2 4 ) ) PUNCT cana-3155 2 5 488 488 NUM cana-3155 2 6 https://internationalpubls.com https://internationalpubls.com NOUN cana-3155 2 7 class class NOUN cana-3155 2 8 of of ADP cana-3155 2 9 modules module NOUN cana-3155 2 10 for for ADP cana-3155 2 11 which which PRON cana-3155 2 12 strongly strongly ADV cana-3155 2 13 hopfian hopfian ADJ cana-3155 2 14 modules module NOUN cana-3155 2 15 are be AUX cana-3155 2 16 noetherian noetherian ADJ cana-3155 2 17 mankagna mankagna PROPN cana-3155 2 18 albert albert PROPN cana-3155 2 19 diompy1 diompy1 PROPN cana-3155 2 20 , , PUNCT cana-3155 2 21 ousseynou ousseynou NOUN cana-3155 2 22 bousso2 bousso2 PROPN cana-3155 2 23 , , PUNCT cana-3155 2 24 oumar oumar PROPN cana-3155 2 25 diankha3 diankha3 VERB cana-3155 2 26 1,2,3department 1,2,3department NUM cana-3155 2 27 of of ADP cana-3155 2 28 mathematics mathematic NOUN cana-3155 2 29 and and CCONJ cana-3155 2 30 computer computer NOUN cana-3155 2 31 science science NOUN cana-3155 2 32 , , PUNCT cana-3155 2 33 university university NOUN cana-3155 2 34 of of ADP cana-3155 2 35 cheikh cheikh PROPN cana-3155 2 36 anta anta PROPN cana-3155 2 37 diop diop PROPN cana-3155 2 38 , , PUNCT cana-3155 2 39 dakar dakar NOUN cana-3155 2 40 , , PUNCT cana-3155 2 41 senegal senegal ADJ cana-3155 2 42 email email NOUN cana-3155 2 43 id id NOUN cana-3155 2 44 : : PUNCT cana-3155 2 45 ousseynou1.bousso@ucad.edu.sn2 ousseynou1.bousso@ucad.edu.sn2 PROPN cana-3155 2 46 , , PUNCT cana-3155 2 47 oumar.diankha@ucad.edu.sn3 oumar.diankha@ucad.edu.sn3 PRON cana-3155 2 48 corresponding corresponding ADJ cana-3155 2 49 author author NOUN cana-3155 2 50 : : PUNCT cana-3155 2 51 albertdiompy@yahoo.fr albertdiompy@yahoo.fr PROPN cana-3155 2 52 article article PROPN cana-3155 2 53 history history NOUN cana-3155 2 54 : : PUNCT cana-3155 2 55 received receive VERB cana-3155 2 56 : : PUNCT cana-3155 2 57 13 13 NUM cana-3155 2 58 - - SYM cana-3155 2 59 10 10 NUM cana-3155 2 60 - - PUNCT cana-3155 2 61 2024 2024 NUM cana-3155 2 62 revised revise VERB cana-3155 2 63 : : PUNCT cana-3155 2 64 28 28 NUM cana-3155 2 65 - - SYM cana-3155 2 66 11 11 NUM cana-3155 2 67 - - PUNCT cana-3155 2 68 2024 2024 NUM cana-3155 2 69 accepted accept VERB cana-3155 2 70 : : PUNCT cana-3155 2 71 09 09 NUM cana-3155 2 72 - - SYM cana-3155 2 73 12 12 NUM cana-3155 2 74 - - PUNCT cana-3155 2 75 2024 2024 NUM cana-3155 2 76 abstract abstract NOUN cana-3155 2 77 : : PUNCT cana-3155 2 78 let let VERB cana-3155 2 79 r r PRON cana-3155 2 80 be be AUX cana-3155 2 81 an an DET cana-3155 2 82 arbitrary arbitrary ADJ cana-3155 2 83 ring ring NOUN cana-3155 2 84 and and CCONJ cana-3155 2 85 m m VERB cana-3155 2 86 a a DET cana-3155 2 87 left left ADJ cana-3155 2 88 r r NOUN cana-3155 2 89 - - PUNCT cana-3155 2 90 module module NOUN cana-3155 2 91 . . PUNCT cana-3155 3 1 in in ADP cana-3155 3 2 this this DET cana-3155 3 3 paper paper NOUN cana-3155 3 4 we we PRON cana-3155 3 5 introduce introduce VERB cana-3155 3 6 the the DET cana-3155 3 7 modules module NOUN cana-3155 3 8 m m VERB cana-3155 3 9 such such ADJ cana-3155 3 10 that that SCONJ cana-3155 3 11 every every DET cana-3155 3 12 strongly strongly ADV cana-3155 3 13 hopfian hopfian ADJ cana-3155 3 14 module module NOUN cana-3155 3 15 in in ADP cana-3155 3 16 Οƒ[m Οƒ[m ADJ cana-3155 3 17 ] ] PUNCT cana-3155 3 18 is be AUX cana-3155 3 19 noetherian noetherian ADJ cana-3155 3 20 . . PUNCT cana-3155 4 1 these these DET cana-3155 4 2 modules module NOUN cana-3155 4 3 will will AUX cana-3155 4 4 be be AUX cana-3155 4 5 called call VERB cana-3155 4 6 sf sf NOUN cana-3155 4 7 - - PUNCT cana-3155 4 8 modules module NOUN cana-3155 4 9 . . PUNCT cana-3155 5 1 we we PRON cana-3155 5 2 characterize characterize VERB cana-3155 5 3 such such ADJ cana-3155 5 4 modules module NOUN cana-3155 5 5 and and CCONJ cana-3155 5 6 study study VERB cana-3155 5 7 their their PRON cana-3155 5 8 properties property NOUN cana-3155 5 9 . . PUNCT cana-3155 6 1 relationships relationship NOUN cana-3155 6 2 between between ADP cana-3155 6 3 sf sf NOUN cana-3155 6 4 - - PUNCT cana-3155 6 5 modules module NOUN cana-3155 6 6 and and CCONJ cana-3155 6 7 other other ADJ cana-3155 6 8 classes class NOUN cana-3155 6 9 of of ADP cana-3155 6 10 modules module NOUN cana-3155 6 11 are be AUX cana-3155 6 12 given give VERB cana-3155 6 13 . . PUNCT cana-3155 7 1 keywords keyword NOUN cana-3155 7 2 : : PUNCT cana-3155 7 3 strongly strongly ADV cana-3155 7 4 hopfian hopfian ADJ cana-3155 7 5 module module NOUN cana-3155 7 6 , , PUNCT cana-3155 7 7 𝑆𝐹-module 𝑆𝐹-module PROPN cana-3155 7 8 , , PUNCT cana-3155 7 9 perfect perfect ADJ cana-3155 7 10 module module NOUN cana-3155 7 11 , , PUNCT cana-3155 7 12 locally locally ADV cana-3155 7 13 noetherian noetherian ADJ cana-3155 7 14 module module NOUN cana-3155 7 15 ; ; PUNCT cana-3155 7 16 hollow hollow ADJ cana-3155 7 17 module module NOUN cana-3155 7 18 ; ; PUNCT cana-3155 7 19 semiartinian semiartinian ADJ cana-3155 7 20 module module NOUN cana-3155 7 21 ; ; PUNCT cana-3155 7 22 𝛱-semiartinian 𝛱-semiartinian ADJ cana-3155 7 23 module module NOUN cana-3155 7 24 . . PUNCT cana-3155 8 1 ams am NOUN cana-3155 8 2 subject subject ADJ cana-3155 8 3 classification classification NOUN cana-3155 8 4 : : PUNCT cana-3155 8 5 13c05 13c05 NUM cana-3155 8 6 , , PUNCT cana-3155 8 7 13e05 13e05 NUM cana-3155 8 8 , , PUNCT cana-3155 8 9 13f10 13f10 NUM cana-3155 8 10 1 1 NUM cana-3155 8 11 . . PUNCT cana-3155 9 1 introduction introduction NOUN cana-3155 9 2 the the DET cana-3155 9 3 study study NOUN cana-3155 9 4 of of ADP cana-3155 9 5 modules module NOUN cana-3155 9 6 by by ADP cana-3155 9 7 properties property NOUN cana-3155 9 8 of of ADP cana-3155 9 9 their their PRON cana-3155 9 10 endomorphisms endomorphism NOUN cana-3155 9 11 has have AUX cana-3155 9 12 long long ADV cana-3155 9 13 been be AUX cana-3155 9 14 of of ADP cana-3155 9 15 interest interest NOUN cana-3155 9 16 . . PUNCT cana-3155 10 1 throughout throughout ADP cana-3155 10 2 in in ADP cana-3155 10 3 this this DET cana-3155 10 4 paper paper NOUN cana-3155 10 5 , , PUNCT cana-3155 10 6 rings ring NOUN cana-3155 10 7 are be AUX cana-3155 10 8 considered consider VERB cana-3155 10 9 associative associative ADJ cana-3155 10 10 , , PUNCT cana-3155 10 11 non non X cana-3155 10 12 necessarily necessarily ADV cana-3155 10 13 commutative commutative ADJ cana-3155 10 14 with with ADP cana-3155 10 15 identity identity NOUN cana-3155 10 16 1 1 NUM cana-3155 10 17 β‰  β‰  PROPN cana-3155 10 18 0 0 NUM cana-3155 10 19 , , PUNCT cana-3155 10 20 all all DET cana-3155 10 21 modules module NOUN cana-3155 10 22 are be AUX cana-3155 10 23 unitary unitary ADJ cana-3155 10 24 left leave VERB cana-3155 10 25 𝑅-modules 𝑅-modules PROPN cana-3155 10 26 and and CCONJ cana-3155 10 27 𝑅-mod 𝑅-mod PROPN cana-3155 10 28 denotes denote NOUN cana-3155 10 29 the the DET cana-3155 10 30 category category NOUN cana-3155 10 31 of of ADP cana-3155 10 32 left leave VERB cana-3155 10 33 unitary unitary ADJ cana-3155 10 34 𝑅modules 𝑅modules PROPN cana-3155 10 35 . . PUNCT cana-3155 11 1 we we PRON cana-3155 11 2 denote denote VERB cana-3155 11 3 by by ADP cana-3155 11 4 𝜎[𝑀 𝜎[𝑀 PROPN cana-3155 11 5 ] ] PUNCT cana-3155 11 6 the the DET cana-3155 11 7 full full ADJ cana-3155 11 8 subcategory subcategory NOUN cana-3155 11 9 of of ADP cana-3155 11 10 𝑅-modules 𝑅-modules PROPN cana-3155 11 11 whose whose DET cana-3155 11 12 objects object NOUN cana-3155 11 13 are be AUX cana-3155 11 14 all all PRON cana-3155 11 15 𝑅-mod 𝑅-mod ADV cana-3155 11 16 subgenerated subgenerate VERB cana-3155 11 17 by by ADP cana-3155 11 18 𝑀. 𝑀. PROPN cana-3155 11 19 a a DET cana-3155 11 20 𝑅-module 𝑅-module PROPN cana-3155 11 21 𝑀 𝑀 PROPN cana-3155 11 22 is be AUX cana-3155 11 23 noetherian noetherian ADJ cana-3155 11 24 ( ( PUNCT cana-3155 11 25 resp resp NOUN cana-3155 11 26 . . PUNCT cana-3155 12 1 artinian artinian PROPN cana-3155 12 2 ) ) PUNCT cana-3155 12 3 if if SCONJ cana-3155 12 4 any any DET cana-3155 12 5 ascending ascending NOUN cana-3155 12 6 ( ( PUNCT cana-3155 12 7 resp resp NOUN cana-3155 12 8 . . PUNCT cana-3155 13 1 descending descend VERB cana-3155 13 2 ) ) PUNCT cana-3155 13 3 chain chain NOUN cana-3155 13 4 of of ADP cana-3155 13 5 submodules submodule NOUN cana-3155 13 6 of of ADP cana-3155 13 7 𝑀 𝑀 PROPN cana-3155 13 8 is be AUX cana-3155 13 9 stationary stationary ADJ cana-3155 13 10 . . PUNCT cana-3155 14 1 a a DET cana-3155 14 2 𝑅-module 𝑅-module PROPN cana-3155 14 3 𝑀 𝑀 PROPN cana-3155 14 4 is be AUX cana-3155 14 5 called call VERB cana-3155 14 6 hopfian hopfian ADJ cana-3155 14 7 , , PUNCT cana-3155 14 8 if if SCONJ cana-3155 14 9 any any DET cana-3155 14 10 surjective surjective ADJ cana-3155 14 11 𝑅-homomorphism 𝑅-homomorphism PROPN cana-3155 14 12 𝑓 𝑓 X cana-3155 14 13 : : PUNCT cana-3155 14 14 𝑀 𝑀 PROPN cana-3155 14 15 β†’ β†’ SYM cana-3155 14 16 𝑀 𝑀 PROPN cana-3155 14 17 is be AUX cana-3155 14 18 an an DET cana-3155 14 19 isomorphism isomorphism NOUN cana-3155 14 20 . . PUNCT cana-3155 15 1 an an DET cana-3155 15 2 object object NOUN cana-3155 15 3 𝑁 𝑁 PROPN cana-3155 15 4 of of ADP cana-3155 15 5 𝜎[m 𝜎[m NOUN cana-3155 15 6 ] ] PUNCT cana-3155 15 7 is be AUX cana-3155 15 8 said say VERB cana-3155 15 9 to to PART cana-3155 15 10 be be AUX cana-3155 15 11 strongly strongly ADV cana-3155 15 12 hopfian hopfian ADJ cana-3155 15 13 , , PUNCT cana-3155 15 14 if if SCONJ cana-3155 15 15 for for ADP cana-3155 15 16 every every DET cana-3155 15 17 𝑅-endomorphism 𝑅-endomorphism PROPN cana-3155 15 18 of of ADP cana-3155 15 19 𝑁 𝑁 PROPN cana-3155 15 20 , , PUNCT cana-3155 15 21 the the DET cana-3155 15 22 chain chain NOUN cana-3155 15 23 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN cana-3155 16 1 βŠ† βŠ† NUM cana-3155 16 2 πΎπ‘’π‘Ÿπ‘“2 πΎπ‘’π‘Ÿπ‘“2 PROPN cana-3155 16 3 βŠ† βŠ† NUM cana-3155 16 4 β‹― β‹― ADP cana-3155 16 5 βŠ† βŠ† NUM cana-3155 16 6 πΎπ‘’π‘Ÿπ‘“π‘› πΎπ‘’π‘Ÿπ‘“π‘› PROPN cana-3155 16 7 βŠ† βŠ† NUM cana-3155 16 8 β‹― β‹― NOUN cana-3155 16 9 is be AUX cana-3155 16 10 stabilizes stabilize VERB cana-3155 16 11 . . PUNCT cana-3155 17 1 a a DET cana-3155 17 2 ring ring NOUN cana-3155 17 3 𝑅 𝑅 PROPN cana-3155 17 4 is be AUX cana-3155 17 5 said say VERB cana-3155 17 6 𝑆𝐹-ring 𝑆𝐹-re VERB cana-3155 17 7 , , PUNCT cana-3155 17 8 if if SCONJ cana-3155 17 9 every every DET cana-3155 17 10 strongly strongly ADV cana-3155 17 11 hopfian hopfian ADJ cana-3155 17 12 𝑅-module 𝑅-module PROPN cana-3155 17 13 is be AUX cana-3155 17 14 noetherian noetherian ADJ cana-3155 17 15 . . PUNCT cana-3155 18 1 let let VERB cana-3155 18 2 𝑅 𝑅 NOUN cana-3155 18 3 be be AUX cana-3155 18 4 a a DET cana-3155 18 5 commutative commutative ADJ cana-3155 18 6 ring ring NOUN cana-3155 18 7 , , PUNCT cana-3155 18 8 a a DET cana-3155 18 9 𝑅-module 𝑅-module PROPN cana-3155 18 10 𝑀 𝑀 PROPN cana-3155 18 11 is be AUX cana-3155 18 12 said say VERB cana-3155 18 13 𝐹𝐺𝑆module 𝐹𝐺𝑆module NOUN cana-3155 18 14 if if SCONJ cana-3155 18 15 every every DET cana-3155 18 16 hopfian hopfian ADJ cana-3155 18 17 object object NOUN cana-3155 18 18 of of ADP cana-3155 18 19 𝜎[𝑀 𝜎[𝑀 PROPN cana-3155 18 20 ] ] PUNCT cana-3155 18 21 is be AUX cana-3155 18 22 finitely finitely ADV cana-3155 18 23 generated generate VERB cana-3155 18 24 . . PUNCT cana-3155 19 1 a a DET cana-3155 19 2 𝑅-module 𝑅-module PROPN cana-3155 19 3 𝑀 𝑀 PROPN cana-3155 19 4 is be AUX cana-3155 19 5 called call VERB cana-3155 19 6 endo endo NOUN cana-3155 19 7 - - PUNCT cana-3155 19 8 noetherian noetherian NOUN cana-3155 19 9 if if SCONJ cana-3155 19 10 for for ADP cana-3155 19 11 any any DET cana-3155 19 12 family family NOUN cana-3155 19 13 ( ( PUNCT cana-3155 19 14 𝑓𝑖)𝑖β‰₯1 𝑓𝑖)𝑖β‰₯1 NOUN cana-3155 19 15 of of ADP cana-3155 19 16 endomorphisms endomorphism NOUN cana-3155 19 17 of of ADP cana-3155 19 18 𝑀 𝑀 PROPN cana-3155 19 19 , , PUNCT cana-3155 19 20 the the DET cana-3155 19 21 sequence sequence NOUN cana-3155 19 22 πΎπ‘’π‘Ÿ(𝑓1 πΎπ‘’π‘Ÿ(𝑓1 NOUN cana-3155 19 23 ) ) PUNCT cana-3155 19 24 βŠ† βŠ† NUM cana-3155 19 25 πΎπ‘’π‘Ÿ(𝑓2 πΎπ‘’π‘Ÿ(𝑓2 ADJ cana-3155 19 26 ) ) PUNCT cana-3155 19 27 βŠ† βŠ† NUM cana-3155 19 28 β‹― β‹― ADP cana-3155 19 29 βŠ† βŠ† NUM cana-3155 19 30 πΎπ‘’π‘Ÿπ‘“π‘› πΎπ‘’π‘Ÿπ‘“π‘› PROPN cana-3155 19 31 βŠ† βŠ† NUM cana-3155 19 32 β‹― β‹― NOUN cana-3155 19 33 stabilizes stabilize VERB cana-3155 19 34 . . PUNCT cana-3155 20 1 a a DET cana-3155 20 2 𝑅-module 𝑅-module PROPN cana-3155 20 3 𝑀 𝑀 PROPN cana-3155 20 4 is be AUX cana-3155 20 5 said say VERB cana-3155 20 6 𝐸𝐾𝐹𝑁-module 𝐸𝐾𝐹𝑁-module ADJ cana-3155 20 7 if if SCONJ cana-3155 20 8 every every DET cana-3155 20 9 endo endo NOUN cana-3155 20 10 - - PUNCT cana-3155 20 11 noetherian noetherian ADJ cana-3155 20 12 object object NOUN cana-3155 20 13 of of ADP cana-3155 20 14 𝜎[𝑀 𝜎[𝑀 PROPN cana-3155 20 15 ] ] PUNCT cana-3155 