id sid tid token lemma pos cana-727 1 1 communications communication NOUN cana-727 1 2 on on ADP cana-727 1 3 applied apply VERB cana-727 1 4 nonlinear nonlinear ADJ cana-727 1 5 analysis analysis NOUN cana-727 1 6 issn issn NOUN cana-727 1 7 : : PUNCT cana-727 1 8 1074 1074 NUM cana-727 1 9 - - PUNCT cana-727 1 10 133x 133x NUM cana-727 1 11 vol vol NOUN cana-727 1 12 31 31 NUM cana-727 1 13 no no NOUN cana-727 1 14 . . PUNCT cana-727 2 1 3s 3s NUM cana-727 2 2 ( ( PUNCT cana-727 2 3 2024 2024 NUM cana-727 2 4 ) ) PUNCT cana-727 2 5 9 9 NUM cana-727 2 6 https://internationalpubls.com https://internationalpubls.com X cana-727 2 7 an an DET cana-727 2 8 approach approach NOUN cana-727 2 9 to to ADP cana-727 2 10 goldie goldie PROPN cana-727 2 11 extending extend VERB cana-727 2 12 modules module NOUN cana-727 2 13 on on ADP cana-727 2 14 the the DET cana-727 2 15 class class NOUN cana-727 2 16 of of ADP cana-727 2 17 cyclic cyclic ADJ cana-727 2 18 submodules submodule NOUN cana-727 2 19 abdallah abdallah PROPN cana-727 2 20 shihadeh shihadeh PROPN cana-727 2 21 department department PROPN cana-727 2 22 of of ADP cana-727 2 23 mathematics mathematic NOUN cana-727 2 24 , , PUNCT cana-727 2 25 faculty faculty NOUN cana-727 2 26 of of ADP cana-727 2 27 science science NOUN cana-727 2 28 , , PUNCT cana-727 2 29 the the DET cana-727 2 30 hashemite hashemite PROPN cana-727 2 31 university university NOUN cana-727 2 32 , , PUNCT cana-727 2 33 zarqa zarqa NOUN cana-727 2 34 13133 13133 NUM cana-727 2 35 , , PUNCT cana-727 2 36 po po PROPN cana-727 2 37 box box PROPN cana-727 2 38 330127 330127 NUM cana-727 2 39 , , PUNCT cana-727 2 40 jordan jordan PROPN cana-727 2 41 abdallaha_ka@hu.edu.jo abdallaha_ka@hu.edu.jo PROPN cana-727 2 42 article article NOUN cana-727 2 43 history history NOUN cana-727 2 44 : : PUNCT cana-727 2 45 received receive VERB cana-727 2 46 : : PUNCT cana-727 2 47 12 12 NUM cana-727 2 48 - - PUNCT cana-727 2 49 04 04 NUM cana-727 2 50 - - PUNCT cana-727 2 51 2024 2024 NUM cana-727 2 52 revised revise VERB cana-727 2 53 : : PUNCT cana-727 2 54 18 18 NUM cana-727 2 55 - - SYM cana-727 2 56 05 05 NUM cana-727 2 57 - - PUNCT cana-727 2 58 2024 2024 NUM cana-727 2 59 accepted accept VERB cana-727 2 60 : : PUNCT cana-727 2 61 01 01 NUM cana-727 2 62 - - SYM cana-727 2 63 06 06 NUM cana-727 2 64 - - PUNCT cana-727 2 65 2024 2024 NUM cana-727 2 66 abstract abstract NOUN cana-727 2 67 : : PUNCT cana-727 2 68 in in ADP cana-727 2 69 this this DET cana-727 2 70 article article NOUN cana-727 2 71 , , PUNCT cana-727 2 72 we we PRON cana-727 2 73 provide provide VERB cana-727 2 74 a a DET cana-727 2 75 class class NOUN cana-727 2 76 of of ADP cana-727 2 77 modules module NOUN cana-727 2 78 that that PRON cana-727 2 79 is be AUX cana-727 2 80 comparable comparable ADJ cana-727 2 81 to to ADP cana-727 2 82 𝐺𝑧-extending 𝐺𝑧-extende VERB cana-727 2 83 and and CCONJ cana-727 2 84 πΊπ‘Ÿ-extending πΊπ‘Ÿ-extende VERB cana-727 2 85 modules module NOUN cana-727 2 86 . . PUNCT cana-727 3 1 we we PRON cana-727 3 2 specify specify VERB cana-727 3 3 what what PRON