20 16 is be AUX cana-3155 20 17 noetherian noetherian ADJ cana-3155 20 18 . . PUNCT cana-3155 21 1 a a DET cana-3155 21 2 ring ring NOUN cana-3155 21 3 𝑅 𝑅 PROPN cana-3155 21 4 is be AUX cana-3155 21 5 said say VERB cana-3155 21 6 𝑆-ring 𝑆-ring PROPN cana-3155 21 7 , , PUNCT cana-3155 21 8 if if SCONJ cana-3155 21 9 every every DET cana-3155 21 10 hopfian hopfian NOUN cana-3155 21 11 𝑅-module 𝑅-module PROPN cana-3155 21 12 is be AUX cana-3155 21 13 noetherian noetherian ADJ cana-3155 21 14 and and CCONJ cana-3155 21 15 an an DET cana-3155 21 16 𝑅-module 𝑅-module PROPN cana-3155 21 17 𝑀 𝑀 PROPN cana-3155 21 18 is be AUX cana-3155 21 19 said say VERB cana-3155 21 20 𝑆-module 𝑆-module PROPN cana-3155 21 21 if if SCONJ cana-3155 21 22 every every DET cana-3155 21 23 hopfian hopfian ADJ cana-3155 21 24 object object NOUN cana-3155 21 25 of of ADP cana-3155 21 26 𝜎[𝑀 𝜎[𝑀 PROPN cana-3155 21 27 ] ] PUNCT cana-3155 21 28 is be AUX cana-3155 21 29 noetherian noetherian ADJ cana-3155 21 30 . . PUNCT cana-3155 22 1 an an DET cana-3155 22 2 module module NOUN cana-3155 22 3 𝑀 𝑀 PROPN cana-3155 22 4 is be AUX cana-3155 22 5 hollow hollow ADJ cana-3155 22 6 , , PUNCT cana-3155 22 7 if if SCONJ cana-3155 22 8 𝑀 𝑀 PROPN cana-3155 22 9 β‰  β‰  PROPN cana-3155 22 10 0 0 NUM cana-3155 22 11 and and CCONJ cana-3155 22 12 submodule submodule NOUN cana-3155 22 13 of of ADP cana-3155 22 14 𝑀 𝑀 PROPN cana-3155 22 15 is be AUX cana-3155 22 16 a a DET cana-3155 22 17 small small ADJ cana-3155 22 18 submodule submodule NOUN cana-3155 22 19 of of ADP cana-3155 22 20 𝑀 𝑀 PROPN cana-3155 22 21 . . PUNCT cana-3155 23 1 all all DET cana-3155 23 2 noetherian noetherian ADJ cana-3155 23 3 module module NOUN cana-3155 23 4 is be AUX cana-3155 23 5 strongly strongly ADV cana-3155 23 6 hopfian hopfian ADJ cana-3155 23 7 but but CCONJ cana-3155 23 8 converse converse NOUN cana-3155 23 9 is be AUX cana-3155 23 10 not not PART cana-3155 23 11 always always ADV cana-3155 23 12 true true ADJ cana-3155 23 13 . . PUNCT cana-3155 24 1 for for ADP cana-3155 24 2 example example NOUN cana-3155 24 3 , , PUNCT cana-3155 24 4 the the DET cana-3155 24 5 β„€-module β„€-module PROPN cana-3155 24 6 𝑀 𝑀 NOUN cana-3155 24 7 = = PRON cana-3155 24 8 β¨π‘βˆˆπ‘ƒβ„€π‘ β¨π‘βˆˆπ‘ƒβ„€π‘ NOUN cana-3155 24 9 is be AUX cana-3155 24 10 strongly strongly ADV cana-3155 24 11 hopfian hopfian ADJ cana-3155 24 12 but but CCONJ cana-3155 24 13 it it PRON cana-3155 24 14 is be AUX cana-3155 24 15 not not PART cana-3155 24 16 noetherian noetherian ADJ cana-3155 24 17 , , PUNCT cana-3155 24 18 where where SCONJ cana-3155 24 19 𝑃 𝑃 NOUN cana-3155 24 20 is be AUX cana-3155 24 21 the the DET cana-3155 24 22 set set NOUN cana-3155 24 23 of of ADP cana-3155 24 24 all all DET cana-3155 24 25 primes prime NOUN cana-3155 24 26 . . PUNCT cana-3155 25 1 a a DET cana-3155 25 2 module module NOUN cana-3155 25 3 is be AUX cana-3155 25 4 named name VERB cana-3155 25 5 𝑆𝐹-module 𝑆𝐹-module PROPN cana-3155 25 6 if if SCONJ cana-3155 25 7 every every DET cana-3155 25 8 strongly strongly ADV cana-3155 25 9 object object NOUN cana-3155 25 10 of of ADP cana-3155 25 11 𝜎[m 𝜎[m NOUN cana-3155 25 12 ] ] PUNCT cana-3155 25 13 is be AUX cana-3155 25 14 noetherian noetherian ADJ cana-3155 25 15 in in ADP cana-3155 25 16 this this DET cana-3155 25 17 paper paper NOUN cana-3155 25 18 , , PUNCT cana-3155 25 19 first first ADV cana-3155 25 20 we we PRON cana-3155 25 21 present present VERB cana-3155 25 22 preliminary preliminary ADJ cana-3155 25 23 results result NOUN cana-3155 25 24 and and CCONJ cana-3155 25 25 some some DET cana-3155 25 26 fundamental fundamental ADJ cana-3155 25 27 properties property NOUN cana-3155 25 28 of of ADP cana-3155 25 29 𝑆𝐹-modules 𝑆𝐹-module NOUN cana-3155 25 30 . . PUNCT cana-3155 26 1 secondly secondly ADV cana-3155 26 2 we we PRON cana-3155 26 3 characterize characterize VERB cana-3155 26 4 the the DET cana-3155 26 5 class class NOUN cana-3155 26 6 of of ADP cana-3155 26 7 finitely finitely ADV cana-3155 26 8 generated generate VERB cana-3155 26 9 and and CCONJ cana-3155 26 10 hollow hollow ADJ cana-3155 26 11 𝑆𝐹-modules 𝑆𝐹-module NOUN cana-3155 26 12 . . PUNCT cana-3155 27 1 additionally additionally ADV cana-3155 27 2 , , PUNCT cana-3155 27 3 we we PRON cana-3155 27 4 communications communication VERB cana-3155 27 5 on on ADP cana-3155 27 6 applied apply VERB cana-3155 27 7 nonlinear nonlinear ADJ cana-3155 27 8 analysis analysis NOUN cana-3155 27 9 issn issn NOUN cana-3155 27 10 : : PUNCT cana-3155 27 11 1074 1074 NUM cana-3155 27 12 - - PUNCT cana-3155 27 13 133x 133x NUM cana-3155 27 14 vol vol NOUN cana-3155 27 15 32 32 NUM cana-3155 27 16 no no NOUN cana-3155 27 17 . . PUNCT cana-3155 28 1 5s 5s NUM cana-3155 28 2 ( ( PUNCT cana-3155 28 3 2025 2025 NUM cana-3155 28 4 ) ) PUNCT cana-3155 28 5 489 489 NUM cana-3155 28 6 https://internationalpubls.com https://internationalpubls.com X cana-3155 28 7 prove prove VERB cana-3155 28 8 that that SCONJ cana-3155 28 9 in in ADP cana-3155 28 10 the the DET cana-3155 28 11 setting setting NOUN cana-3155 28 12 of of ADP cana-3155 28 13 finitely finitely ADV cana-3155 28 14 generated generate VERB cana-3155 28 15 𝑆𝐹-modules 𝑆𝐹-module NOUN cana-3155 28 16 , , PUNCT cana-3155 28 17 noetherian noetherian ADJ cana-3155 28 18 module module NOUN cana-3155 28 19 , , PUNCT cana-3155 28 20 artinian artinian ADJ cana-3155 28 21 module module NOUN cana-3155 28 22 and and CCONJ cana-3155 28 23 semiartinian semiartinian ADJ cana-3155 28 24 module module NOUN cana-3155 28 25 are be AUX cana-3155 28 26 equivalent equivalent ADJ cana-3155 28 27 . . PUNCT cana-3155 29 1 2 2 X cana-3155 29 2 . . X cana-3155 29 3 some some DET cana-3155 29 4 properties property NOUN cana-3155 29 5 of of ADP cana-3155 29 6 sf sf NOUN cana-3155 29 7 - - PUNCT cana-3155 29 8 modules module NOUN cana-3155 29 9 lemma lemma PROPN cana-3155 29 10 2.1 2.1 NUM cana-3155 29 11 . . PUNCT cana-3155 30 1 for for ADP cana-3155 30 2 a a DET cana-3155 30 3 ring ring NOUN cana-3155 30 4 r r NOUN cana-3155 30 5 we we PRON cana-3155 30 6 have have VERB cana-3155 30 7 : : PUNCT cana-3155 30 8 1 1 X cana-3155 30 9 . . X cana-3155 30 10 every every DET cana-3155 30 11 noetherian noetherian ADJ cana-3155 30 12 r r NOUN cana-3155 30 13 - - PUNCT cana-3155 30 14 module module NOUN cana-3155 30 15 is be AUX cana-3155 30 16 endo endo NOUN cana-3155 30 17 - - PUNCT cana-3155 30 18 noetherian noetherian ADJ cana-3155 30 19 . . PUNCT cana-3155 31 1 2 2 X cana-3155 31 2 . . X cana-3155 31 3 every every DET cana-3155 31 4 endo endo NOUN cana-3155 31 5 - - PUNCT cana-3155 31 6 noetherian noetherian ADJ cana-3155 31 7 r r NOUN cana-3155 31 8 - - PUNCT cana-3155 31 9 module module NOUN cana-3155 31 10 is be AUX cana-3155 31 11 strongly strongly ADV cana-3155 31 12 hopfian hopfian ADJ cana-3155 31 13 . . PUNCT cana-3155 32 1 3 3 X cana-3155 32 2 . . X cana-3155 32 3 every every PRON cana-3155 32 4 strongly strongly ADV cana-3155 32 5 hopfian hopfian ADJ cana-3155 32 6 r r NOUN cana-3155 32 7 - - PUNCT cana-3155 32 8 module module NOUN cana-3155 32 9 is be AUX cana-3155 32 10 hopfian hopfian ADJ cana-3155 32 11 . . PUNCT cana-3155 33 1 proposition proposition NOUN cana-3155 33 2 2.2 2.2 NUM cana-3155 33 3 . . PUNCT cana-3155 34 1 if if SCONJ cana-3155 34 2 m m NOUN cana-3155 34 3 be be VERB cana-3155 34 4 a a DET cana-3155 34 5 sf sf NOUN cana-3155 34 6 - - PUNCT cana-3155 34 7 module module NOUN cana-3155 34 8 . . PUNCT cana-3155 35 1 then then ADV cana-3155 35 2 we we PRON cana-3155 35 3 have have VERB cana-3155 35 4 the the DET cana-3155 35 5 following follow VERB cana-3155 35 6 properties property NOUN cana-3155 35 7 : : PUNCT cana-3155 36 1 1 1 X cana-3155 36 2 . . X cana-3155 37 1 every every DET cana-3155 37 2 submodule submodule NOUN cana-3155 37 3 of of ADP cana-3155 37 4 a a DET cana-3155 37 5 strongly strongly ADV cana-3155 37 6 hopfian hopfian ADJ cana-3155 37 7 module module NOUN cana-3155 37 8 in in ADP cana-3155 37 9 Οƒ[m Οƒ[m ADJ cana-3155 37 10 ] ] PUNCT cana-3155 37 11 is be AUX cana-3155 37 12 strongly strongly ADV cana-3155 37 13 hopfian hopfian ADJ cana-3155 37 14 . . PUNCT cana-3155 38 1 2 2 X cana-3155 38 2 . . X cana-3155 39 1 every every DET cana-3155 39 2 quotient quotient NOUN cana-3155 39 3 of of ADP cana-3155 39 4 a a DET cana-3155 39 5 strongly strongly ADV cana-3155 39 6 hopfian hopfian ADJ cana-3155 39 7 module module NOUN cana-3155 39 8 in in ADP cana-3155 39 9 Οƒ[m Οƒ[m ADJ cana-3155 39 10 ] ] PUNCT cana-3155 39 11 is be AUX cana-3155 39 12 strongly strongly ADV cana-3155 39 13 hopfian hopfian ADJ cana-3155 39 14 . . PUNCT cana-3155 40 1 proof proof NOUN cana-3155 40 2 . . PUNCT cana-3155 41 1 1 1 X cana-3155 41 2 ) ) PUNCT cana-3155 41 3 let let VERB cana-3155 41 4 n n PRON cana-3155 41 5 be be AUX cana-3155 41 6 a a DET cana-3155 41 7 submodule submodule NOUN cana-3155 41 8 of of ADP cana-3155 41 9 strongly strongly ADV cana-3155 41 10 hopfian hopfian ADJ cana-3155 41 11 module module NOUN cana-3155 41 12 k k PROPN cana-3155 41 13 in in ADP cana-3155 41 14 Οƒ[m Οƒ[m NOUN cana-3155 41 15 ] ] PUNCT cana-3155 41 16 . . PUNCT cana-3155 42 1 as as SCONJ cana-3155 42 2 m m PROPN cana-3155 42 3 is be AUX cana-3155 42 4 an an DET cana-3155 42 5 sf sf NOUN cana-3155 42 6 - - PUNCT cana-3155 42 7 module module NOUN cana-3155 42 8 , , PUNCT cana-3155 42 9 then then ADV cana-3155 42 10 k k PROPN cana-3155 42 11 is be AUX cana-3155 42 12 noetherian noetherian ADJ cana-3155 42 13 . . PUNCT cana-3155 43 1 since since SCONJ cana-3155 43 2 submodule submodule NOUN cana-3155 43 3 of of ADP cana-3155 43 4 noetherian noetherian ADJ cana-3155 43 5 module module NOUN cana-3155 43 6 is be AUX cana-3155 43 7 noetherian noetherian ADJ cana-3155 43 8 so so SCONJ cana-3155 43 9 n n NOUN cana-3155 43 10 is be AUX cana-3155 43 11 noetherian noetherian ADJ cana-3155 43 12 . . PUNCT cana-3155 44 1 therefore therefore ADV cana-3155 44 2 n n PROPN cana-3155 44 3 is be AUX cana-3155 44 4 strongly strongly ADV cana-3155 44 5 hopfian hopfian ADJ cana-3155 44 6 beacause beacause NOUN cana-3155 44 7 every every DET cana-3155 44 8 noetherian noetherian ADJ cana-3155 44 9 module module NOUN cana-3155 44 10 is be AUX cana-3155 44 11 strongly strongly ADV cana-3155 44 12 hopfian hopfian ADJ cana-3155 44 13 . . PUNCT cana-3155 45 1 2 2 X cana-3155 45 2 ) ) PUNCT cana-3155 45 3 result result NOUN cana-3155 45 4 from from ADP cana-3155 45 5 the the DET cana-3155 45 6 fact fact NOUN cana-3155 45 7 that that SCONJ cana-3155 45 8 any any DET cana-3155 45 9 quotient quotient NOUN cana-3155 45 10 of of ADP cana-3155 45 11 a a DET cana-3155 45 12 noetherian noetherian ADJ cana-3155 45 13 module module NOUN cana-3155 45 14 is be AUX cana-3155 45 15 noetherian noetherian ADJ cana-3155 45 16 . . PUNCT cana-3155 46 1 β—» β—» PROPN cana-3155 46 2 proposition proposition NOUN cana-3155 46 3 2.3 2.3 NUM cana-3155 46 4 . . PUNCT cana-3155 47 1 let let VERB cana-3155 47 2 r r PRON cana-3155 47 3 be be AUX cana-3155 47 4 a a DET cana-3155 47 5 ring ring NOUN cana-3155 47 6 . . PUNCT cana-3155 48 1 the the DET cana-3155 48 2 following follow VERB cana-3155 48 3 assertions assertion NOUN cana-3155 48 4 are be AUX cana-3155 48 5 equivalent equivalent ADJ cana-3155 48 6 : : PUNCT cana-3155 48 7 1 1 X cana-3155 48 8 . . X cana-3155 48 9 r r NOUN cana-3155 48 10 is be AUX cana-3155 48 11 sf sf NOUN cana-3155 48 12 - - PUNCT cana-3155 48 13 ring ring NOUN cana-3155 48 14 . . PUNCT cana-3155 49 1 2 2 X cana-3155 49 2 . . X cana-3155 49 3 every every DET cana-3155 49 4 r r NOUN cana-3155 49 5 - - PUNCT cana-3155 49 6 module module NOUN cana-3155 49 7 is be AUX cana-3155 49 8 a a DET cana-3155 49 9 sf sf NOUN cana-3155 49 10 - - PUNCT cana-3155 49 11 module module NOUN cana-3155 49 12 . . PUNCT cana-3155 50 1 proof proof NOUN cana-3155 50 2 . . PUNCT cana-3155 51 1 1 1 NUM cana-3155 51 2 ) ) PUNCT cana-3155 51 3 ⟹ ⟹ NUM cana-3155 51 4 2 2 NUM cana-3155 51 5 ) ) PUNCT cana-3155 51 6 . . PUNCT cana-3155 52 1 let let VERB cana-3155 52 2 m m PRON cana-3155 52 3 a a DET cana-3155 52 4 r r NOUN cana-3155 52 5 - - PUNCT cana-3155 52 6 module module NOUN cana-3155 52 7 and and CCONJ cana-3155 52 8 n n DET cana-3155 52 9 a a DET cana-3155 52 10 strongly strongly ADV cana-3155 52 11 hopfian hopfian ADJ cana-3155 52 12 objet objet NOUN cana-3155 52 13 of of ADP cana-3155 52 14 Οƒ[m Οƒ[m NOUN cana-3155 52 15 ] ] PUNCT cana-3155 52 16 . . PUNCT cana-3155 53 1 since since SCONJ cana-3155 53 2 Οƒ[m Οƒ[m NOUN cana-3155 53 3 ] ] PUNCT cana-3155 53 4 is be AUX cana-3155 53 5 the the DET cana-3155 53 6 full full ADJ cana-3155 53 7 subcategory subcategory NOUN cana-3155 53 8 of of ADP cana-3155 53 9 r r NOUN cana-3155 53 10 - - PUNCT cana-3155 53 11 mod mod NOUN cana-3155 53 12 then then ADV cana-3155 53 13 n n PROPN cana-3155 53 14 