cana-727 3 4 a a DET cana-727 3 5 module module NOUN cana-727 3 6 m m NOUN cana-727 3 7 is be AUX cana-727 3 8 as as SCONJ cana-727 3 9 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 3 10 if if SCONJ cana-727 3 11 and and CCONJ cana-727 3 12 only only ADV cana-727 3 13 if if SCONJ cana-727 3 14 for for ADP cana-727 3 15 each each DET cana-727 3 16 cyclic cyclic ADJ cana-727 3 17 submodule submodule NOUN cana-727 3 18 a a PRON cana-727 3 19 of of ADP cana-727 3 20 m m PROPN cana-727 3 21 , , PUNCT cana-727 3 22 there there PRON cana-727 3 23 exists exist VERB cana-727 3 24 a a DET cana-727 3 25 direct direct ADJ cana-727 3 26 summand summand NOUN cana-727 3 27 d d PROPN cana-727 3 28 of of ADP cana-727 3 29 m m PRON cana-727 3 30 such such ADJ cana-727 3 31 that that SCONJ cana-727 3 32 a∩ a∩ PROPN cana-727 3 33 𝐷 𝐷 PROPN cana-727 3 34 is be AUX cana-727 3 35 essential essential ADJ cana-727 3 36 in in ADP cana-727 3 37 both both CCONJ cana-727 3 38 a a PRON cana-727 3 39 and and CCONJ cana-727 3 40 d. d. NOUN cana-727 3 41 we we PRON cana-727 3 42 look look VERB cana-727 3 43 into into ADP cana-727 3 44 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 3 45 modules module NOUN cana-727 3 46 and and CCONJ cana-727 3 47 locate locate VERB cana-727 3 48 this this DET cana-727 3 49 inference inference NOUN cana-727 3 50 between between ADP cana-727 3 51 the the DET cana-727 3 52 other other ADJ cana-727 3 53 extending extend VERB cana-727 3 54 properties property NOUN cana-727 3 55 . . PUNCT cana-727 4 1 we we PRON cana-727 4 2 present present VERB cana-727 4 3 some some PRON cana-727 4 4 of of ADP cana-727 4 5 characterizations characterization NOUN cana-727 4 6 of of ADP cana-727 4 7 𝐺𝑝extending 𝐺𝑝extende VERB cana-727 4 8 condition condition NOUN cana-727 4 9 . . PUNCT cana-727 5 1 we we PRON cana-727 5 2 show show VERB cana-727 5 3 that that SCONJ cana-727 5 4 the the DET cana-727 5 5 direct direct ADJ cana-727 5 6 sum sum NOUN cana-727 5 7 of of ADP cana-727 5 8 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 5 9 need need AUX cana-727 5 10 not not PART cana-727 5 11 be be AUX cana-727 5 12 𝐺𝑝extending 𝐺𝑝extende VERB cana-727 5 13 and and CCONJ cana-727 5 14 deal deal VERB cana-727 5 15 with with ADP cana-727 5 16 decompositions decomposition NOUN cana-727 5 17 for for ADP cana-727 5 18 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 5 19 concept concept NOUN cana-727 5 20 . . PUNCT cana-727 5 21 " " PUNCT cana-727 6 1 keywords keyword NOUN cana-727 6 2 : : PUNCT cana-727 6 3 cyclic cyclic ADJ cana-727 6 4 submodules submodule NOUN cana-727 6 5 , , PUNCT cana-727 6 6 p p NOUN cana-727 6 7 - - PUNCT cana-727 6 8 extending extend VERB cana-727 6 9 modules module NOUN cana-727 6 10 , , PUNCT cana-727 6 11 goldie goldie PROPN cana-727 6 12 extending extend VERB cana-727 6 13 modules module NOUN cana-727 6 14 , , PUNCT cana-727 6 15 𝐺𝑝extending 𝐺𝑝extending NOUN cana-727 6 16 . . PUNCT cana-727 7 1 1 1 X cana-727 7 2 . . X cana-727 7 3 introduction introduction NOUN cana-727 7 4 throughout throughout ADP cana-727 7 5 this this DET cana-727 7 6 paper paper NOUN cana-727 7 7 , , PUNCT cana-727 7 8 all all DET cana-727 7 9 rings ring NOUN cana-727 7 10 are be AUX cana-727 7 11 associative associative ADJ cana-727 7 12 with with ADP cana-727 7 13 unitary unitary ADJ cana-727 7 14 , , PUNCT cana-727 7 15 r r NOUN cana-727 7 16 denotes denote NOUN cana-727 7 17 such such DET cana-727 7 18 a a DET cana-727 7 19 ring ring NOUN cana-727 7 20 , , PUNCT cana-727 7 21 and and CCONJ cana-727 7 22 all all DET cana-727 7 23 modules module NOUN cana-727 7 24 are be AUX cana-727 7 25 unital unital ADJ cana-727 7 26 right right ADJ cana-727 7 27 rmodules rmodule NOUN cana-727 7 28 . . PUNCT cana-727 8 1 in in ADP cana-727 8 2 the the DET cana-727 8 3 spirit spirit NOUN cana-727 8 4 of of ADP cana-727 8 5 [ [ X cana-727 8 6 1 1 NUM cana-727 8 7 ] ] PUNCT cana-727 8 8 , , PUNCT cana-727 8 9 for for ADP cana-727 8 10 a a DET cana-727 8 11 module module NOUN cana-727 8 12 m m NOUN cana-727 8 13 , , PUNCT cana-727 8 14 think think VERB cana-727 8 15 of of ADP cana-727 8 16 the the DET cana-727 8 17 following follow VERB cana-727 8 18 relations relation NOUN cana-727 8 19 on on ADP cana-727 8 20 the the DET cana-727 8 21 set set NOUN cana-727 8 22 of of ADP cana-727 8 23 submodules submodule NOUN cana-727 8 24 of of ADP cana-727 8 25 m m NOUN cana-727 8 26 : : PUNCT cana-727 8 27 " " PUNCT cana-727 8 28 𝐴𝛼𝐡 𝐴𝛼𝐡 NOUN cana-727 8 29 if if SCONJ cana-727 8 30 and and CCONJ cana-727 8 31 only only ADV cana-727 8 32 if if SCONJ cana-727 8 33 there there PRON cana-727 8 34 exists exist VERB cana-727 8 35 a a DET cana-727 8 36 submodule submodule NOUN cana-727 8 37 c c PROPN cana-727 8 38 of of ADP cana-727 8 39 m m PRON cana-727 8 40 such such ADJ cana-727 8 41 that that SCONJ cana-727 8 42 𝐴 𝐴 PROPN cana-727 8 43 ≀e ≀e VERB cana-727 8 44 c c PROPN cana-727 8 45 and and CCONJ cana-727 8 46 𝐡 𝐡 PROPN cana-727 8 47 ≀e ≀e PROPN cana-727 8 48 c. c. PROPN cana-727 8 49 𝐴𝛽𝐡 𝐴𝛽𝐡 PROPN cana-727 9 1 if if SCONJ cana-727 9 2 and and CCONJ cana-727 9 3 only only ADV cana-727 9 4 if if SCONJ cana-727 9 5 𝐴 𝐴 PROPN cana-727 9 6 ∩ ∩ NOUN cana-727 9 7 𝐡 𝐡 NOUN cana-727 9 8 ≀e ≀e VERB cana-727 9 9 a a DET cana-727 9 10 and and CCONJ cana-727 9 11 𝐴 𝐴 PROPN cana-727 9 12 ∩ ∩ NOUN cana-727 9 13 𝐡 𝐡 PROPN cana-727 9 14 ≀e ≀e PROPN cana-727 9 15 b b PROPN cana-727 9 16 . . PUNCT cana-727 10 1 recall recall VERB cana-727 10 2 that that SCONJ cana-727 10 3 Ξ² Ξ² PROPN cana-727 10 4 is be AUX cana-727 10 5 an an DET cana-727 10 6 equivalence equivalence NOUN cana-727 10 7 relation relation NOUN cana-727 10 8 . . PUNCT cana-727 11 1 " " PUNCT cana-727 11 2 it it PRON cana-727 11 3 is be AUX cana-727 11 4 clear clear ADJ cana-727 11 5 that that SCONJ cana-727 11 6 a a DET cana-727 11 7 module module NOUN cana-727 11 8 m m NOUN cana-727 11 9 is be AUX cana-727 11 10 extending extend VERB cana-727 11 11 ( ( PUNCT cana-727 11 12 or or CCONJ cana-727 11 13 cs cs ADJ cana-727 11 14 ) ) PUNCT cana-727 11 15 if if SCONJ cana-727 11 16 and and CCONJ cana-727 11 17 only only ADV cana-727 11 18 if if SCONJ cana-727 11 19 for for