is be AUX cana-3155 53 15 a a DET cana-3155 53 16 strongly strongly ADV cana-3155 53 17 hopfian hopfian ADJ cana-3155 53 18 r r NOUN cana-3155 53 19 - - PUNCT cana-3155 53 20 module module NOUN cana-3155 53 21 . . PUNCT cana-3155 54 1 as as SCONJ cana-3155 54 2 r r NOUN cana-3155 54 3 is be AUX cana-3155 54 4 a a DET cana-3155 54 5 sf sf NOUN cana-3155 54 6 - - PUNCT cana-3155 54 7 ring ring NOUN cana-3155 54 8 then then ADV cana-3155 54 9 n n PRON cana-3155 54 10 is be AUX cana-3155 54 11 noetherian noetherian ADJ cana-3155 54 12 . . PUNCT cana-3155 55 1 2 2 NUM cana-3155 55 2 ) ) PUNCT cana-3155 55 3 ⟹ ⟹ NUM cana-3155 55 4 1 1 X cana-3155 55 5 ) ) PUNCT cana-3155 55 6 suppose suppose VERB cana-3155 55 7 that that SCONJ cana-3155 55 8 every every DET cana-3155 55 9 r r NOUN cana-3155 55 10 - - PUNCT cana-3155 55 11 module module NOUN cana-3155 55 12 is be AUX cana-3155 55 13 sf sf NOUN cana-3155 55 14 - - PUNCT cana-3155 55 15 module module NOUN cana-3155 55 16 . . PUNCT cana-3155 56 1 let let VERB cana-3155 56 2 k k PRON cana-3155 56 3 be be AUX cana-3155 56 4 a a DET cana-3155 56 5 strongly strongly ADV cana-3155 56 6 hofian hofian ADJ cana-3155 56 7 r r NOUN cana-3155 56 8 - - PUNCT cana-3155 56 9 module module NOUN cana-3155 56 10 . . PUNCT cana-3155 57 1 since since SCONJ cana-3155 57 2 k k PROPN cana-3155 57 3 ∈ ∈ PROPN cana-3155 57 4 Οƒ[k Οƒ[k PROPN cana-3155 57 5 ] ] PUNCT cana-3155 57 6 then then ADV cana-3155 57 7 k k PROPN cana-3155 57 8 is be AUX cana-3155 57 9 noetherian noetherian ADJ cana-3155 57 10 . . PUNCT cana-3155 58 1 hence hence ADV cana-3155 58 2 r r NOUN cana-3155 58 3 is be AUX cana-3155 58 4 a a DET cana-3155 58 5 sf sf NOUN cana-3155 58 6 - - PUNCT cana-3155 58 7 ring ring NOUN cana-3155 58 8 . . PUNCT cana-3155 59 1 β—» β—» PROPN cana-3155 59 2 remark remark VERB cana-3155 59 3 2.4 2.4 NUM cana-3155 59 4 . . NOUN cana-3155 60 1 1 1 NUM cana-3155 60 2 . . X cana-3155 61 1 every every DET cana-3155 61 2 s s NOUN cana-3155 61 3 - - PUNCT cana-3155 61 4 module module NOUN cana-3155 61 5 is be AUX cana-3155 61 6 sf sf NOUN cana-3155 61 7 - - PUNCT cana-3155 61 8 module module NOUN cana-3155 61 9 . . PUNCT cana-3155 62 1 2 2 X cana-3155 62 2 . . X cana-3155 62 3 every every DET cana-3155 62 4 sf sf NOUN cana-3155 62 5 - - PUNCT cana-3155 62 6 module module NOUN cana-3155 62 7 is be AUX cana-3155 62 8 a a DET cana-3155 62 9 ekfn ekfn NOUN cana-3155 62 10 - - PUNCT cana-3155 62 11 module module NOUN cana-3155 62 12 . . PUNCT cana-3155 63 1 proposition proposition NOUN cana-3155 63 2 2.5 2.5 NUM cana-3155 63 3 . . PUNCT cana-3155 64 1 let let VERB cana-3155 64 2 r r PRON cana-3155 64 3 be be AUX cana-3155 64 4 a a DET cana-3155 64 5 commutative commutative ADJ cana-3155 64 6 ring ring NOUN cana-3155 64 7 and and CCONJ cana-3155 64 8 m m DET cana-3155 64 9 a a DET cana-3155 64 10 finitely finitely ADV cana-3155 64 11 generated generate VERB cana-3155 64 12 r r NOUN cana-3155 64 13 - - PUNCT cana-3155 64 14 module module NOUN cana-3155 64 15 . . PUNCT cana-3155 65 1 if if SCONJ cana-3155 65 2 m m NOUN cana-3155 65 3 is be AUX cana-3155 65 4 a a DET cana-3155 65 5 sfmodule sfmodule NOUN cana-3155 65 6 , , PUNCT cana-3155 65 7 then then ADV cana-3155 65 8 every every DET cana-3155 65 9 object object NOUN cana-3155 65 10 of of ADP cana-3155 65 11 Οƒ[m Οƒ[m ADV cana-3155 65 12 ] ] PUNCT cana-3155 65 13 has have VERB cana-3155 65 14 a a DET cana-3155 65 15 projective projective ADJ cana-3155 65 16 cover cover NOUN cana-3155 65 17 . . PUNCT cana-3155 66 1 proof proof NOUN cana-3155 66 2 . . PUNCT cana-3155 67 1 if if SCONJ cana-3155 67 2 m m PROPN cana-3155 67 3 is be AUX cana-3155 67 4 sf sf NOUN cana-3155 67 5 - - PUNCT cana-3155 67 6 module module NOUN cana-3155 67 7 , , PUNCT cana-3155 67 8 then then ADV cana-3155 67 9 by by ADP cana-3155 67 10 remark remark NOUN cana-3155 67 11 2.4 2.4 NUM cana-3155 67 12 . . PUNCT cana-3155 68 1 m m PROPN cana-3155 68 2 is be AUX cana-3155 68 3 ekfn ekfn NOUN cana-3155 68 4 - - PUNCT cana-3155 68 5 module module NOUN cana-3155 68 6 and and CCONJ cana-3155 68 7 by by ADP cana-3155 68 8 proposition proposition NOUN cana-3155 68 9 3.4 3.4 NUM cana-3155 68 10 . . PUNCT cana-3155 68 11 of of ADP cana-3155 68 12 [ [ X cana-3155 68 13 5 5 NUM cana-3155 68 14 ] ] PUNCT cana-3155 68 15 , , PUNCT cana-3155 68 16 every every DET cana-3155 68 17 object object NOUN cana-3155 68 18 of of ADP cana-3155 68 19 Οƒ[m Οƒ[m ADV cana-3155 68 20 ] ] PUNCT cana-3155 68 21 has have VERB cana-3155 68 22 a a DET cana-3155 68 23 projective projective ADJ cana-3155 68 24 cover cover NOUN cana-3155 68 25 . . PUNCT cana-3155 69 1 β—» β—» PROPN cana-3155 69 2 proposition proposition NOUN cana-3155 69 3 2.6 2.6 NUM cana-3155 69 4 . . PUNCT cana-3155 70 1 for for ADP cana-3155 70 2 a a DET cana-3155 70 3 r r NOUN cana-3155 70 4 - - PUNCT cana-3155 70 5 module module NOUN cana-3155 70 6 m m NOUN cana-3155 70 7 , , PUNCT cana-3155 70 8 the the DET cana-3155 70 9 following follow VERB cana-3155 70 10 properties property NOUN cana-3155 70 11 are be AUX cana-3155 70 12 equivalent equivalent ADJ cana-3155 70 13 : : PUNCT cana-3155 70 14 1 1 X cana-3155 70 15 . . X cana-3155 70 16 m m PROPN cana-3155 70 17 is be AUX cana-3155 70 18 an an DET cana-3155 70 19 sf sf NOUN cana-3155 70 20 - - PUNCT cana-3155 70 21 module module NOUN cana-3155 70 22 . . PUNCT cana-3155 71 1 2 2 X cana-3155 71 2 . . X cana-3155 71 3 every every DET cana-3155 71 4 module module NOUN cana-3155 71 5 in in ADP cana-3155 71 6 Οƒ[m Οƒ[m ADJ cana-3155 71 7 ] ] PUNCT cana-3155 71 8 is be AUX cana-3155 71 9 an an DET cana-3155 71 10 sf sf NOUN cana-3155 71 11 - - PUNCT cana-3155 71 12 module module NOUN cana-3155 71 13 . . PUNCT cana-3155 72 1 communications communication NOUN cana-3155 72 2 on on ADP cana-3155 72 3 applied apply VERB cana-3155 72 4 nonlinear nonlinear ADJ cana-3155 72 5 analysis analysis NOUN cana-3155 72 6 issn issn NOUN cana-3155 72 7 : : PUNCT cana-3155 72 8 1074 1074 NUM cana-3155 72 9 - - PUNCT cana-3155 72 10 133x 133x NUM cana-3155 72 11 vol vol NOUN cana-3155 72 12 32 32 NUM cana-3155 72 13 no no NOUN cana-3155 72 14 . . PUNCT cana-3155 73 1 5s 5s NUM cana-3155 73 2 ( ( PUNCT cana-3155 73 3 2025 2025 NUM cana-3155 73 4 ) ) PUNCT cana-3155 73 5 490 490 NUM cana-3155 73 6 https://internationalpubls.com https://internationalpubls.com X cana-3155 73 7 proof proof NOUN cana-3155 73 8 . . PUNCT cana-3155 74 1 1)⟹ 1)⟹ NUM cana-3155 74 2 2 2 NUM cana-3155 74 3 ): ): PUNCT cana-3155 74 4 let let VERB cana-3155 74 5 n n PRON cana-3155 74 6 ∈ ∈ PROPN cana-3155 74 7 Οƒ[m Οƒ[m X cana-3155 74 8 ] ] PUNCT cana-3155 74 9 then then ADV cana-3155 74 10 Οƒ[n Οƒ[n VERB cana-3155 74 11 ] ] PUNCT cana-3155 74 12 is be AUX cana-3155 74 13 the the DET cana-3155 74 14 smallest small ADJ cana-3155 74 15 category category NOUN cana-3155 74 16 of of ADP cana-3155 74 17 Οƒ[m Οƒ[m X cana-3155 74 18 ] ] PUNCT cana-3155 74 19 containing contain VERB cana-3155 74 20 n n CCONJ cana-3155 75 1 and and CCONJ cana-3155 75 2 it it PRON cana-3155 75 3 is be AUX cana-3155 75 4 a a DET cana-3155 75 5 full full ADJ cana-3155 75 6 subcategory subcategory NOUN cana-3155 75 7 of of ADP cana-3155 75 8 Οƒ[m Οƒ[m NOUN cana-3155 75 9 ] ] PUNCT cana-3155 75 10 . . PUNCT cana-3155 76 1 if if SCONJ cana-3155 76 2 k k PROPN cana-3155 76 3 is be AUX cana-3155 76 4 a a DET cana-3155 76 5 strongly strongly ADV cana-3155 76 6 hopfian hopfian ADJ cana-3155 76 7 object object NOUN cana-3155 76 8 of of ADP cana-3155 76 9 Οƒ[n Οƒ[n NOUN cana-3155 76 10 ] ] PUNCT cana-3155 76 11 , , PUNCT cana-3155 76 12 then then ADV cana-3155 76 13 k k PROPN cana-3155 76 14 ∈ ∈ PROPN cana-3155 76 15 Οƒ[m Οƒ[m X cana-3155 76 16 ] ] PUNCT cana-3155 76 17 and and CCONJ cana-3155 76 18 since since SCONJ cana-3155 76 19 m m PROPN cana-3155 76 20 is be AUX cana-3155 76 21 an an DET cana-3155 76 22 sf sf PROPN cana-3155 76 23 -module -module PROPN cana-3155 76 24 then then ADV cana-3155 76 25 k k PROPN cana-3155 76 26 is be AUX cana-3155 76 27 noetherian noetherian ADJ cana-3155 76 28 . . PUNCT cana-3155 77 1 2)⟹ 2)⟹ NUM cana-3155 77 2 1 1 NUM cana-3155 77 3 ): ): PUNCT cana-3155 77 4 it it PRON cana-3155 77 5 ’s ’ VERB cana-3155 77 6 obvious obvious ADJ cana-3155 77 7 because because SCONJ cana-3155 77 8 m m PROPN cana-3155 77 9 ∈ ∈ NOUN cana-3155 77 10 Οƒ[m Οƒ[m NOUN cana-3155 77 11 ] ] PUNCT cana-3155 77 12 . . PUNCT cana-3155 78 1 β—» β—» PROPN cana-3155 78 2 proposition proposition NOUN cana-3155 78 3 2.7 2.7 NUM cana-3155 78 4 . . PUNCT cana-3155 79 1 over over ADP cana-3155 79 2 r r NOUN cana-3155 79 3 - - PUNCT cana-3155 79 4 artinian artinian ADJ cana-3155 79 5 ring ring NOUN cana-3155 79 6 , , PUNCT cana-3155 79 7 all all DET cana-3155 79 8 r r NOUN cana-3155 79 9 - - PUNCT cana-3155 79 10 module module NOUN cana-3155 79 11 is be AUX cana-3155 79 12 sf sf NOUN cana-3155 79 13 - - PUNCT cana-3155 79 14 module module NOUN cana-3155 79 15 . . PUNCT cana-3155 80 1 proof proof NOUN cana-3155 80 2 . . PUNCT cana-3155 81 1 let let VERB cana-3155 81 2 m m PRON cana-3155 81 3 be be AUX cana-3155 81 4 a a DET cana-3155 81 5 r r NOUN cana-3155 81 6 - - PUNCT cana-3155 81 7 module module NOUN cana-3155 81 8 and and CCONJ cana-3155 81 9 let let VERB cana-3155 81 10 n n PRON cana-3155 81 11 be be AUX cana-3155 81 12 a a DET cana-3155 81 13 strongly strongly ADV cana-3155 81 14 hopfian hopfian ADJ cana-3155 81 15 module module NOUN cana-3155 81 16 in in ADP cana-3155 81 17 Οƒ[m Οƒ[m NOUN cana-3155 81 18 ] ] PUNCT cana-3155 81 19 . . PUNCT cana-3155 82 1 since since SCONJ cana-3155 82 2 r r NOUN cana-3155 82 3 is be AUX cana-3155 82 4 artinian artinian ADJ cana-3155 82 5 ring ring NOUN cana-3155 82 6 then then ADV cana-3155 82 7 according accord VERB cana-3155 82 8 to to ADP cana-3155 82 9 31.5 31.5 NUM cana-3155 82 10 of of ADP cana-3155 82 11 [ [ X cana-3155 82 12 11 11 NUM cana-3155 82 13 ] ] PUNCT cana-3155 82 14 , , PUNCT cana-3155 82 15 Οƒ[m Οƒ[m X cana-3155 82 16 ] ] X cana-3155 82 17 = = SYM cana-3155 82 18 r r NOUN cana-3155 82 19 / / SYM cana-3155 82 20 ann(m)-mod ann(m)-mod NOUN cana-3155 82 21 . . PUNCT cana-3155 83 1 hence hence ADV cana-3155 83 2 every every DET cana-3155 83 3 module module NOUN cana-3155 83 4 in in ADP cana-3155 83 5 Οƒ[m Οƒ[m ADJ cana-3155 83 6 ] ] PUNCT cana-3155 83 7 is be AUX cana-3155 83 8 an an DET cana-3155 83 9 r r NOUN cana-3155 83 10 / / SYM cana-3155 83 11 ann(m)-module ann(m)-module NOUN cana-3155 83 12 therefore therefore ADV cana-3155 83 13 n n PROPN cana-3155 83 14 is be AUX cana-3155 83 15 a a DET cana-3155 83 16 ideal ideal NOUN cana-3155 83 17 of of ADP cana-3155 83 18 r r NOUN cana-3155 83 19 / / SYM cana-3155 83 20 ann(m ann(m PROPN cana-3155 83 21 ) ) PUNCT cana-3155 83 22 . . PUNCT cana-3155 84 1 as as SCONJ cana-3155 84 2 r r NOUN cana-3155 84 3 is be AUX cana-3155 84 4 artinian artinian ADJ cana-3155 84 5 then then ADV cana-3155 84 6 r r PROPN cana-3155 84 7 / / SYM cana-3155 84 8 ann(m ann(m NOUN cana-3155 84 9 ) ) PUNCT cana-3155 84 10 is be AUX cana-3155 84 11 artinian artinian ADJ cana-3155 84 12 and and CCONJ cana-3155 84 13 so so ADV cana-3155 84 14 n n ADV cana-3155 84 15 is be AUX cana-3155 84 16 finitely finitely ADV cana-3155 84 17 generated generate VERB cana-3155 84 18 . . PUNCT cana-3155 85 1 since since SCONJ cana-3155 85 2 over over ADP cana-3155 85 3 artinian artinian ADJ cana-3155 85 4 ring ring NOUN cana-3155 85 5 , , PUNCT cana-3155 85 6 finitely finitely ADV cana-3155 85 7 generated generate VERB cana-3155 85 8 and and CCONJ cana-3155 85 9 noetherian noetherian ADJ cana-3155 85 10 are be AUX cana-3155 85 11 equivalent equivalent ADJ cana-3155 85 12 then then ADV cana-3155 85 13 n n PRON cana-3155 85 14 is be AUX cana-3155 85 15 noetherian noetherian ADJ cana-3155 85 16 . . PUNCT cana-3155 86 1 β—» β—» PROPN cana-3155 86 2 proposition proposition NOUN cana-3155 86 3 2.8 2.8 NUM cana-3155 86 4 . . PUNCT cana-3155 87 1 let let VERB cana-3155 87 2 m m PRON cana-3155 87 3 be be AUX cana-3155 87 4 a a DET cana-3155 87 5 r r NOUN cana-3155 87 6 - - PUNCT cana-3155 87 7 module module NOUN cana-3155 87 8 . . PUNCT cana-3155 88 1 if if SCONJ cana-3155 88 2 every every DET cana-3155 88 3 module module NOUN cana-3155 88 4 in in ADP cana-3155 88 5 Οƒ[m Οƒ[m ADJ cana-3155 88 6 ] ] PUNCT cana-3155 88 7 is be AUX cana-3155 88 8 injective injective ADJ cana-3155 88 9 , , PUNCT cana-3155 88 10 then then ADV cana-3155 88 11 m m VERB cana-3155 88 12 is be AUX cana-3155 88 13 an an DET cana-3155 88 14 sfmodule sfmodule NOUN cana-3155 88 15 . . PUNCT cana-3155 89 1 proof proof NOUN cana-3155 89 2 . . PUNCT cana-3155 90 1 suppose suppose VERB cana-3155 90 2 that that SCONJ cana-3155 90 3 every every DET cana-3155 90 4 module module NOUN cana-3155 90 5 in in ADP cana-3155 90 6 Οƒ[m Οƒ[m ADJ cana-3155 90 7 ] ] PUNCT cana-3155 90 8 is be AUX cana-3155 90 9 injective injective ADJ cana-3155 90 10 . . PUNCT cana-3155 91 1 let let VERB cana-3155 91 2 k k PRON cana-3155 91 3 be be AUX cana-3155 91 4 a a DET cana-3155 91 5 strongly strongly ADV cana-3155 91 6 hopfian hopfian ADJ cana-3155 91 7 object object NOUN cana-3155 91 8 of of ADP cana-3155 91 9 Οƒ[m Οƒ[m ADJ cana-3155 91 10 ] ] PUNCT cana-3155 91 11 then then ADV cana-3155 91 12 k k PROPN cana-3155 91 13 is be AUX cana-3155 91 14 hopfian hopfian ADJ cana-3155 91 15 . . PUNCT cana-3155 92 1 since since SCONJ cana-3155 92 2 by by ADP cana-3155 92 3 hypothesis hypothesis NOUN cana-3155 92 4 k k PROPN cana-3155 92 5 is be AUX cana-3155 92 6 injective injective ADJ cana-3155 92 7 then then ADV cana-3155 92 8 according accord VERB cana-3155 92 9 to to ADP cana-3155 92 10 theorem theorem ADJ cana-3155 92 11 3.5 3.5 NUM cana-3155 92 12 . . PUNCT cana-3155 92 13 of of ADP cana-3155 92 14 [ [ X cana-3155 92 15 9 9 NUM cana-3155 92 16 ] ] PUNCT cana-3155 92 17 , , PUNCT cana-3155 92 18 k k PROPN cana-3155 92 19 is be AUX cana-3155 92 20 noetherian noetherian ADJ cana-3155 92 21 and and CCONJ cana-3155 92 22 therefore therefore ADV cana-3155 92 23 m m PROPN cana-3155 92 24 is be AUX cana-3155 92 25 an an DET cana-3155 92 26 sf sf NOUN cana-3155 92 27 - - PUNCT cana-3155 92 28 module module NOUN cana-3155 92 29 . . PUNCT cana-3155 93 1 β—» β—» PROPN cana-3155 93 2 3 3 X cana-3155 93 3 . . X cana-3155 93 4 characterization characterization NOUN cana-3155 93 5 of of ADP cana-3155 93 6 sf sf NOUN cana-3155 93 7 - - PUNCT cana-3155 93 8 modules module NOUN cana-3155 93 9 definition definition NOUN cana-3155 93 10 3.1 3.1 NUM cana-3155 93 11 . . PUNCT cana-3155 94 1 let let VERB cana-3155 94 2 m m PRON cana-3155 94 3 be be AUX cana-3155 94 4 an an DET cana-3155 94 5 r r NOUN cana-3155 94 6 - - PUNCT cana-3155 94 7 module module NOUN cana-3155 94 8 . . PUNCT cana-3155 95 1 a a DET cana-3155 95 2 module module NOUN cana-3155 95 3 n n CCONJ cana-3155 95 4 in in ADP cana-3155 95 5 Οƒ[m Οƒ[m ADJ cana-3155 95 6 ] ] PUNCT cana-3155 95 7 is be AUX cana-3155 95 8 semiperfect semiperfect ADJ cana-3155 95 9 in in ADP cana-3155 95 10 Οƒ[m Οƒ[m X cana-3155 95 11 ] ] PUNCT cana-3155 95 12 if if SCONJ cana-3155 95 13 every every DET cana-3155 95 14 factor factor NOUN cana-3155 95 15 module module NOUN cana-3155 95 16 of of ADP cana-3155 95 17 n n PROPN cana-3155 95 18 has have VERB cana-3155 95 19 a a DET cana-3155 95 20 projective projective ADJ cana-3155 95 21 cover cover NOUN cana-3155 95 22 in in ADP cana-3155 95 23 Οƒ[m Οƒ[m NOUN cana-3155 95 24 ] ] PUNCT cana-3155 95 25 . . PUNCT cana-3155 96 1 definition definition NOUN cana-3155 96 2 3.2 3.2 NUM cana-3155 96 3 . . PUNCT cana-3155 97 1 n n PRON cana-3155 97 2 is be AUX cana-3155 97 3 perfect perfect ADJ cana-3155 97 4 in in ADP cana-3155 97 5 Οƒ[m Οƒ[m X cana-3155 97 6 ] ] PUNCT cana-3155 97 7 if if SCONJ cana-3155 97 8 , , PUNCT cana-3155 97 9 for for ADP cana-3155 97 10 every every DET cana-3155 97 11 index index NOUN cana-3155 97 12 set set VERB cana-3155 97 13 Ξ» Ξ» PROPN cana-3155 97 14 , , PUNCT cana-3155 97 15 the the DET cana-3155 97 16 sum sum NOUN cana-3155 97 17 n n CCONJ cana-3155 97 18 ( ( PUNCT cana-3155 97 19 Ξ» Ξ» X cana-3155 97 20 ) ) PUNCT cana-3155 97 21 is be AUX cana-3155 97 22 semiperfect semiperfect ADJ cana-3155 97 23 in in ADP cana-3155 97 24 Οƒ[m Οƒ[m NOUN cana-3155 97 25 ] ] PUNCT cana-3155 97 26 . . PUNCT cana-3155 98 1 lemma lemma PROPN cana-3155 98 2 3.3 3.3 NUM cana-3155 98 3 . . PUNCT cana-3155 99 1 ( ( PUNCT cana-3155 99 2 see see VERB cana-3155 99 3 43.11 43.11 NUM cana-3155 99 4 in in ADP cana-3155 99 5 [ [ X cana-3155 99 6 11 11 NUM cana-3155 99 7 ] ] NUM cana-3155 99 8 ) ) PUNCT cana-3155 99 9 . . PUNCT cana-3155 100 1 let let VERB cana-3155 100 2 r r PRON cana-3155 100 3 be be AUX cana-3155 100 4 a a DET cana-3155 100 5 commutative commutative ADJ cana-3155 100 6 ring ring NOUN cana-3155 100 7 , , PUNCT cana-3155 100 8 m m VERB cana-3155 100 9 a a DET cana-3155 100 10 finitely finitely ADV cana-3155 100 11 generated generate VERB cana-3155 100 12 , , PUNCT cana-3155 100 13 self self NOUN cana-3155 100 14 - - PUNCT cana-3155 100 15 projective projective ADJ cana-3155 100 16 r r NOUN cana-3155 100 17 - - PUNCT cana-3155 100 18 module module NOUN cana-3155 100 19 . . PUNCT cana-3155 101 1 then then ADV cana-3155 101 2 the the DET cana-3155 101 3 following follow VERB cana-3155 101 4 statements statement NOUN cana-3155 101 5 are be AUX cana-3155 101 6 equivalent equivalent ADJ cana-3155 101 7 : : PUNCT cana-3155 101 8 1 1 X cana-3155 101 9 . . X cana-3155 101 10 m m VERB cana-3155 101 11 perfect perfect ADJ cana-3155 101 12 in in ADP cana-3155 101 13 Οƒ[m Οƒ[m X cana-3155 101 14 ] ] X cana-3155 101 15 2 2 NUM cana-3155 101 16 . . X cana-3155 101 17 οΏ½ οΏ½ NOUN cana-3155 101 18 Μ… Μ… NOUN cana-3155 101 19 οΏ½ οΏ½ NOUN cana-3155 101 20 = = SYM cana-3155 101 21 r r NOUN cana-3155 101 22 / / SYM cana-3155 101 23 an(m an(m NOUN cana-3155 101 24 ) ) PUNCT cana-3155 101 25 is be AUX cana-3155 101 26 a a DET cana-3155 101 27 perfect perfect ADJ cana-3155 101 28 ring ring NOUN cana-3155 101 29 . . PUNCT cana-3155 102 1 theorem theorem VERB cana-3155 102 2 3.4 3.4 NUM cana-3155 102 3 . . PUNCT cana-3155 103 1 let let VERB cana-3155 103 2 r r PRON cana-3155 103 3 be be AUX cana-3155 103 4 a a DET cana-3155 103 5 commutative commutative ADJ cana-3155 103 6 ring ring NOUN cana-3155 103 7 , , PUNCT cana-3155 103 8 m m VERB cana-3155 103 9 a a DET cana-3155 103 10 finitely finitely ADV cana-3155 103 11 generated generate VERB cana-3155 103 12 , , PUNCT cana-3155 103 13 self self NOUN cana-3155 103 14 - - PUNCT cana-3155 103 15 projective projective ADJ cana-3155 103 16 r r NOUN cana-3155 103 17 - - PUNCT cana-3155 103 18 module module NOUN cana-3155 103 19 . . PUNCT cana-3155 104 1 then then ADV cana-3155 104 2 the the DET cana-3155 104 3 following follow VERB cana-3155 104 4 statements statement NOUN cana-3155 104 5 are be AUX cana-3155 104 6 equivalent equivalent ADJ cana-3155 104 7 : : PUNCT cana-3155 104 8 1 1 X cana-3155 104 9 . . X cana-3155 104 10 m m PROPN cana-3155 104 11 is be AUX cana-3155 104 12 a a DET cana-3155 104 13 sf sf NOUN cana-3155 104 14 - - PUNCT cana-3155 104 15 module module NOUN cana-3155 104 16 ; ; PUNCT cana-3155 104 17 2 2 X cana-3155 104 18 . . X cana-3155 104 19 m m VERB cana-3155 104 20 perfect perfect ADJ cana-3155 104 21 in in ADP cana-3155 104 22 Οƒ[m Οƒ[m NOUN cana-3155 104 23 ] ] PUNCT cana-3155 104 24 ; ; PUNCT cana-3155 104 25 3 3 X cana-3155 104 26 . . X cana-3155 105 1 all all DET cana-3155 105 2 m m PROPN cana-3155 105 3 - - PUNCT cana-3155 105 4 generated generate VERB cana-3155 105 5 flat flat ADJ cana-3155 105 6 module module NOUN cana-3155 105 7 in in ADP cana-3155 105 8 Οƒ[m Οƒ[m ADJ cana-3155 105 9 ] ] PUNCT cana-3155 105 10 is be AUX cana-3155 105 11 projective projective ADJ cana-3155 105 12 in in ADP cana-3155 105 13 Οƒ[m Οƒ[m NOUN cana-3155 105 14 ] ] PUNCT cana-3155 105 15 . . PUNCT cana-3155 106 1 proof proof NOUN cana-3155 106 2 . . PUNCT cana-3155 107 1 according accord VERB cana-3155 107 2 to to ADP cana-3155 107 3 43.8 43.8 NUM cana-3155 107 4 in in ADP cana-3155 107 5 [ [ X cana-3155 107 6 11 11 NUM cana-3155 107 7 ] ] PUNCT cana-3155 107 8 , , PUNCT cana-3155 107 9 we we PRON cana-3155 107 10 have have VERB cana-3155 107 11 the the DET cana-3155 107 12 equivalence equivalence NOUN cana-3155 107 13 of of ADP cana-3155 107 14 assertions assertion NOUN cana-3155 107 15 ( ( PUNCT cana-3155 107 16 2 2 NUM cana-3155 107 17 ) ) PUNCT cana-3155 107 18 and and CCONJ cana-3155 107 19 ( ( PUNCT cana-3155 107 20 3 3 NUM cana-3155 107 21 ) ) PUNCT cana-3155 107 22 . . PUNCT cana-3155 108 1 now now ADV cana-3155 108 2 let let VERB cana-3155 108 3 ’s ’s NOUN cana-3155 108 4 prove prove VERB cana-3155 108 5 that that SCONJ cana-3155 108 6 1 1 X cana-3155 108 7 ) ) PUNCT cana-3155 108 8 is be AUX cana-3155 108 9 equivalent equivalent ADJ cana-3155 108 10 to to ADP cana-3155 108 11 2 2 NUM cana-3155 108 12 ) ) PUNCT cana-3155 108 13 . . PUNCT cana-3155 109 1 1)⟹ 1)⟹ NUM cana-3155 109 2 2 2 NUM cana-3155 109 3 ): ): PUNCT cana-3155 109 4 m m AUX cana-3155 109 5 finitely finitely ADV cana-3155 109 6 generated generate VERB cana-3155 109 7 sf sf NOUN cana-3155 109 8 - - PUNCT cana-3155 109 9 module module NOUN cana-3155 109 10 implies imply VERB cana-3155 109 11 Οƒ[m Οƒ[m X cana-3155 109 12 ] ] X cana-3155 109 13 = = SYM cana-3155 109 14 r r X cana-3155 109 15 / / SYM cana-3155 109 16 an(m)-mod an(m)-mod NOUN cana-3155 109 17 and and CCONJ cana-3155 109 18 m m PROPN cana-3155 109 19 β‰… β‰… PROPN cana-3155 109 20 r r NOUN cana-3155 109 21 / / SYM cana-3155 109 22 an(m an(m NOUN cana-3155 109 23 ) ) PUNCT cana-3155 109 24 is be AUX cana-3155 109 25 an an DET cana-3155 109 26 artinian artinian ADJ cana-3155 109 27 principal principal NOUN cana-3155 109 28 ideal ideal PROPN cana-3155 109 29 ring ring NOUN cana-3155 109 30 . . PUNCT cana-3155 110 1 since since SCONJ cana-3155 110 2 every every DET cana-3155 110 3 artinian artinian ADJ cana-3155 110 4 ring ring NOUN cana-3155 110 5 is be AUX cana-3155 110 6 perfect perfect ADJ cana-3155 110 7 ring ring NOUN cana-3155 110 8 then then ADV cana-3155 110 9 m m VERB cana-3155 110 10 is be AUX cana-3155 110 11 perfect perfect ADJ cana-3155 110 12 and and CCONJ cana-3155 110 13 so so ADV cana-3155 110 14 r r NOUN cana-3155 110 15 / / SYM cana-3155 110 16 an(m an(m NUM cana-3155 110 17 ) ) PUNCT cana-3155 111 1 is be AUX cana-3155 111 2 a a DET cana-3155 111 3 perfect perfect ADJ cana-3155 111 4 ring ring NOUN cana-3155 111 5 . . PUNCT cana-3155 112 1 referring refer VERB cana-3155 112 2 to to ADP cana-3155 112 3 43.11 43.11 NUM cana-3155 112 4 in in ADP cana-3155 112 5 [ [ X cana-3155 112 6 11 11 NUM cana-3155 112 7 ] ] PUNCT cana-3155 112 8 , , PUNCT cana-3155 112 9 m m VERB cana-3155 112 10 is be AUX cana-3155 112 11 perfect perfect ADJ cana-3155 112 12 in in ADP cana-3155 112 13 Οƒ[m Οƒ[m NOUN cana-3155 112 14 ] ] PUNCT cana-3155 112 15 . . PUNCT cana-3155 113 1 2)⟹ 2)⟹ NUM cana-3155 113 2 1 1 NUM cana-3155 113 3 ) ) PUNCT cana-3155 113 4 if if SCONJ cana-3155 113 5 m m NOUN cana-3155 113 6 perfect perfect ADJ cana-3155 113 7 in in ADP cana-3155 113 8 Οƒ[m Οƒ[m X cana-3155 113 9 ] ] PUNCT cana-3155 113 10 then then ADV cana-3155 113 11 by by ADP cana-3155 113 12 43.11 43.11 NUM cana-3155 113 13 of of ADP cana-3155 113 14 [ [ X cana-3155 113 15 11 11 NUM cana-3155 113 16 ] ] PUNCT cana-3155 113 17 , , PUNCT cana-3155 113 18 r r NOUN cana-3155 113 19 / / SYM cana-3155 113 20 an(m an(m NOUN cana-3155 113 21 ) ) PUNCT cana-3155 113 22 is be AUX cana-3155 113 23 a a DET cana-3155 113 24 perfect perfect ADJ cana-3155 113 25 ring ring NOUN cana-3155 113 26 . . PUNCT cana-3155 114 1 since since SCONJ cana-3155 114 2 every every DET cana-3155 114 3 perfect perfect ADJ cana-3155 114 4 communications communication NOUN cana-3155 114 5 on on ADP cana-3155 114 6 applied apply VERB cana-3155 114 7 nonlinear nonlinear ADJ cana-3155 114 8 analysis analysis NOUN cana-3155 114 9 issn issn NOUN cana-3155 114 10 : : PUNCT cana-3155 114 11 1074 1074 NUM cana-3155 114 12 - - PUNCT cana-3155 114 13 133x 133x NUM cana-3155 114 14 vol vol NOUN cana-3155 114 15 32 32 NUM cana-3155 114 16 no no NOUN cana-3155 114 17 . . PUNCT cana-3155 115 1 5s 5s NUM cana-3155 115 2 ( ( PUNCT cana-3155 115 3 2025 2025 NUM cana-3155 115 4 ) ) PUNCT cana-3155 115 5 491 491 NUM cana-3155 115 6 https://internationalpubls.com https://internationalpubls.com X cana-3155 115 7 ring ring NOUN cana-3155 115 8 is be AUX cana-3155 115 9 semiperfect semiperfect ADJ cana-3155 115 10 and and CCONJ cana-3155 115 11 every every DET cana-3155 115 12 semiperfect semiperfect ADJ cana-3155 115 13 ring ring NOUN cana-3155 115 14 is be AUX cana-3155 115 15 semilocal semilocal ADJ cana-3155 115 16 , , PUNCT cana-3155 115 17 then then ADV cana-3155 115 18 r r PROPN cana-3155 115 19 / / SYM cana-3155 115 20 ann(m ann(m NOUN cana-3155 115 21 ) ) PUNCT cana-3155 115 22 is be AUX cana-3155 115 23 a a DET cana-3155 115 24 semilocal semilocal ADJ cana-3155 115 25 ring ring NOUN cana-3155 115 26 . . PUNCT cana-3155 116 1 by by ADP cana-3155 116 2 theorem theorem NOUN cana-3155 116 3 3.2 3.2 NUM cana-3155 116 4 in in ADP cana-3155 116 5 [ [ X cana-3155 116 6 4 4 NUM cana-3155 116 7 ] ] PUNCT cana-3155 116 8 , , PUNCT cana-3155 116 9 we we PRON cana-3155 116 10 deduce deduce VERB cana-3155 116 11 that that SCONJ cana-3155 116 12 r r NOUN cana-3155 116 13 / / SYM cana-3155 116 14 ann(m ann(m NOUN cana-3155 116 15 ) ) PUNCT cana-3155 116 16 is be AUX cana-3155 116 17 an an DET cana-3155 116 18 sf sf NOUN cana-3155 116 19 - - PUNCT cana-3155 116 20 ring ring NOUN cana-3155 116 21 . . PUNCT