ADP cana-727 11 20 each each DET cana-727 11 21 submodule submodule NOUN cana-727 11 22 a a PRON cana-727 11 23 of of ADP cana-727 11 24 m m PROPN cana-727 11 25 , , PUNCT cana-727 11 26 there there PRON cana-727 11 27 is be VERB cana-727 11 28 a a DET cana-727 11 29 direct direct ADJ cana-727 11 30 summand summand NOUN cana-727 11 31 d d PROPN cana-727 11 32 of of ADP cana-727 11 33 m m PRON cana-727 11 34 such such ADJ cana-727 11 35 that that DET cana-727 11 36 aΞ±d aΞ±d NOUN cana-727 11 37 , , PUNCT cana-727 11 38 ( ( PUNCT cana-727 11 39 see see VERB cana-727 11 40 [ [ X cana-727 11 41 1,2 1,2 NUM cana-727 11 42 ] ] PUNCT cana-727 11 43 ) ) PUNCT cana-727 11 44 . . PUNCT cana-727 12 1 further far ADV cana-727 12 2 a a DET cana-727 12 3 module module NOUN cana-727 12 4 m m VERB cana-727 12 5 is be AUX cana-727 12 6 called call VERB cana-727 12 7 goldie goldie PROPN cana-727 12 8 extending extending NOUN cana-727 12 9 module module NOUN cana-727 12 10 ( ( PUNCT cana-727 12 11 or or CCONJ cana-727 12 12 g g NOUN cana-727 12 13 - - PUNCT cana-727 12 14 extending extending NOUN cana-727 12 15 ) ) PUNCT cana-727 12 16 if if SCONJ cana-727 13 1 and and CCONJ cana-727 13 2 only only ADV cana-727 13 3 if if SCONJ cana-727 13 4 for for ADP cana-727 13 5 each each DET cana-727 13 6 submodule submodule NOUN cana-727 13 7 a a PRON cana-727 13 8 of of ADP cana-727 13 9 m m PROPN cana-727 13 10 , , PUNCT cana-727 13 11 there there PRON cana-727 13 12 is be VERB cana-727 13 13 a a DET cana-727 13 14 direct direct ADJ cana-727 13 15 summand summand NOUN cana-727 13 16 d d PROPN cana-727 13 17 of of ADP cana-727 13 18 m m PRON cana-727 13 19 such such ADJ cana-727 13 20 that that DET cana-727 13 21 aΞ²d aΞ²d NOUN cana-727 13 22 or or CCONJ cana-727 13 23 equivalently equivalently ADV cana-727 13 24 , , PUNCT cana-727 13 25 for for ADP cana-727 13 26 each each DET cana-727 13 27 closed closed ADJ cana-727 13 28 submodule submodule NOUN cana-727 13 29 a a PRON cana-727 13 30 in in ADP cana-727 13 31 m m PROPN cana-727 13 32 , , PUNCT cana-727 13 33 there there PRON cana-727 13 34 is be VERB cana-727 13 35 a a DET cana-727 13 36 direct direct ADJ cana-727 13 37 summand summand NOUN cana-727 13 38 d d PROPN cana-727 13 39 of of ADP cana-727 13 40 m m PRON cana-727 13 41 such such ADJ cana-727 13 42 that that DET cana-727 13 43 aΞ²d aΞ²d NOUN cana-727 13 44 ( ( PUNCT cana-727 13 45 see see VERB cana-727 13 46 [ [ X cana-727 13 47 1 1 NUM cana-727 13 48 ] ] NUM cana-727 13 49 ) ) PUNCT cana-727 13 50 . . PUNCT cana-727 14 1 obviously obviously ADV cana-727 14 2 , , PUNCT cana-727 14 3 every every DET cana-727 14 4 extending extend VERB cana-727 14 5 module module NOUN cana-727 14 6 is be AUX cana-727 14 7 g g NOUN cana-727 14 8 - - PUNCT cana-727 14 9 extending extending ADJ cana-727 14 10 . . PUNCT cana-727 15 1 as as ADP cana-727 15 2 a a DET cana-727 15 3 generalization generalization NOUN cana-727 15 4 of of ADP cana-727 15 5 cs c NOUN cana-727 15 6 - - NOUN cana-727 15 7 modules module NOUN cana-727 15 8 is be AUX cana-727 15 9 p p NOUN cana-727 15 10 - - PUNCT