cana-3155 117 1 as as SCONJ cana-3155 117 2 m m PROPN cana-3155 117 3 is be AUX cana-3155 117 4 finitely finitely ADV cana-3155 117 5 generated generate VERB cana-3155 117 6 Οƒ[m Οƒ[m NOUN cana-3155 117 7 ] ] X cana-3155 117 8 = = SYM cana-3155 117 9 r r NOUN cana-3155 117 10 / / SYM cana-3155 117 11 ann(m)-mod ann(m)-mod PUNCT cana-3155 117 12 and and CCONJ cana-3155 117 13 so so ADV cana-3155 117 14 every every DET cana-3155 117 15 module module NOUN cana-3155 117 16 in in ADP cana-3155 117 17 Οƒ[m Οƒ[m ADJ cana-3155 117 18 ] ] PUNCT cana-3155 117 19 is be AUX cana-3155 117 20 a a DET cana-3155 117 21 r r NOUN cana-3155 117 22 / / SYM cana-3155 117 23 ann(m)-module ann(m)-module NOUN cana-3155 117 24 . . PUNCT cana-3155 118 1 if if SCONJ cana-3155 118 2 n n PRON cana-3155 118 3 is be AUX cana-3155 118 4 a a DET cana-3155 118 5 strongly strongly ADV cana-3155 118 6 hopfian hopfian ADJ cana-3155 118 7 module module NOUN cana-3155 118 8 in in ADP cana-3155 118 9 Οƒ[m Οƒ[m X cana-3155 118 10 ] ] PUNCT cana-3155 118 11 then then ADV cana-3155 118 12 n n PRON cana-3155 118 13 is be AUX cana-3155 118 14 noetherian noetherian ADJ cana-3155 118 15 because because SCONJ cana-3155 118 16 r r NOUN cana-3155 118 17 / / SYM cana-3155 118 18 ann(m ann(m NOUN cana-3155 118 19 ) ) PUNCT cana-3155 118 20 is be AUX cana-3155 118 21 an an DET cana-3155 118 22 sf sf NOUN cana-3155 118 23 - - PUNCT cana-3155 118 24 ring ring NOUN cana-3155 118 25 and and CCONJ cana-3155 118 26 n n NOUN cana-3155 118 27 is be AUX cana-3155 118 28 a a DET cana-3155 118 29 module module NOUN cana-3155 118 30 of of ADP cana-3155 118 31 r r PROPN cana-3155 118 32 / / SYM cana-3155 118 33 ann(m ann(m PROPN cana-3155 118 34 ) ) PUNCT cana-3155 118 35 . . PUNCT cana-3155 119 1 therefore therefore ADV cana-3155 119 2 m m PROPN cana-3155 119 3 is be AUX cana-3155 119 4 an an DET cana-3155 119 5 sf sf NOUN cana-3155 119 6 - - PUNCT cana-3155 119 7 module module NOUN cana-3155 119 8 . . PUNCT cana-3155 120 1 β—» β—» PROPN cana-3155 120 2 nb nb PROPN cana-3155 120 3 : : PUNCT cana-3155 120 4 we we PRON cana-3155 120 5 denote denote VERB cana-3155 120 6 by by ADP cana-3155 120 7 max(m max(m PROPN cana-3155 120 8 ) ) PUNCT cana-3155 120 9 , , PUNCT cana-3155 120 10 the the DET cana-3155 120 11 set set NOUN cana-3155 120 12 of of ADP cana-3155 120 13 maximal maximal ADJ cana-3155 120 14 submodules submodule NOUN cana-3155 120 15 of of ADP cana-3155 120 16 a a DET cana-3155 120 17 module module NOUN cana-3155 120 18 m. m. NOUN cana-3155 120 19 corollary corollary NOUN cana-3155 120 20 3.5 3.5 NUM cana-3155 120 21 . . PUNCT cana-3155 121 1 let let AUX cana-3155 121 2 r r PRON cana-3155 121 3 be be AUX cana-3155 121 4 a a DET cana-3155 121 5 commutative commutative ADJ cana-3155 121 6 ring ring NOUN cana-3155 121 7 and and CCONJ cana-3155 121 8 m m VERB cana-3155 121 9 a a DET cana-3155 121 10 self self NOUN cana-3155 121 11 - - PUNCT cana-3155 121 12 projective projective ADJ cana-3155 121 13 hollow hollow ADJ cana-3155 121 14 module module NOUN cana-3155 121 15 and and CCONJ cana-3155 121 16 max(m max(m PROPN cana-3155 121 17 ) ) PUNCT cana-3155 121 18 β‰  β‰  PROPN cana-3155 121 19 βˆ…. βˆ…. VERB cana-3155 121 20 if if SCONJ cana-3155 121 21 m m NOUN cana-3155 121 22 is be AUX cana-3155 121 23 a a DET cana-3155 121 24 sf sf NOUN cana-3155 121 25 - - PUNCT cana-3155 121 26 module module NOUN cana-3155 121 27 , , PUNCT cana-3155 121 28 then then ADV cana-3155 121 29 s s PART cana-3155 121 30 = = SYM cana-3155 121 31 endr(m endr(m PROPN cana-3155 121 32 ) ) PUNCT cana-3155 121 33 satisfies satisfy VERB cana-3155 121 34 the the DET cana-3155 121 35 descending descend VERB cana-3155 121 36 chain chain NOUN cana-3155 121 37 conditions condition NOUN cana-3155 121 38 for for ADP cana-3155 121 39 cyclic cyclic ADJ cana-3155 121 40 ideals ideal NOUN cana-3155 121 41 . . PUNCT cana-3155 122 1 proof proof NOUN cana-3155 122 2 . . PUNCT cana-3155 123 1 assume assume VERB cana-3155 123 2 m m PRON cana-3155 123 3 a a DET cana-3155 123 4 projective projective ADJ cana-3155 123 5 hollow hollow ADJ cana-3155 123 6 module module NOUN cana-3155 123 7 and and CCONJ cana-3155 123 8 max(m max(m PROPN cana-3155 123 9 ) ) PUNCT cana-3155 123 10 β‰  β‰  PROPN cana-3155 123 11 βˆ… βˆ… NOUN cana-3155 123 12 then then ADV cana-3155 123 13 according accord VERB cana-3155 123 14 to to ADP cana-3155 123 15 theorem theorem VERB cana-3155 123 16 2.2 2.2 NUM cana-3155 123 17 of of ADP cana-3155 123 18 [ [ X cana-3155 123 19 2 2 NUM cana-3155 123 20 ] ] PUNCT cana-3155 123 21 , , PUNCT cana-3155 123 22 m m PROPN cana-3155 123 23 is be AUX cana-3155 123 24 a a DET cana-3155 123 25 finitely finitely ADV cana-3155 123 26 generated generate VERB cana-3155 123 27 local local ADJ cana-3155 123 28 module module NOUN cana-3155 123 29 . . PUNCT cana-3155 124 1 then then ADV cana-3155 124 2 m m PROPN cana-3155 124 3 is be AUX cana-3155 124 4 finitely finitely ADV cana-3155 124 5 generated generate VERB cana-3155 124 6 self self NOUN cana-3155 124 7 - - PUNCT cana-3155 124 8 projective projective ADJ cana-3155 124 9 . . PUNCT cana-3155 125 1 hence hence ADV cana-3155 125 2 if if SCONJ cana-3155 125 3 m m NOUN cana-3155 125 4 is be AUX cana-3155 125 5 a a DET cana-3155 125 6 sf sf NOUN cana-3155 125 7 - - PUNCT cana-3155 125 8 module module NOUN cana-3155 125 9 then then ADV cana-3155 125 10 by by ADP cana-3155 125 11 theorem theorem NOUN cana-3155 125 12 3.4 3.4 NUM cana-3155 125 13 . . PUNCT cana-3155 126 1 m m VERB cana-3155 126 2 perfect perfect ADJ cana-3155 126 3 in in ADP cana-3155 126 4 Οƒ[m Οƒ[m X cana-3155 126 5 ] ] PUNCT cana-3155 126 6 and and CCONJ cana-3155 126 7 referring refer VERB cana-3155 126 8 to to ADP cana-3155 126 9 43.4 43.4 NUM cana-3155 126 10 of of ADP cana-3155 126 11 [ [ X cana-3155 126 12 11 11 NUM cana-3155 126 13 ] ] PUNCT cana-3155 126 14 , , PUNCT cana-3155 126 15 s s NOUN cana-3155 126 16 = = X cana-3155 126 17 endr(m endr(m PROPN cana-3155 126 18 ) ) PUNCT cana-3155 126 19 satisfies satisfy VERB cana-3155 126 20 the the DET cana-3155 126 21 descending descend VERB cana-3155 126 22 chain chain NOUN cana-3155 126 23 conditions condition NOUN cana-3155 126 24 for for ADP cana-3155 126 25 cyclic cyclic ADJ cana-3155 126 26 ideals ideal NOUN cana-3155 126 27 . . PUNCT cana-3155 127 1 β—» β—» PROPN cana-3155 127 2 definition definition NOUN cana-3155 127 3 3.6 3.6 NUM cana-3155 127 4 . . PUNCT cana-3155 128 1 a a DET cana-3155 128 2 module module NOUN cana-3155 128 3 m m VERB cana-3155 128 4 is be AUX cana-3155 128 5 called call VERB cana-3155 128 6 semiartinian semiartinian ADJ cana-3155 128 7 if if SCONJ cana-3155 128 8 every every DET cana-3155 128 9 nonzero nonzero ADJ cana-3155 128 10 homomorphic homomorphic ADJ cana-3155 128 11 image image NOUN cana-3155 128 12 of of ADP cana-3155 128 13 m m PROPN cana-3155 128 14 has have VERB cana-3155 128 15 nonzero nonzero PROPN cana-3155 128 16 socle socle NOUN cana-3155 128 17 . . PUNCT cana-3155 129 1 definition definition NOUN cana-3155 129 2 3.7 3.7 NUM cana-3155 129 3 . . PUNCT cana-3155 130 1 a a DET cana-3155 130 2 module module NOUN cana-3155 130 3 m m VERB cana-3155 130 4 is be AUX cana-3155 130 5 called call VERB cana-3155 130 6 Ο€ Ο€ NOUN cana-3155 130 7 - - NOUN cana-3155 130 8 semiartinian semiartinian ADJ cana-3155 130 9 if if SCONJ cana-3155 130 10 the the DET cana-3155 130 11 direct direct ADJ cana-3155 130 12 product product NOUN cana-3155 130 13 mi mi PROPN cana-3155 130 14 is be AUX cana-3155 130 15 a a DET cana-3155 130 16 semiartinian semiartinian ADJ cana-3155 130 17 module module NOUN cana-3155 130 18 for for ADP cana-3155 130 19 every every DET cana-3155 130 20 non non ADJ cana-3155 130 21 empty empty ADJ cana-3155 130 22 set set ADJ cana-3155 130 23 i. i. NOUN cana-3155 130 24 definition definition NOUN cana-3155 130 25 3.8 3.8 NUM cana-3155 130 26 . . PUNCT cana-3155 131 1 the the DET cana-3155 131 2 ring ring NOUN cana-3155 131 3 r r NOUN cana-3155 131 4 is be AUX cana-3155 131 5 called call VERB cana-3155 131 6 strongly strongly ADV cana-3155 131 7 Ο€ Ο€ NOUN cana-3155 131 8 - - NOUN cana-3155 131 9 regular regular ADJ cana-3155 131 10 if if SCONJ cana-3155 131 11 for for ADP cana-3155 131 12 each each DET cana-3155 131 13 a a DET cana-3155 131 14 ∈ ∈ PROPN cana-3155 131 15 r r NOUN cana-3155 131 16 , , PUNCT cana-3155 131 17 there there PRON cana-3155 131 18 is be VERB cana-3155 131 19 an an DET cana-3155 131 20 integer integer NOUN cana-3155 131 21 n n PRON cana-3155 131 22 β‰₯ β‰₯ NOUN cana-3155 131 23 1 1 NUM cana-3155 131 24 and and CCONJ cana-3155 131 25 b b X cana-3155 131 26 ∈ ∈ NOUN cana-3155 131 27 r r NOUN cana-3155 131 28 such such ADJ cana-3155 131 29 that that SCONJ cana-3155 131 30 an an DET cana-3155 131 31 = = PUNCT cana-3155 131 32 an+1b an+1b NOUN cana-3155 131 33 . . PUNCT cana-3155 132 1 m m PROPN cana-3155 132 2 is be AUX cana-3155 132 3 called call VERB cana-3155 132 4 fitting fitting ADJ cana-3155 132 5 module module NOUN cana-3155 132 6 if if SCONJ cana-3155 132 7 every every DET cana-3155 132 8 endomorphism endomorphism NOUN cana-3155 132 9 of of ADP cana-3155 132 10 m m PROPN cana-3155 132 11 satifies satifie NOUN cana-3155 132 12 fitting fitting PROPN cana-3155 132 13 ’s ’s PART cana-3155 132 14 lemma lemma PROPN cana-3155 132 15 ( ( PUNCT cana-3155 132 16 i.e. i.e. X cana-3155 132 17 , , PUNCT cana-3155 132 18 there there PRON cana-3155 132 19 exists exist VERB cana-3155 132 20 an an DET cana-3155 132 21 integer integer NOUN cana-3155 132 22 n n PRON cana-3155 132 23 β‰₯ β‰₯ NOUN cana-3155 132 24 1 1 NUM cana-3155 132 25 such such ADJ cana-3155 132 26 that that SCONJ cana-3155 132 27 m m NOUN cana-3155 132 28 = = VERB cana-3155 132 29 kerf kerf NOUN cana-3155 132 30 n n PROPN cana-3155 132 31 βŠ• βŠ• PROPN cana-3155 132 32 imf imf PROPN cana-3155 132 33 n n PRON cana-3155 132 34 ) ) PUNCT cana-3155 132 35 . . PUNCT cana-3155 133 1 theorem theorem ADJ cana-3155 133 2 3.9 3.9 NUM cana-3155 133 3 . . PUNCT cana-3155 134 1 let let VERB cana-3155 134 2 r r PRON cana-3155 134 3 be be AUX cana-3155 134 4 a a DET cana-3155 134 5 ring ring NOUN cana-3155 134 6 and and CCONJ cana-3155 134 7 m m DET cana-3155 134 8 a a DET cana-3155 134 9 finitely finitely ADV cana-3155 134 10 generated generate VERB cana-3155 134 11 r r NOUN cana-3155 134 12 - - PUNCT cana-3155 134 13 module module NOUN cana-3155 134 14 . . PUNCT cana-3155 135 1 if if SCONJ cana-3155 135 2 m m PROPN cana-3155 135 3 is be AUX cana-3155 135 4 sf sf NOUN cana-3155 135 5 - - PUNCT cana-3155 135 6 module module NOUN cana-3155 135 7 then then ADV cana-3155 135 8 the the DET cana-3155 135 9 following following ADJ cana-3155 135 10 statements statement NOUN cana-3155 135 11 are be AUX cana-3155 135 12 equivalent equivalent ADJ cana-3155 135 13 : : PUNCT cana-3155 135 14 1 1 X cana-3155 135 15 . . X cana-3155 135 16 m m PROPN cana-3155 135 17 is be AUX cana-3155 135 18 artinian artinian ADJ cana-3155 135 19 ; ; PUNCT cana-3155 135 20 2 2 X cana-3155 135 21 . . X cana-3155 135 22 m m PROPN cana-3155 135 23 is be AUX cana-3155 135 24 semiartinian semiartinian ADJ cana-3155 135 25 module module NOUN cana-3155 135 26 ; ; PUNCT cana-3155 135 27 3 3 X cana-3155 135 28 . . X cana-3155 136 1 every every DET cana-3155 136 2 module module NOUN cana-3155 136 3 in in ADP cana-3155 136 4 Οƒ[m Οƒ[m ADJ cana-3155 136 5 ] ] PUNCT cana-3155 136 6 is be AUX cana-3155 136 7 semiartinian semiartinian ADJ cana-3155 136 8 ; ; PUNCT cana-3155 136 9 4 4 X cana-3155 136 10 . . X cana-3155 136 11 m m PROPN cana-3155 136 12 is be AUX cana-3155 136 13 Ο€ Ο€ ADJ cana-3155 136 14 - - ADJ cana-3155 136 15 semiartinian semiartinian ADJ cana-3155 136 16 module module NOUN cana-3155 136 17 ; ; PUNCT cana-3155 136 18 5 5 X cana-3155 136 19 . . X cana-3155 136 20 m m PROPN cana-3155 136 21 is be AUX cana-3155 136 22 noetherian noetherian ADJ cana-3155 136 23 . . PUNCT cana-3155 137 1 proof proof NOUN cana-3155 137 2 . . PUNCT cana-3155 138 1 1)⟹ 1)⟹ NUM cana-3155 138 2 2 2 NUM cana-3155 138 3 ): ): PUNCT cana-3155 138 4 it it PRON cana-3155 138 5 ’s ’ VERB cana-3155 138 6 obvious obvious ADJ cana-3155 138 7 . . PUNCT cana-3155 139 1 2)⇔ 2)⇔ NUM cana-3155 140 1 3 3 NUM cana-3155 140 2 ): ): PUNCT cana-3155 140 3 if if SCONJ cana-3155 140 4 m m NOUN cana-3155 140 5 is be AUX cana-3155 140 6 semiartinian semiartinian ADJ cana-3155 140 7 module module NOUN cana-3155 140 8 , , PUNCT cana-3155 140 9 then then ADV cana-3155 140 10 by by ADP cana-3155 140 11 corollary corollary ADJ cana-3155 140 12 2.13 2.13 NUM cana-3155 140 13 . . PUNCT cana-3155 141 1 in in ADP cana-3155 141 2 [ [ X cana-3155 141 3 8 8 NUM cana-3155 141 4 ] ] PUNCT cana-3155 141 5 , , PUNCT cana-3155 141 6 r r NOUN cana-3155 141 7 / / SYM cana-3155 141 8 an(m an(m NOUN cana-3155 141 9 ) ) PUNCT cana-3155 141 10 is be AUX cana-3155 141 11 a a DET cana-3155 141 12 semiartinian semiartinian ADJ cana-3155 141 13 ring ring NOUN cana-3155 141 14 . . PUNCT cana-3155 142 1 let let VERB cana-3155 142 2 n n PRON cana-3155 142 3 an an DET cana-3155 142 4 object object NOUN cana-3155 142 5 of of ADP cana-3155 142 6 Οƒ[m Οƒ[m ADJ cana-3155 142 7 ] ] PUNCT cana-3155 142 8 then then ADV cana-3155 142 9 since since SCONJ cana-3155 142 10 m m PROPN cana-3155 142 11 is be AUX cana-3155 142 12 finitely finitely ADV cana-3155 142 13 generated generate VERB cana-3155 142 14 Οƒ[m Οƒ[m NOUN cana-3155 142 15 ] ] X cana-3155 142 16 = = SYM cana-3155 142 17 r r NOUN cana-3155 142 18 / / SYM cana-3155 142 19 ann(m)-mod ann(m)-mod PRON cana-3155 142 20 and and CCONJ cana-3155 142 21 therefore therefore ADV cana-3155 142 22 n n PRON cana-3155 142 23 is be AUX cana-3155 142 24 a a DET cana-3155 142 25 module module NOUN cana-3155 142 26 r r NOUN cana-3155 142 27 / / SYM cana-3155 142 28 ann(m)-module ann(m)-module NOUN cana-3155 142 29 . . PUNCT cana-3155 143 1 it it PRON cana-3155 143 2 is be AUX cana-3155 143 3 well well ADV cana-3155 143 4 know know VERB cana-3155 143 5 a a DET cana-3155 143 6 ring ring NOUN cana-3155 143 7 r r NOUN cana-3155 143 8 is be AUX cana-3155 143 9 semiartinian semiartinian ADJ cana-3155 144 1 if if SCONJ cana-3155 144 2 and and CCONJ cana-3155 144 3 only only ADV cana-3155 144 4 if if SCONJ cana-3155 144 5 every every DET cana-3155 144 6 r r NOUN cana-3155 144 7 - - PUNCT cana-3155 144 8 module module NOUN cana-3155 144 9 is be AUX cana-3155 144 10 semiartinian semiartinian ADJ cana-3155 144 11 . . PUNCT cana-3155 145 1 since since SCONJ cana-3155 145 2 r r NOUN cana-3155 145 3 / / SYM cana-3155 145 4 ann(m ann(m NOUN cana-3155 145 5 ) ) PUNCT cana-3155 145 6 is be AUX cana-3155 145 7 a a DET cana-3155 145 8 semiartinian semiartinian ADJ cana-3155 145 9 ring ring NOUN cana-3155 145 10 then then ADV cana-3155 145 11 every every DET cana-3155 145 12 r r NOUN cana-3155 145 13 / / SYM cana-3155 145 14 ann(m)-module ann(m)-module NOUN cana-3155 145 15 is be AUX cana-3155 145 16 semiartinian semiartinian ADJ cana-3155 145 17 and and CCONJ cana-3155 145 18 hence hence ADV cana-3155 145 19 n n PRON cana-3155 145 20 is be AUX cana-3155 145 21 semiartinian semiartinian ADJ cana-3155 145 22 . . PUNCT cana-3155 146 1 the the DET cana-3155 146 2 converse converse NOUN cana-3155 146 3 is be AUX cana-3155 146 4 trivial trivial ADJ cana-3155 146 5 . . PUNCT cana-3155 147 1 2)⇔ 2)⇔ NUM cana-3155 148 1 4 4 X cana-3155 148 2 ) ) PUNCT cana-3155 148 3 result result NOUN cana-3155 148 4 from from ADP cana-3155 148 5 corollary corollary ADJ cana-3155 148 6 3.3 3.3 NUM cana-3155 148 7 . . PUNCT cana-3155 149 1 of of ADP cana-3155 149 2 [ [ X cana-3155 149 3 8 8 NUM cana-3155 149 4 ] ] PUNCT cana-3155 149 5 communications communication NOUN cana-3155 149 6 on on ADP cana-3155 149 7 applied apply VERB cana-3155 149 8 nonlinear nonlinear ADJ cana-3155 149 9 analysis analysis NOUN cana-3155 149 10 issn issn NOUN cana-3155 149 11 : : PUNCT cana-3155 149 12 1074 1074 NUM cana-3155 149 13 - - PUNCT cana-3155 149 14 133x 133x NUM cana-3155 149 15 vol vol NOUN cana-3155 149 16 32 32 NUM cana-3155 149 17 no no NOUN cana-3155 149 18 . . PUNCT cana-3155 150 1 5s 5s NUM cana-3155 150 2 ( ( PUNCT cana-3155 150 3 2025 2025 NUM cana-3155 150 4 ) ) PUNCT cana-3155 150 5 492 492 NUM cana-3155 151 1 https://internationalpubls.com https://internationalpubls.com X cana-3155 151 2 2)⟹ 2)⟹ NUM cana-3155 151 3 5 5 NUM cana-3155 151 4 ) ) PUNCT cana-3155 151 5 if if SCONJ cana-3155 151 6 m m NOUN cana-3155 151 7 is be AUX cana-3155 151 8 semiartinian semiartinian ADJ cana-3155 151 9 then then ADV cana-3155 151 10 according accord VERB cana-3155 151 11 to to ADP cana-3155 151 12 corollary corollary NOUN cana-3155 151 13 2.13 2.13 NUM cana-3155 151 14 . . PUNCT cana-3155 152 1 in in ADP cana-3155 152 2 [ [ X cana-3155 152 3 8 8 NUM cana-3155 152 4 ] ] PUNCT cana-3155 152 5 , , PUNCT cana-3155 152 6 endr(m endr(m PROPN cana-3155 152 7 ) ) PUNCT cana-3155 152 8 is be AUX cana-3155 152 9 a a DET cana-3155 152 10 strongly strongly ADV cana-3155 152 11 Ο€ Ο€ ADJ cana-3155 152 12 - - ADJ cana-3155 152 13 regular regular ADJ cana-3155 152 14 ring ring NOUN cana-3155 152 15 . . PUNCT cana-3155 153 1 therefore therefore ADV cana-3155 153 2 by by ADP cana-3155 153 3 proposition proposition NOUN cana-3155 153 4 2.7 2.7 NUM cana-3155 153 5 of of ADP cana-3155 153 6 [ [ X cana-3155 153 7 6 6 NUM cana-3155 153 8 ] ] PUNCT cana-3155 153 9 , , PUNCT cana-3155 153 10 m m VERB cana-3155 153 11 is be AUX cana-3155 153 12 fitting fitting ADJ cana-3155 153 13 module module NOUN cana-3155 153 14 and and CCONJ cana-3155 153 15 so so ADV cana-3155 153 16 an an DET cana-3155 153 17 strongly strongly ADV cana-3155 153 18 hopfian hopfian ADJ cana-3155 153 19 module module NOUN cana-3155 153 20 . . PUNCT cana-3155 154 1 since since SCONJ cana-3155 154 2 by by ADP cana-3155 154 3 hypothesis hypothesis NOUN cana-3155 154 4 m m NOUN cana-3155 154 5 is be AUX cana-3155 154 6 a a DET cana-3155 154 7 sf sf NOUN cana-3155 154 8 - - PUNCT cana-3155 154 9 module module NOUN cana-3155 154 10 so so ADV cana-3155 154 11 m m VERB cana-3155 154 12 is be AUX cana-3155 154 13 noetherian noetherian ADJ cana-3155 154 14 . . PUNCT cana-3155 155 1 5)⟹ 5)⟹ NUM cana-3155 155 2 1 1 NUM cana-3155 155 3 ) ) PUNCT cana-3155 155 4 m m AUX cana-3155 155 5 finitely finitely ADV cana-3155 155 6 generated generate VERB cana-3155 155 7 and and CCONJ cana-3155 155 8 sf sf NOUN cana-3155 155 9 - - PUNCT cana-3155 155 10 module module NOUN cana-3155 155 11 implies imply VERB cana-3155 155 12 m m VERB cana-3155 155 13 β‰… β‰… ADJ cana-3155 155 14 r r NOUN cana-3155 155 15 / / SYM cana-3155 155 16 an(m an(m NOUN cana-3155 155 17 ) ) PUNCT cana-3155 155 18 is be AUX cana-3155 155 19 artinian artinian ADJ cana-3155 155 20 principal principal ADJ cana-3155 155 21 ideal ideal NOUN cana-3155 155 22 and and CCONJ cana-3155 155 23 over over ADP cana-3155 155 24 artinian artinian ADJ cana-3155 155 25 principal principal ADJ cana-3155 155 26 ideal ideal NOUN cana-3155 155 27 ring ring NOUN cana-3155 155 28 noetherian noetherian ADJ cana-3155 155 29 module module NOUN cana-3155 155 30 and and CCONJ cana-3155 155 31 artinian artinian ADJ cana-3155 155 32 module module NOUN cana-3155 155 33 coincide coincide NOUN cana-3155 155 34 . . PUNCT cana-3155 156 1 β—» β—» PROPN cana-3155 156 2 lemma lemma PROPN cana-3155 156 3 3.10 3.10 NUM cana-3155 156 4 . . PUNCT cana-3155 157 1 ( ( PUNCT cana-3155 157 2 see see VERB cana-3155 157 3 proposition proposition NOUN cana-3155 157 4 14 14 NUM cana-3155 157 5 of of ADP cana-3155 157 6 hollow hollow ADJ cana-3155 157 7 and and CCONJ cana-3155 157 8 semihollow semihollow NOUN cana-3155 157 9 modules module NOUN cana-3155 157 10 ) ) PUNCT cana-3155 157 11 . . PUNCT cana-3155 158 1 let let VERB cana-3155 158 2 n n PRON cana-3155 158 3 be be AUX cana-3155 158 4 a a DET cana-3155 158 5 proper proper ADJ cana-3155 158 6 submodule submodule NOUN cana-3155 158 7 of of ADP cana-3155 158 8 a a DET cana-3155 158 9 module module NOUN cana-3155 158 10 m. m. NOUN cana-3155 158 11 if if SCONJ cana-3155 158 12 m m NOUN cana-3155 158 13 is be AUX cana-3155 158 14 a a DET cana-3155 158 15 hollow hollow ADJ cana-3155 158 16 module module NOUN cana-3155 158 17 and and CCONJ cana-3155 158 18 m m PROPN cana-3155 158 19 / / SYM cana-3155 158 20 n n PROPN cana-3155 158 21 is be AUX cana-3155 158 22 finitely finitely ADV cana-3155 158 23 generated generate VERB cana-3155 158 24 , , PUNCT cana-3155 158 25 then then ADV cana-3155 158 26 m m VERB cana-3155 158 27 is be AUX cana-3155 158 28 finitely finitely ADV cana-3155 158 29 generated generate VERB cana-3155 158 30 . . PUNCT cana-3155 159 1 theorem theorem PROPN cana-3155 159 2 3.11 3.11 NUM cana-3155 159 3 . . PUNCT cana-3155 160 1 let let VERB cana-3155 160 2 r r PRON cana-3155 160 3 be be AUX cana-3155 160 4 a a DET cana-3155 160 5 commutative commutative ADJ cana-3155 160 6 ring ring NOUN cana-3155 160 7 and and CCONJ cana-3155 160 8 m m VERB cana-3155 160 9 a a DET cana-3155 160 10 hollow hollow ADJ cana-3155 160 11 module module NOUN cana-3155 160 12 . . PUNCT cana-3155 161 1 we we PRON cana-3155 161 2 suppose suppose VERB cana-3155 161 3 that that SCONJ cana-3155 161 4 for for SCONJ cana-3155 161 5 every every DET cana-3155 161 6 proper proper ADJ cana-3155 161 7 submodule submodule NOUN cana-3155 161 8 n n PROPN cana-3155 161 9 of of ADP cana-3155 161 10 m m PROPN cana-3155 161 11 , , PUNCT cana-3155 161 12 m m PROPN cana-3155 161 13 / / SYM cana-3155 161 14 n n PROPN cana-3155 161 15 is be AUX cana-3155 161 16 finitely finitely ADV cana-3155 161 17 generated generate VERB cana-3155 161 18 . . PUNCT cana-3155 162 1 then then ADV cana-3155 162 2 the the DET cana-3155 162 3 following follow VERB cana-3155 162 4 conditions condition NOUN cana-3155 162 5 are be AUX cana-3155 162 6 equivalent equivalent ADJ cana-3155 162 7 : : PUNCT cana-3155 162 8 1 1 X cana-3155 162 9 . . X cana-3155 162 10 m m PROPN cana-3155 162 11 is be AUX cana-3155 162 12 a a DET cana-3155 162 13 sf sf NOUN cana-3155 162 14 - - PUNCT cana-3155 162 15 module module NOUN cana-3155 162 16 ; ; PUNCT cana-3155 162 17 2 2 X cana-3155 162 18 . . X cana-3155 162 19 m m PROPN cana-3155 162 20 is be AUX cana-3155 162 21 a a DET cana-3155 162 22 locally locally ADV cana-3155 162 23 noetherian noetherian ADJ cana-3155 162 24 module module NOUN cana-3155 162 25 ; ; PUNCT cana-3155 162 26 3 3 X cana-3155 162 27 . . X cana-3155 162 28 m m PROPN cana-3155 162 29 is be AUX cana-3155 162 30 noetherian noetherian ADJ cana-3155 162 31 module module NOUN cana-3155 162 32 ; ; PUNCT cana-3155 162 33 proof proof NOUN cana-3155 162 34 . . PUNCT cana-3155 163 1 1)⟹ 1)⟹ NUM cana-3155 163 2 2 2 NUM cana-3155 163 3 ): ): PUNCT cana-3155 163 4 let let VERB cana-3155 163 5 m m PRON cana-3155 163 6 be be AUX cana-3155 163 7 a a DET cana-3155 163 8 sf sf NOUN cana-3155 163 9 - - PUNCT cana-3155 163 10 module module NOUN cana-3155 163 11 then then ADV cana-3155 163 12 by by ADP cana-3155 163 13 remark remark NOUN cana-3155 163 14 2.4 2.4 NUM cana-3155 163 15 . . PUNCT cana-3155 164 1 m m PROPN cana-3155 164 2 is be AUX cana-3155 164 3 a a DET cana-3155 164 4 ekfn ekfn NOUN cana-3155 164 5 - - PUNCT cana-3155 164 6 module module NOUN cana-3155 164 7 . . PUNCT cana-3155 165 1 by by ADP cana-3155 165 2 hypothesis hypothesis NOUN cana-3155 165 3 , , PUNCT cana-3155 165 4 it it PRON cana-3155 165 5 results result VERB cana-3155 165 6 from from ADP cana-3155 165 7 lemma lemma PROPN cana-3155 165 8 3.10 3.10 NUM cana-3155 165 9 . . PUNCT cana-3155 166 1 that that SCONJ cana-3155 166 2 m m PROPN cana-3155 166 3 is be AUX cana-3155 166 4 finitely finitely ADV cana-3155 166 5 generated generate VERB cana-3155 166 6 . . PUNCT cana-3155 167 1 hence hence ADV cana-3155 167 2 by by ADP cana-3155 167 3 theorem theorem NOUN cana-3155 167 4 3 3 NUM cana-3155 167 5 . . NOUN cana-3155 167 6 of of ADP cana-3155 167 7 [ [ X cana-3155 167 8 5 5 NUM cana-3155 167 9 ] ] PUNCT cana-3155 167 10 , , PUNCT cana-3155 167 11 m m VERB cana-3155 167 12 is be AUX cana-3155 167 13 a a DET cana-3155 167 14 locally locally ADV cana-3155 167 15 noetherian noetherian ADJ cana-3155 167 16 module module NOUN cana-3155 167 17 . . PUNCT cana-3155 168 1 2 2 X cana-3155 168 2 ) ) PUNCT cana-3155 168 3 ⇔ ⇔ X cana-3155 168 4 3 3 NUM cana-3155 168 5 ) ) PUNCT cana-3155 168 6 result result NOUN cana-3155 168 7 from from ADP cana-3155 168 8 corollary corollary ADJ cana-3155 168 9 2.3 2.3 NUM cana-3155 168 10 . . PUNCT cana-3155 169 1 in in ADP cana-3155 169 2 [ [ X cana-3155 169 3 7 7 X cana-3155 169 4 ] ] PUNCT cana-3155 169 5 now now ADV cana-3155 169 6 we we PRON cana-3155 169 7 prove prove VERB cana-3155 169 8 that that SCONJ cana-3155 169 9 2 2 NUM cana-3155 169 10 ) ) PUNCT cana-3155 169 11 ⟹ ⟹ NUM cana-3155 169 12 1 1 NUM cana-3155 169 13 ): ): PUNCT cana-3155 169 14 let let VERB cana-3155 169 15 n n PRON cana-3155 169 16 ∈ ∈ PROPN cana-3155 169 17 Οƒ[m Οƒ[m PART cana-3155 169 18 ] ] X cana-3155 169 19 a a DET cana-3155 169 20 strongly strongly ADV cana-3155 169 21 hopfian hopfian ADJ cana-3155 169 22 module module NOUN cana-3155 169 23 . . PUNCT cana-3155 170 1 since since SCONJ cana-3155 170 2 m m PROPN cana-3155 170 3 is be AUX cana-3155 170 4 locally locally ADV cana-3155 170 5 noetherian noetherian ADJ cana-3155 170 6 then then ADV cana-3155 170 7 according accord VERB cana-3155 170 8 to to ADP cana-3155 170 9 corollary corollary ADJ cana-3155 170 10 2.3 2.3 NUM cana-3155 170 11 . . PUNCT cana-3155 171 1 in in ADP cana-3155 171 2 [ [ X cana-3155 171 3 7 7 NUM cana-3155 171 4 ] ] PUNCT cana-3155 171 5 , , PUNCT