cana-727 15 11 extending extend VERB cana-727 15 12 ( ( PUNCT cana-727 15 13 see see VERB cana-727 15 14 [ [ X cana-727 15 15 3,4 3,4 NUM cana-727 15 16 ] ] NUM cana-727 15 17 ) ) PUNCT cana-727 15 18 . . PUNCT cana-727 16 1 recall recall VERB cana-727 16 2 that that SCONJ cana-727 16 3 a a DET cana-727 16 4 module module NOUN cana-727 16 5 m m VERB cana-727 16 6 is be AUX cana-727 16 7 called call VERB cana-727 16 8 pextending pextende VERB cana-727 16 9 if if SCONJ cana-727 16 10 every every DET cana-727 16 11 cyclic cyclic ADJ cana-727 16 12 submodule submodule NOUN cana-727 16 13 of of ADP cana-727 16 14 m m PROPN cana-727 16 15 is be AUX cana-727 16 16 essential essential ADJ cana-727 16 17 in in ADP cana-727 16 18 a a DET cana-727 16 19 direct direct ADJ cana-727 16 20 summand summand NOUN cana-727 16 21 of of ADP cana-727 16 22 m. m. NOUN cana-727 16 23 " " PUNCT cana-727 16 24 " " PUNCT cana-727 16 25 in in ADP cana-727 16 26 this this DET cana-727 16 27 paper paper NOUN cana-727 16 28 , , PUNCT cana-727 16 29 we we PRON cana-727 16 30 study study VERB cana-727 16 31 a a DET cana-727 16 32 module module NOUN cana-727 16 33 condition condition NOUN cana-727 16 34 including include VERB cana-727 16 35 the the DET cana-727 16 36 Ξ² Ξ² PROPN cana-727 16 37 relation relation NOUN cana-727 16 38 on on ADP cana-727 16 39 the the DET cana-727 16 40 set set NOUN cana-727 16 41 of of ADP cana-727 16 42 all all DET cana-727 16 43 cyclic cyclic ADJ cana-727 16 44 submodules submodule NOUN cana-727 16 45 of of ADP cana-727 16 46 a a DET cana-727 16 47 module module NOUN cana-727 16 48 . . PUNCT cana-727 17 1 we we PRON cana-727 17 2 call call VERB cana-727 17 3 a a DET cana-727 17 4 module module NOUN cana-727 17 5 m m VERB cana-727 17 6 is be AUX cana-727 17 7 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 17 8 if if SCONJ cana-727 17 9 for for ADP cana-727 17 10 every every DET cana-727 17 11 cyclic cyclic ADJ cana-727 17 12 submodule submodule NOUN cana-727 17 13 a a PRON cana-727 17 14 of of ADP cana-727 17 15 m m PROPN cana-727 17 16 , , PUNCT cana-727 17 17 there there PRON cana-727 17 18 is be VERB cana-727 17 19 a a DET cana-727 17 20 direct direct ADJ cana-727 17 21 summand summand NOUN cana-727 17 22 d d PROPN cana-727 17 23 of of ADP cana-727 17 24 m m PRON cana-727 17 25 such such ADJ cana-727 17 26 that that DET cana-727 17 27 aΞ²d aΞ²d NOUN cana-727 17 28 . . PUNCT cana-727 18 1 a a DET cana-727 18 2 ring ring NOUN cana-727 18 3 r r NOUN cana-727 18 4 is be AUX cana-727 18 5 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 18 6 if if SCONJ cana-727 18 7 𝑅𝑅 𝑅𝑅 PROPN cana-727 18 8 is be AUX cana-727 18 9 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 18 10 module module NOUN cana-727 18 11 . . PUNCT cana-727 19 1 it it PRON cana-727 19 2 is be AUX cana-727 19 3 clear clear ADJ cana-727 19 4 that that SCONJ cana-727 19 5 the the DET cana-727 19 6 class class NOUN cana-727 19 7 of of ADP cana-727 19 8 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 19 9 modules module NOUN cana-727 19 10 property property NOUN