cana-3155 171 6 r r NOUN cana-3155 171 7 / / SYM cana-3155 171 8 ann(m ann(m NOUN cana-3155 171 9 ) ) PUNCT cana-3155 171 10 is be AUX cana-3155 171 11 a a DET cana-3155 171 12 noetherian noetherian ADJ cana-3155 171 13 ring ring NOUN cana-3155 171 14 . . PUNCT cana-3155 172 1 since since SCONJ cana-3155 172 2 m m PROPN cana-3155 172 3 is be AUX cana-3155 172 4 finitely finitely ADV cana-3155 172 5 generated generate VERB cana-3155 172 6 Οƒ[m Οƒ[m NOUN cana-3155 172 7 ] ] X cana-3155 172 8 = = SYM cana-3155 172 9 r r NOUN cana-3155 172 10 / / SYM cana-3155 172 11 ann(m)-mod ann(m)-mod PRON cana-3155 172 12 and and CCONJ cana-3155 172 13 m m PROPN cana-3155 172 14 β‰… β‰… PROPN cana-3155 172 15 r r PROPN cana-3155 172 16 / / SYM cana-3155 172 17 ann(m ann(m NOUN cana-3155 172 18 ) ) PUNCT cana-3155 172 19 is be AUX cana-3155 172 20 finitely finitely ADV cana-3155 172 21 generated generate VERB cana-3155 172 22 and and CCONJ cana-3155 172 23 noetherian noetherian ADJ cana-3155 172 24 . . PUNCT cana-3155 173 1 so so ADV cana-3155 173 2 n n PRON cana-3155 173 3 ∈ ∈ PROPN cana-3155 173 4 Οƒ[m Οƒ[m NOUN cana-3155 173 5 ] ] PUNCT cana-3155 173 6 implies imply VERB cana-3155 173 7 that that SCONJ cana-3155 173 8 n n X cana-3155 173 9 is be AUX cana-3155 173 10 an an DET cana-3155 173 11 ideal ideal NOUN cana-3155 173 12 of of ADP cana-3155 173 13 r r NOUN cana-3155 173 14 / / SYM cana-3155 173 15 ann(m ann(m NOUN cana-3155 173 16 ) ) PUNCT cana-3155 173 17 and and CCONJ cana-3155 173 18 therefore therefore ADV cana-3155 173 19 a a DET cana-3155 173 20 submodule submodule NOUN cana-3155 173 21 of of ADP cana-3155 173 22 m. m. NOUN cana-3155 173 23 it it PRON cana-3155 173 24 ’s ’ VERB cana-3155 173 25 well well ADV cana-3155 173 26 know know VERB cana-3155 173 27 over over ADP cana-3155 173 28 noetherian noetherian ADJ cana-3155 173 29 ring ring NOUN cana-3155 173 30 , , PUNCT cana-3155 173 31 every every DET cana-3155 173 32 submodule submodule NOUN cana-3155 173 33 of of ADP cana-3155 173 34 finitely finitely ADV cana-3155 173 35 generated generate VERB cana-3155 173 36 module module NOUN cana-3155 173 37 is be AUX cana-3155 173 38 finitely finitely ADV cana-3155 173 39 generated generate VERB cana-3155 173 40 . . PUNCT cana-3155 174 1 hence hence ADV cana-3155 174 2 n n ADV cana-3155 174 3 is be AUX cana-3155 174 4 noetherian noetherian ADJ cana-3155 174 5 because because SCONJ cana-3155 174 6 over over ADP cana-3155 174 7 noetherian noetherian ADJ cana-3155 174 8 ring ring NOUN cana-3155 174 9 , , PUNCT cana-3155 174 10 finitely finitely ADV cana-3155 174 11 generated generate VERB cana-3155 174 12 and and CCONJ cana-3155 174 13 noetherian noetherian ADJ cana-3155 174 14 module module NOUN cana-3155 174 15 coincide coincide NOUN cana-3155 174 16 . . PUNCT cana-3155 175 1 β—» β—» PROPN cana-3155 175 2 corollary corollary ADJ cana-3155 175 3 3.12 3.12 NUM cana-3155 175 4 . . PUNCT cana-3155 176 1 let let VERB cana-3155 176 2 r r PRON cana-3155 176 3 be be AUX cana-3155 176 4 a a DET cana-3155 176 5 commutative commutative ADJ cana-3155 176 6 ring ring NOUN cana-3155 176 7 and and CCONJ cana-3155 176 8 m m VERB cana-3155 176 9 a a DET cana-3155 176 10 hollow hollow ADJ cana-3155 176 11 module module NOUN cana-3155 176 12 . . PUNCT cana-3155 177 1 we we PRON cana-3155 177 2 suppose suppose VERB cana-3155 177 3 that that SCONJ cana-3155 177 4 for for SCONJ cana-3155 177 5 every every DET cana-3155 177 6 proper proper ADJ cana-3155 177 7 submodule submodule NOUN cana-3155 177 8 n n PROPN cana-3155 177 9 of of ADP cana-3155 177 10 m m PROPN cana-3155 177 11 , , PUNCT cana-3155 177 12 m m PROPN cana-3155 177 13 / / SYM cana-3155 177 14 n n PROPN cana-3155 177 15 is be AUX cana-3155 177 16 finitely finitely ADV cana-3155 177 17 generated generate VERB cana-3155 177 18 . . PUNCT cana-3155 178 1 then then ADV cana-3155 178 2 the the DET cana-3155 178 3 following follow VERB cana-3155 178 4 conditions condition NOUN cana-3155 178 5 are be AUX cana-3155 178 6 equivalent equivalent ADJ cana-3155 178 7 : : PUNCT cana-3155 178 8 1 1 X cana-3155 178 9 . . X cana-3155 178 10 m m PROPN cana-3155 178 11 is be AUX cana-3155 178 12 a a DET cana-3155 178 13 sf sf NOUN cana-3155 178 14 - - PUNCT cana-3155 178 15 module module NOUN cana-3155 178 16 , , PUNCT cana-3155 178 17 2 2 NUM cana-3155 178 18 . . X cana-3155 179 1 every every DET cana-3155 179 2 finitely finitely ADV cana-3155 179 3 generated generate VERB cana-3155 179 4 module module NOUN cana-3155 179 5 in in ADP cana-3155 179 6 Οƒ[m Οƒ[m ADJ cana-3155 179 7 ] ] PUNCT cana-3155 179 8 is be AUX cana-3155 179 9 noetherian noetherian ADJ cana-3155 179 10 . . PUNCT cana-3155 180 1 3 3 X cana-3155 180 2 . . X cana-3155 180 3 every every DET cana-3155 180 4 finitely finitely ADV cana-3155 180 5 generated generate VERB cana-3155 180 6 module module NOUN cana-3155 180 7 is be AUX cana-3155 180 8 finitely finitely ADV cana-3155 180 9 presented present VERB cana-3155 180 10 in in ADP cana-3155 180 11 Οƒ[m Οƒ[m NOUN cana-3155 180 12 ] ] PUNCT cana-3155 180 13 . . PUNCT cana-3155 181 1 4 4 X cana-3155 181 2 . . X cana-3155 181 3 every every DET cana-3155 181 4 direct direct ADJ cana-3155 181 5 sum sum NOUN cana-3155 181 6 of of ADP cana-3155 181 7 m m ADJ cana-3155 181 8 - - ADJ cana-3155 181 9 injective injective ADJ cana-3155 181 10 module module NOUN cana-3155 181 11 in in ADP cana-3155 181 12 Οƒ[m Οƒ[m ADJ cana-3155 181 13 ] ] PUNCT cana-3155 181 14 is be AUX cana-3155 181 15 m m NOUN cana-3155 181 16 - - PUNCT cana-3155 181 17 injective injective ADJ cana-3155 181 18 . . PUNCT cana-3155 182 1 proof proof NOUN cana-3155 182 2 . . PUNCT cana-3155 183 1 by by ADP cana-3155 183 2 hypothesis hypothesis NOUN cana-3155 183 3 , , PUNCT cana-3155 183 4 it it PRON cana-3155 183 5 results result VERB cana-3155 183 6 from from ADP cana-3155 183 7 lemma lemma PROPN cana-3155 183 8 3.10 3.10 NUM cana-3155 183 9 . . PUNCT cana-3155 184 1 that that PRON cana-3155 184 2 , , PUNCT cana-3155 184 3 m m VERB cana-3155 184 4 is be AUX cana-3155 184 5 finitely finitely ADV cana-3155 184 6 generated generate VERB cana-3155 184 7 and and CCONJ cana-3155 184 8 according accord VERB cana-3155 184 9 to to ADP cana-3155 184 10 the the DET cana-3155 184 11 theorem theorem NOUN cana-3155 184 12 3.11 3.11 NUM cana-3155 184 13 . . PUNCT cana-3155 185 1 m m PROPN cana-3155 185 2 is be AUX cana-3155 185 3 a a DET cana-3155 185 4 sf sf NOUN cana-3155 185 5 - - PUNCT cana-3155 185 6 module module NOUN cana-3155 185 7 if if SCONJ cana-3155 185 8 and and CCONJ cana-3155 185 9 only only ADV cana-3155 185 10 if if SCONJ cana-3155 185 11 , , PUNCT cana-3155 185 12 m m VERB cana-3155 185 13 is be AUX cana-3155 185 14 a a DET cana-3155 185 15 locally locally ADV cana-3155 185 16 noetherian noetherian ADJ cana-3155 185 17 module module NOUN cana-3155 185 18 ; ; PUNCT cana-3155 185 19 and and CCONJ cana-3155 185 20 referring refer VERB cana-3155 185 21 to to ADP cana-3155 185 22 27.3 27.3 NUM cana-3155 185 23 of of ADP cana-3155 185 24 [ [ X cana-3155 185 25 11 11 NUM cana-3155 185 26 ] ] PUNCT cana-3155 185 27 , , PUNCT cana-3155 185 28 we we PRON cana-3155 185 29 have have VERB cana-3155 185 30 the the DET cana-3155 185 31 result result NOUN cana-3155 185 32 . . PUNCT cana-3155 186 1 β—» β—» PROPN cana-3155 186 2 theorem theorem VERB cana-3155 186 3 4 4 X cana-3155 186 4 . . PUNCT cana-3155 187 1 let let VERB cana-3155 187 2 m m PRON cana-3155 187 3 be be AUX cana-3155 187 4 a a DET cana-3155 187 5 local local ADJ cana-3155 187 6 r r NOUN cana-3155 187 7 - - PUNCT cana-3155 187 8 module module NOUN cana-3155 187 9 , , PUNCT cana-3155 187 10 then then ADV cana-3155 187 11 the the DET cana-3155 187 12 following following NOUN cana-3155 187 13 are be AUX cana-3155 187 14 equivalent equivalent ADJ cana-3155 187 15 : : PUNCT cana-3155 187 16 1 1 X cana-3155 187 17 . . X cana-3155 187 18 m m PROPN cana-3155 187 19 is be AUX cana-3155 187 20 a a DET cana-3155 187 21 s s NOUN cana-3155 187 22 - - NOUN cana-3155 187 23 module module NOUN cana-3155 187 24 ; ; PUNCT cana-3155 187 25 2 2 X cana-3155 187 26 . . X cana-3155 187 27 m m PROPN cana-3155 187 28 is be AUX cana-3155 187 29 a a DET cana-3155 187 30 sf sf NOUN cana-3155 187 31 - - PUNCT cana-3155 187 32 module module NOUN cana-3155 187 33 ; ; PUNCT cana-3155 187 34 3 3 X cana-3155 187 35 . . X cana-3155 187 36 m m PROPN cana-3155 187 37 is be AUX cana-3155 187 38 of of ADP cana-3155 187 39 finite finite ADJ cana-3155 187 40 length length NOUN cana-3155 187 41 and and CCONJ cana-3155 187 42 every every DET cana-3155 187 43 submodule submodule NOUN cana-3155 187 44 of of ADP cana-3155 187 45 m m PROPN cana-3155 187 46 is be AUX cana-3155 187 47 cyclic cyclic ADJ cana-3155 187 48 ; ; PUNCT cana-3155 187 49 4 4 X cana-3155 187 50 . . X cana-3155 188 1 m m PROPN cana-3155 188 2 is be AUX cana-3155 188 3 of of ADP cana-3155 188 4 finite finite ADJ cana-3155 188 5 representation representation NOUN cana-3155 188 6 type type NOUN cana-3155 188 7 ; ; PUNCT cana-3155 188 8 communications communication NOUN cana-3155 188 9 on on ADP cana-3155 188 10 applied apply VERB cana-3155 188 11 nonlinear nonlinear ADJ cana-3155 188 12 analysis analysis NOUN cana-3155 188 13 issn issn NOUN cana-3155 188 14 : : PUNCT cana-3155 188 15 1074 1074 NUM cana-3155 188 16 - - PUNCT cana-3155 188 17 133x 133x NUM cana-3155 188 18 vol vol NOUN cana-3155 188 19 32 32 NUM cana-3155 188 20 no no NOUN cana-3155 188 21 . . PUNCT cana-3155 189 1 5s 5s NUM cana-3155 189 2 ( ( PUNCT cana-3155 189 3 2025 2025 NUM cana-3155 189 4 ) ) PUNCT cana-3155 189 5 493 493 NUM cana-3155 189 6 https://internationalpubls.com https://internationalpubls.com X cana-3155 189 7 5 5 NUM cana-3155 189 8 . . PUNCT cana-3155 190 1 m m PROPN cana-3155 190 2 is be AUX cana-3155 190 3 fgs fgs NOUN cana-3155 190 4 - - PUNCT cana-3155 190 5 module module NOUN cana-3155 190 6 ; ; PUNCT cana-3155 190 7 proof proof NOUN cana-3155 190 8 . . PUNCT cana-3155 191 1 1 1 NUM cana-3155 191 2 ) ) PUNCT cana-3155 191 3 ⟹ ⟹ NUM cana-3155 191 4 2 2 NUM cana-3155 191 5 ) ) PUNCT cana-3155 191 6 . . PUNCT cana-3155 192 1 result result VERB cana-3155 192 2 from from ADP cana-3155 192 3 remark remark NOUN cana-3155 192 4 2.4 2.4 NUM cana-3155 192 5 . . NOUN cana-3155 192 6 2 2 NUM cana-3155 192 7 ) ) PUNCT cana-3155 192 8 ⇔ ⇔ X cana-3155 192 9 3 3 NUM cana-3155 192 10 ) ) PUNCT cana-3155 192 11 ⇔ ⇔ X cana-3155 192 12 4 4 NUM cana-3155 192 13 ) ) PUNCT cana-3155 192 14 since since SCONJ cana-3155 192 15 m m PROPN cana-3155 192 16 is be AUX cana-3155 192 17 local local ADJ cana-3155 192 18 sf sf NOUN cana-3155 192 19 - - PUNCT cana-3155 192 20 module module NOUN cana-3155 192 21 then then ADV cana-3155 192 22 m m VERB cana-3155 192 23 is be AUX cana-3155 192 24 a a DET cana-3155 192 25 finite finite NOUN cana-3155 192 26 generated generate VERB cana-3155 192 27 sf sf NOUN cana-3155 192 28 - - PUNCT cana-3155 192 29 module module NOUN cana-3155 192 30 . . PUNCT cana-3155 193 1 by by ADP cana-3155 193 2 lemma lemma PROPN cana-3155 193 3 2.1 2.1 NUM cana-3155 193 4 . . PUNCT cana-3155 194 1 and and CCONJ cana-3155 194 2 lemma lemma PROPN cana-3155 194 3 3.3 3.3 NUM cana-3155 194 4 . . PUNCT cana-3155 195 1 m m PROPN cana-3155 195 2 is be AUX cana-3155 195 3 isomorphic isomorphic ADJ cana-3155 195 4 to to ADP cana-3155 195 5 r r NOUN cana-3155 195 6 / / SYM cana-3155 195 7 ann(m ann(m NOUN cana-3155 195 8 ) ) PUNCT cana-3155 195 9 who who PRON cana-3155 195 10 is be AUX cana-3155 195 11 a a DET cana-3155 195 12 principal principal ADJ cana-3155 195 13 ideal ideal ADJ cana-3155 195 14 ring ring NOUN cana-3155 195 15 . . PUNCT cana-3155 196 1 this this DET cana-3155 196 2 double double ADJ cana-3155 196 3 equivalence equivalence NOUN cana-3155 196 4 result result NOUN cana-3155 196 5 from from ADP cana-3155 196 6 theorem theorem NOUN cana-3155 196 7 9 9 NUM cana-3155 196 8 in in ADP cana-3155 196 9 [ [ X cana-3155 196 10 10 10 NUM cana-3155 196 11 ] ] PUNCT cana-3155 196 12 . . PUNCT cana-3155 197 1 4 4 NUM cana-3155 197 2 ) ) PUNCT cana-3155 197 3 ⟹ ⟹ NUM cana-3155 197 4 5 5 NUM cana-3155 197 5 ) ) PUNCT cana-3155 197 6 result result NOUN cana-3155 197 7 from from ADP cana-3155 197 8 theorem theorem NOUN cana-3155 197 9 1 1 NUM cana-3155 197 10 in in ADP cana-3155 197 11 [ [ X cana-3155 197 12 3 3 NUM cana-3155 197 13 ] ] PUNCT cana-3155 197 14 . . PUNCT cana-3155 198 1 5 5 NUM cana-3155 198 2 ) ) PUNCT cana-3155 198 3 ⟹ ⟹ NUM cana-3155 198 4 1 1 NUM cana-3155 198 5 ) ) PUNCT cana-3155 198 6 let let VERB cana-3155 198 7 n n PRON cana-3155 198 8 be be AUX cana-3155 198 9 a a DET cana-3155 198 10 hopfian hopfian ADJ cana-3155 198 11 module module NOUN cana-3155 198 12 in in ADP cana-3155 198 13 Οƒ[m Οƒ[m NOUN cana-3155 198 14 ] ] PUNCT cana-3155 