cana-727 19 11 contains contain VERB cana-727 19 12 the the DET cana-727 19 13 type type NOUN cana-727 19 14 of of ADP cana-727 19 15 g g NOUN cana-727 19 16 - - PUNCT cana-727 19 17 extending extend VERB cana-727 19 18 modules module NOUN cana-727 19 19 . . PUNCT cana-727 20 1 the the DET cana-727 20 2 notion notion NOUN cana-727 20 3 of of ADP cana-727 20 4 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 20 5 generalizes generalize VERB cana-727 20 6 both both PRON cana-727 20 7 of of ADP cana-727 20 8 g g NOUN cana-727 20 9 - - PUNCT cana-727 20 10 extending extend VERB cana-727 20 11 , , PUNCT cana-727 20 12 extending extend VERB cana-727 20 13 and and CCONJ cana-727 20 14 p p NOUN cana-727 20 15 - - PUNCT cana-727 20 16 extending extend VERB cana-727 20 17 modules module NOUN cana-727 20 18 . . PUNCT cana-727 20 19 " " PUNCT cana-727 21 1 " " PUNCT cana-727 21 2 in in ADP cana-727 21 3 section section NOUN cana-727 21 4 2 2 NUM cana-727 21 5 , , PUNCT cana-727 21 6 we we PRON cana-727 21 7 consider consider VERB cana-727 21 8 connections connection NOUN cana-727 21 9 between between ADP cana-727 21 10 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 21 11 property property NOUN cana-727 21 12 , , PUNCT cana-727 21 13 p p NOUN cana-727 21 14 - - PUNCT cana-727 21 15 extending extend VERB cana-727 21 16 and and CCONJ cana-727 21 17 gextending gextende VERB cana-727 21 18 conditions condition NOUN cana-727 21 19 . . PUNCT cana-727 22 1 moreover moreover ADV cana-727 22 2 , , PUNCT cana-727 22 3 we we PRON cana-727 22 4 give give VERB cana-727 22 5 sufficient sufficient ADJ cana-727 22 6 circumstances circumstance NOUN cana-727 22 7 under under ADP cana-727 22 8 which which PRON cana-727 22 9 p p NOUN cana-727 22 10 - - PUNCT cana-727 22 11 extending extend VERB cana-727 22 12 and and CCONJ cana-727 22 13 𝐺𝑝extending 𝐺𝑝extending NOUN cana-727 22 14 modules module NOUN cana-727 22 15 are be AUX cana-727 22 16 equivalent equivalent ADJ cana-727 22 17 . . PUNCT cana-727 23 1 communications communication NOUN cana-727 23 2 on on ADP cana-727 23 3 applied apply VERB cana-727 23 4 nonlinear nonlinear ADJ cana-727 23 5 analysis analysis NOUN cana-727 23 6 issn issn NOUN cana-727 23 7 : : PUNCT cana-727 23 8 1074 1074 NUM cana-727 23 9 - - PUNCT cana-727 23 10 133x 133x NUM cana-727 23 11 vol vol NOUN cana-727 23 12 31 31 NUM cana-727 23 13 no no NOUN cana-727 23 14 . . PUNCT cana-727 24 1 3s 3s NUM cana-727 24 2 ( ( PUNCT cana-727 24 3 2024 2024 NUM cana-727 24 4 ) ) PUNCT cana-727 24 5 10 10 NUM cana-727 24 6 https://internationalpubls.com https://internationalpubls.com X cana-727 24 7 section section NOUN cana-727 24 8 3 3 NUM cana-727 24 9 , , PUNCT cana-727 24 10 is be AUX cana-727 24 11 devoted devote VERB cana-727 24 12 to to ADP cana-727 24 13 the the DET cana-727 24 14 characterizations characterization NOUN cana-727 24 15 of of ADP cana-727 24 16 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 24 17 modules module NOUN cana-727 24 18 . . PUNCT cana-727 25 1 since since SCONJ cana-727 25 2 the the DET cana-727 25 3 direct direct ADJ cana-727 25 4 sum sum NOUN cana-727 25 5 of of ADP cana-727 25 6 𝐺𝑝extending 𝐺𝑝extending NOUN cana-727 25 7 modules module NOUN cana-727 25 8 need need AUX cana-727 25 9 not not PART cana-727 25 10 be be AUX cana-727 25 11 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 25 12 , , PUNCT cana-727 25 13 we we PRON cana-727 25 14 focus focus VERB cana-727 25 15 when when SCONJ cana-727 25 16 a a DET cana-727 25 17 direct direct ADJ cana-727 25 18 sum sum NOUN cana-727 25 19 of of ADP cana-727 25 20 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 25 21 modules module NOUN cana-727 25 22 is be AUX cana-727 25 23 also also ADV cana-727 25 24 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 25 25 . . PUNCT cana-727 26 1 also also ADV cana-727 26 2 , , PUNCT cana-727 26 3 we we PRON cana-727 26 4 give give VERB cana-727 26 5 sufficient sufficient ADJ cana-727 26 6 conditions condition NOUN cana-727 26 7 under under ADP cana-727 26 8 which which PRON cana-727 26 9 the the DET cana-727 26 10 direct direct ADJ cana-727 26 11 summand summand NOUN cana-727 26 12 of of ADP cana-727 26 13 𝐺𝑝extending 𝐺𝑝extending PROPN cana-727 26 14 is be AUX cana-727 26 15 also also ADV cana-727 26 16 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 26 17 . . PUNCT cana-727 27 1 these these PRON cana-727 27 2 are be AUX cana-727 27 3 introduced introduce VERB cana-727 27 4 in in ADP cana-727 27 5 section section NOUN cana-727 27 6 4 4 NUM cana-727 27 7 . . PUNCT cana-727 28 1 also also ADV cana-727 28 2 , , PUNCT cana-727 28 3 in in ADP cana-727 28 4 section section NOUN cana-727 28 5 4 4 NUM cana-727 28 6 , , PUNCT cana-727 28 7 we we PRON cana-727 28 8 investigate investigate VERB cana-727 28 9 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 28 10 essential essential ADJ cana-727 28 11 extensions extension NOUN cana-727 28 12 of of ADP cana-727 28 13 a a DET cana-727 28 14 module module NOUN cana-727 28 15 or or CCONJ cana-727 28 16 ring ring NOUN cana-727 28 17 . . PUNCT cana-727 28 18 " " PUNCT cana-727 29 1 following follow VERB cana-727 29 2 [ [ X cana-727 29 3 5 5 NUM cana-727 29 4 ] ] PUNCT cana-727 29 5 , , PUNCT cana-727 29 6 m m VERB cana-727 29 7 is be AUX cana-727 29 8 called call VERB cana-727 29 9 uc uc NOUN cana-727 29 10 - - PUNCT cana-727 29 11 module module NOUN cana-727 29 12 if if SCONJ cana-727 29 13 every every DET cana-727 29 14 submodule submodule NOUN cana-727 29 15 of of ADP cana-727 29 16 m m PROPN cana-727 29 17 has have VERB cana-727 29 18 a a DET cana-727 29 19 unique unique ADJ cana-727 29 20 closure closure NOUN cana-727 29 21 in in ADP cana-727 29 22 m. m. NOUN cana-727 29 23 2 2 NUM cana-727 29 24 . . PUNCT cana-727 29 25 preliminary preliminary ADJ cana-727 29 26 results result NOUN cana-727 29 27 . . PUNCT cana-727 30 1 " " PUNCT cana-727 30 2 the the DET cana-727 30 3 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 30 4 notion notion NOUN cana-727 30 5 is be AUX cana-727 30 6 based base VERB cana-727 30 7 on on ADP cana-727 30 8 two two NUM cana-727 30 9 tools tool NOUN cana-727 30 10 , , PUNCT cana-727 