198 15 . . PUNCT cana-3155 199 1 since since SCONJ cana-3155 199 2 m m PROPN cana-3155 199 3 is be AUX cana-3155 199 4 a a DET cana-3155 199 5 fgs fgs NOUN cana-3155 199 6 - - PUNCT cana-3155 199 7 module module NOUN cana-3155 199 8 then then ADV cana-3155 199 9 n n CCONJ cana-3155 199 10 is be AUX cana-3155 199 11 finite finite NOUN cana-3155 199 12 generated generate VERB cana-3155 199 13 . . PUNCT cana-3155 200 1 from from ADP cana-3155 200 2 proposition proposition NOUN cana-3155 200 3 3 3 NUM cana-3155 200 4 in in ADP cana-3155 200 5 [ [ X cana-3155 200 6 3 3 X cana-3155 200 7 ] ] X cana-3155 200 8 n n PRON cana-3155 200 9 is be AUX cana-3155 200 10 noetherian noetherian ADJ cana-3155 200 11 . . PUNCT cana-3155 201 1 therefore therefore ADV cana-3155 201 2 m m PROPN cana-3155 201 3 is be AUX cana-3155 201 4 a a DET cana-3155 201 5 s s NOUN cana-3155 201 6 - - NOUN cana-3155 201 7 module module NOUN cana-3155 201 8 . . PUNCT cana-3155 202 1 β—» β—» PROPN cana-3155 202 2 acknowledgements acknowledgement VERB cana-3155 202 3 the the DET cana-3155 202 4 authors author NOUN cana-3155 202 5 would would AUX cana-3155 202 6 like like VERB cana-3155 202 7 to to PART cana-3155 202 8 express express VERB cana-3155 202 9 their their PRON cana-3155 202 10 sincere sincere ADJ cana-3155 202 11 thanks thank NOUN cana-3155 202 12 for for ADP cana-3155 202 13 the the DET cana-3155 202 14 referee referee NOUN cana-3155 202 15 for for ADP cana-3155 202 16 his his PRON cana-3155 202 17 / / SYM cana-3155 202 18 her her PRON cana-3155 202 19 helpful helpful ADJ cana-3155 202 20 suggestions suggestion NOUN cana-3155 202 21 and and CCONJ cana-3155 202 22 comments comment NOUN cana-3155 202 23 . . PUNCT cana-3155 203 1 compliance compliance NOUN cana-3155 203 2 with with ADP cana-3155 203 3 ethical ethical ADJ cana-3155 203 4 standards standard NOUN cana-3155 203 5 conflict conflict NOUN cana-3155 203 6 of of ADP cana-3155 203 7 interest interest NOUN cana-3155 203 8 on on ADP cana-3155 203 9 behalf behalf NOUN cana-3155 203 10 of of ADP cana-3155 203 11 all all DET cana-3155 203 12 authors author NOUN cana-3155 203 13 , , PUNCT cana-3155 203 14 the the DET cana-3155 203 15 corresponding corresponding ADJ cana-3155 203 16 author author NOUN cana-3155 203 17 states state VERB cana-3155 203 18 that that SCONJ cana-3155 203 19 there there PRON cana-3155 203 20 is be VERB cana-3155 203 21 no no DET cana-3155 203 22 conflict conflict NOUN cana-3155 203 23 of of ADP cana-3155 203 24 interest interest NOUN cana-3155 203 25 . . PUNCT cana-3155 204 1 references reference NOUN cana-3155 204 2 [ [ X cana-3155 204 3 1 1 NUM cana-3155 204 4 ] ] PUNCT cana-3155 204 5 f. f. PROPN cana-3155 204 6 w. w. PROPN cana-3155 204 7 anderson anderson PROPN cana-3155 204 8 and and CCONJ cana-3155 204 9 k.r k.r PROPN cana-3155 204 10 . . PROPN cana-3155 204 11 fuller full ADJ cana-3155 204 12 : : PUNCT cana-3155 204 13 rings ring NOUN cana-3155 204 14 and and CCONJ cana-3155 204 15 categories category NOUN cana-3155 204 16 of of ADP cana-3155 204 17 modules module NOUN cana-3155 204 18 , , PUNCT cana-3155 204 19 springer springer NOUN cana-3155 204 20 - - PUNCT cana-3155 204 21 verlag verlag PROPN cana-3155 204 22 , , PUNCT cana-3155 204 23 berlin berlin PROPN cana-3155 204 24 1974 1974 NUM cana-3155 204 25 . . PUNCT cana-3155 205 1 [ [ X cana-3155 205 2 2 2 NUM cana-3155 205 3 ] ] PUNCT cana-3155 205 4 a. a. NOUN cana-3155 205 5 azizi azizi PROPN cana-3155 205 6 : : PUNCT cana-3155 205 7 hollow hollow ADJ cana-3155 205 8 modules module NOUN cana-3155 205 9 over over ADP cana-3155 205 10 commutative commutative ADJ cana-3155 205 11 rings ring NOUN cana-3155 205 12 , , PUNCT cana-3155 205 13 palestine palestine PROPN cana-3155 205 14 journal journal PROPN cana-3155 205 15 of of ADP cana-3155 205 16 mathematics mathematics PROPN cana-3155 205 17 vol vol NOUN cana-3155 205 18 . . PUNCT cana-3155 206 1 3(spec 3(spec PROPN cana-3155 206 2 1 1 NUM cana-3155 206 3 ) ) PUNCT cana-3155 206 4 ( ( PUNCT cana-3155 206 5 2014 2014 NUM cana-3155 206 6 ) ) PUNCT cana-3155 206 7 , , PUNCT cana-3155 206 8 449 449 NUM cana-3155 206 9 – – PUNCT cana-3155 206 10 456 456 NUM cana-3155 206 11 . . PUNCT cana-3155 207 1 [ [ X cana-3155 207 2 3 3 X cana-3155 207 3 ] ] PUNCT cana-3155 207 4 a. a. NOUN cana-3155 207 5 ba ba PROPN cana-3155 207 6 , , PUNCT cana-3155 207 7 a. a. NOUN cana-3155 207 8 m m PROPN cana-3155 207 9 diompy diompy ADJ cana-3155 207 10 , , PUNCT cana-3155 207 11 a. a. NOUN cana-3155 207 12 diouf diouf PROPN cana-3155 207 13 and and CCONJ cana-3155 207 14 a. a. PROPN cana-3155 207 15 s s PROPN cana-3155 207 16 diabang diabang PROPN cana-3155 207 17 : : PUNCT cana-3155 207 18 some some DET cana-3155 207 19 results result NOUN cana-3155 207 20 on on ADP cana-3155 207 21 fgs fgs PROPN cana-3155 207 22 -modules -module NOUN cana-3155 207 23 , , PUNCT cana-3155 207 24 journal journal NOUN cana-3155 207 25 of of ADP cana-3155 207 26 mathematics mathematics PROPN cana-3155 207 27 research research NOUN cana-3155 207 28 ; ; PUNCT cana-3155 207 29 vol vol NOUN cana-3155 207 30 . . NOUN cana-3155 207 31 9 9 NUM cana-3155 207 32 , , PUNCT cana-3155 207 33 no no INTJ cana-3155 207 34 . . NOUN cana-3155 207 35 1 1 NUM cana-3155 207 36 ; ; PUNCT cana-3155 207 37 february february PROPN cana-3155 207 38 2017 2017 NUM cana-3155 207 39 . . PUNCT cana-3155 208 1 [ [ X cana-3155 208 2 4 4 NUM cana-3155 208 3 ] ] X cana-3155 208 4 m.a m.a NOUN cana-3155 208 5 diompy diompy NOUN cana-3155 208 6 , , PUNCT cana-3155 208 7 a.s a.s PROPN cana-3155 208 8 diabang diabang PROPN cana-3155 208 9 , , PUNCT cana-3155 208 10 o. o. NOUN cana-3155 208 11 bousso bousso NOUN cana-3155 208 12 and and CCONJ cana-3155 208 13 r.d r.d PRON cana-3155 208 14 diouf diouf PROPN cana-3155 208 15 : : PUNCT cana-3155 208 16 on on ADP cana-3155 208 17 sf sf NOUN cana-3155 208 18 - - PUNCT cana-3155 208 19 rings ring NOUN cana-3155 208 20 , , PUNCT cana-3155 208 21 international international ADJ cana-3155 208 22 journal journal NOUN cana-3155 208 23 of of ADP cana-3155 208 24 algebra algebra PROPN cana-3155 208 25 , , PUNCT cana-3155 208 26 vol.18 vol.18 NOUN cana-3155 208 27 , , PUNCT cana-3155 208 28 2024 2024 NUM cana-3155 208 29 , , PUNCT cana-3155 208 30 no no DET cana-3155 208 31 1 1 NUM cana-3155 208 32 , , PUNCT cana-3155 208 33 1 1 NUM cana-3155 208 34 - - SYM cana-3155 208 35 10 10 NUM cana-3155 208 36 . . PUNCT cana-3155 209 1 [ [ X cana-3155 209 2 5 5 NUM cana-3155 209 3 ] ] X cana-3155 209 4 m.a m.a NOUN cana-3155 209 5 diompy diompy NOUN cana-3155 209 6 , , PUNCT cana-3155 209 7 o. o. NOUN cana-3155 209 8 bousso bousso NOUN cana-3155 209 9 and and CCONJ cana-3155 209 10 r.d r.d PRON cana-3155 209 11 diouf diouf PROPN cana-3155 209 12 : : PUNCT cana-3155 209 13 ekfn ekfn NOUN cana-3155 209 14 - - PUNCT cana-3155 209 15 modules module NOUN cana-3155 209 16 , , PUNCT cana-3155 209 17 utilitas utilitas PROPN cana-3155 209 18 mathematica mathematica PROPN cana-3155 209 19 , , PUNCT cana-3155 209 20 118 118 NUM cana-3155 209 21 , , PUNCT cana-3155 209 22 27 27 NUM cana-3155 209 23 32 32 NUM cana-3155 209 24 ( ( PUNCT cana-3155 209 25 2024 2024 NUM cana-3155 209 26 ) ) PUNCT cana-3155 209 27 . . PUNCT cana-3155 210 1 [ [ X cana-3155 210 2 6 6 NUM cana-3155 210 3 ] ] PUNCT cana-3155 210 4 a. a. NOUN cana-3155 210 5 hmaimou hmaimou PROPN cana-3155 210 6 , , PUNCT cana-3155 210 7 a. a. NOUN cana-3155 210 8 kaidi kaidi PROPN cana-3155 210 9 and and CCONJ cana-3155 210 10 e. e. PROPN cana-3155 210 11 sanchez sanchez PROPN cana-3155 210 12 campos campos PROPN cana-3155 210 13 : : PUNCT cana-3155 210 14 generalized generalize VERB cana-3155 210 15 fitting fitting ADJ cana-3155 210 16 modules module NOUN cana-3155 210 17 and and CCONJ cana-3155 210 18 rings ring NOUN cana-3155 210 19 , , PUNCT cana-3155 210 20 journal journal NOUN cana-3155 210 21 of of ADP cana-3155 210 22 algebra algebra PROPN cana-3155 210 23 , , PUNCT cana-3155 210 24 308 308 NUM cana-3155 210 25 ( ( PUNCT cana-3155 210 26 2007 2007 NUM cana-3155 210 27 ) ) PUNCT cana-3155 210 28 , , PUNCT cana-3155 210 29 199–214 199–214 NUM cana-3155 210 30 . . PUNCT cana-3155 211 1 [ [ X cana-3155 211 2 7 7 X cana-3155 211 3 ] ] X cana-3155 211 4 f. f. PROPN cana-3155 211 5 kourki kourki PROPN cana-3155 211 6 and and CCONJ cana-3155 211 7 r. r. PROPN cana-3155 211 8 tribak tribak PROPN cana-3155 211 9 : : PUNCT cana-3155 211 10 some some DET cana-3155 211 11 results result VERB cana-3155 211 12 on on ADP cana-3155 211 13 locally locally ADV cana-3155 211 14 noetherian noetherian ADJ cana-3155 211 15 modules module NOUN cana-3155 211 16 and and CCONJ cana-3155 211 17 locally locally ADV cana-3155 211 18 artinian artinian ADJ cana-3155 211 19 modules module NOUN cana-3155 211 20 . . PUNCT cana-3155 212 1 kyungpook kyungpook PROPN cana-3155 212 2 math math PROPN cana-3155 212 3 . . PUNCT cana-3155 213 1 j. j. PROPN cana-3155 213 2 58(2018 58(2018 PROPN cana-3155 213 3 ) ) PUNCT cana-3155 213 4 , , PUNCT cana-3155 213 5 1 1 NUM cana-3155 213 6 - - SYM cana-3155 213 7 8 8 NUM cana-3155 213 8 https://doi.org/10.5666/kmj.2018.58.1.1 https://doi.org/10.5666/kmj.2018.58.1.1 NOUN cana-3155 213 9 pissn pissn PROPN cana-3155 213 10 1225 1225 NUM cana-3155 213 11 - - SYM cana-3155 213 12 6951 6951 NUM cana-3155 213 13 eissn eissn PROPN cana-3155 213 14 0454 0454 NUM cana-3155 213 15 - - SYM cana-3155 213 16 8124 8124 NUM cana-3155 213 17 [ [ X cana-3155 213 18 8 8 NUM cana-3155 213 19 ] ] X cana-3155 213 20 f. f. PROPN cana-3155 213 21 kourki kourki PROPN cana-3155 213 22 and and CCONJ cana-3155 213 23 r. r. PROPN cana-3155 213 24 tribak tribak PROPN cana-3155 213 25 : : PUNCT cana-3155 213 26 on on ADP cana-3155 213 27 semiartinian semiartinian ADJ cana-3155 213 28 and and CCONJ cana-3155 213 29 Ο€ Ο€ ADJ cana-3155 213 30 - - ADJ cana-3155 213 31 semiartinian semiartinian ADJ cana-3155 213 32 modules module NOUN cana-3155 213 33 , , PUNCT cana-3155 213 34 palestine palestine PROPN cana-3155 213 35 journal journal PROPN cana-3155 213 36 of of ADP cana-3155 213 37 mathematics mathematics PROPN cana-3155 213 38 vol vol NOUN cana-3155 213 39 . . PUNCT cana-3155 214 1 7(special 7(special ADJ cana-3155 214 2 issue issue NOUN cana-3155 214 3 : : PUNCT cana-3155 214 4 i i PRON cana-3155 214 5 , , PUNCT cana-3155 214 6 2018 2018 NUM cana-3155 214 7 ) ) PUNCT cana-3155 214 8 , , PUNCT cana-3155 214 9 99–107 99–107 PROPN cana-3155 214 10 . . PUNCT cana-3155 215 1 [ [ X cana-3155 215 2 9 9 NUM cana-3155 215 3 ] ] X cana-3155 215 4 f.c f.c PROPN cana-3155 215 5 leary leary PROPN cana-3155 215 6 : : PUNCT cana-3155 215 7 hopfian hopfian PROPN cana-3155 215 8 and and CCONJ cana-3155 215 9 co co ADJ cana-3155 215 10 - - ADJ cana-3155 215 11 hopfian hopfian ADJ cana-3155 215 12 modules module NOUN cana-3155 215 13 over over ADP cana-3155 215 14 artinian artinian ADJ cana-3155 215 15 rings ring NOUN cana-3155 215 16 , , PUNCT cana-3155 215 17 https://arxiv.org/abs/2112.01596v1 https://arxiv.org/abs/2112.01596v1 X cana-3155 215 18 ( ( PUNCT cana-3155 215 19 2021 2021 NUM cana-3155 215 20 ) ) PUNCT cana-3155 215 21 . . PUNCT cana-3155 216 1 [ [ X cana-3155 216 2 10 10 NUM cana-3155 216 3 ] ] PUNCT cana-3155 216 4 sangare sangare ADP cana-3155 216 5 m. m. NOUN cana-3155 216 6 and and CCONJ cana-3155 216 7 kaidi kaidi NOUN cana-3155 216 8 . . PUNCT cana-3155 217 1 a a DET cana-3155 217 2 : : PUNCT cana-3155 217 3 une une PROPN cana-3155 217 4 caracterisation caracterisation NOUN cana-3155 217 5 des des PROPN cana-3155 217 6 anneaux anneaux PROPN cana-3155 217 7 artiniens artinien NOUN cana-3155 217 8 Γ  Γ  X cana-3155 217 9 idΓ©aux idΓ©aux PROPN cana-3155 217 10 principaux principaux PROPN cana-3155 217 11 . . PUNCT cana-3155 218 1 lect lect PROPN cana-3155 218 2 . . PUNCT cana-3155 219 1 note note NOUN cana-3155 219 2 in in ADP cana-3155 219 3 math math NOUN cana-3155 219 4 , , PUNCT cana-3155 219 5 328 328 NUM cana-3155 219 6 . . PUNCT cana-3155 220 1 springer springer NOUN cana-3155 220 2 - - PUNCT cana-3155 220 3 verlag verlag PROPN cana-3155 220 4 . . PUNCT cana-3155 221 1 [ [ X cana-3155 221 2 11 11 NUM cana-3155 221 3 ] ] X cana-3155 221 4 r. r. PROPN cana-3155 221 5 wisbauer wisbauer NOUN cana-3155 221 6 : : PUNCT cana-3155 221 7 foundations foundation NOUN cana-3155 221 8 of of ADP cana-3155 221 9 modules module NOUN cana-3155 221 10 and and CCONJ cana-3155 221 11 rings ring NOUN cana-3155 221 12 theory theory NOUN cana-3155 221 13 . . PUNCT cana-3155 222 1 gordon gordon PROPN cana-3155 222 2 and and CCONJ cana-3155 222 3 breach breach VERB cana-3155 222 4 science science NOUN cana-3155 222 5 publishers publisher NOUN cana-3155 222 6 . . PUNCT cana-3155 223 1 ( ( PUNCT cana-3155 223 2 1991 1991 NUM cana-3155 223 3 ) ) PUNCT cana-3155 223 4 . . PUNCT