30 11 namely namely ADV cana-727 30 12 an an DET cana-727 30 13 equivalence equivalence NOUN cana-727 30 14 relation relation NOUN cana-727 30 15 on on ADP cana-727 30 16 cyclic cyclic ADJ cana-727 30 17 submodules submodule NOUN cana-727 30 18 of of ADP cana-727 30 19 a a DET cana-727 30 20 module module NOUN cana-727 30 21 m. m. NOUN cana-727 30 22 let let VERB cana-727 30 23 us we PRON cana-727 30 24 begin begin VERB cana-727 30 25 by by ADP cana-727 30 26 mentioning mention VERB cana-727 30 27 basic basic ADJ cana-727 30 28 facts fact NOUN cana-727 30 29 about about ADP cana-727 30 30 them they PRON cana-727 30 31 . . PUNCT cana-727 31 1 first first ADV cana-727 31 2 recall recall VERB cana-727 31 3 the the DET cana-727 31 4 following follow VERB cana-727 31 5 relations relation NOUN cana-727 31 6 on on ADP cana-727 31 7 the the DET cana-727 31 8 set set NOUN cana-727 31 9 of of ADP cana-727 31 10 submodules submodule NOUN cana-727 31 11 of of ADP cana-727 31 12 m m PROPN cana-727 31 13 ( ( PUNCT cana-727 31 14 see see VERB cana-727 31 15 [ [ X cana-727 31 16 1 1 NUM cana-727 31 17 ] ] NUM cana-727 31 18 ) ) PUNCT cana-727 31 19 . . PUNCT cana-727 32 1 ( ( PUNCT cana-727 32 2 i i NOUN cana-727 32 3 ) ) PUNCT cana-727 32 4 𝐴𝛼𝐡 𝐴𝛼𝐡 NOUN cana-727 32 5 if if SCONJ cana-727 32 6 and and CCONJ cana-727 32 7 only only ADV cana-727 32 8 if if SCONJ cana-727 32 9 there there PRON cana-727 32 10 exists exist VERB cana-727 32 11 a a DET cana-727 32 12 submodule submodule NOUN cana-727 32 13 c c PROPN cana-727 32 14 of of ADP cana-727 32 15 m m PRON cana-727 32 16 such such ADJ cana-727 32 17 that that SCONJ cana-727 32 18 𝐴 𝐴 PROPN cana-727 32 19 ≀e ≀e VERB cana-727 32 20 c c PROPN cana-727 32 21 and and CCONJ cana-727 32 22 𝐡 𝐡 PROPN cana-727 32 23 ≀e ≀e PROPN cana-727 32 24 c. c. PROPN cana-727 32 25 ( ( PUNCT cana-727 32 26 ii ii NOUN cana-727 32 27 ) ) PUNCT cana-727 32 28 𝐴𝛽𝐡 𝐴𝛽𝐡 PROPN cana-727 32 29 if if SCONJ cana-727 32 30 and and CCONJ cana-727 32 31 only only ADV cana-727 32 32 if if SCONJ cana-727 32 33 𝐴 𝐴 PROPN cana-727 32 34 ∩ ∩ NOUN cana-727 32 35 𝐡 𝐡 NOUN cana-727 32 36 ≀e ≀e VERB cana-727 33 1 a a PRON cana-727 33 2 and and CCONJ cana-727 33 3 𝐴 𝐴 PROPN cana-727 33 4 ∩ ∩ NOUN cana-727 33 5 𝐡 𝐡 PROPN cana-727 33 6 ≀e ≀e PROPN cana-727 33 7 b. b. PROPN cana-727 33 8 observe observe VERB cana-727 33 9 that that SCONJ cana-727 33 10 Ξ± Ξ± NOUN cana-727 33 11 is be AUX cana-727 33 12 reflexive reflexive ADJ cana-727 33 13 and and CCONJ cana-727 33 14 symmetric symmetric ADJ cana-727 33 15 , , PUNCT cana-727 33 16 but but CCONJ cana-727 33 17 it it PRON cana-727 33 18 may may AUX cana-727 33 19 not not PART cana-727 33 20 be be AUX cana-727 33 21 transitive transitive ADJ cana-727 33 22 . . PUNCT cana-727 34 1 however however ADV cana-727 34 2 , , PUNCT cana-727 34 3 Ξ² Ξ² X cana-727 34 4 is be AUX cana-727 34 5 an an DET cana-727 34 6 equivalence equivalence NOUN cana-727 34 7 relation relation NOUN cana-727 34 8 . . PUNCT cana-727 35 1 note note VERB cana-727 35 2 that that SCONJ cana-727 35 3 for for ADP cana-727 35 4 submodules submodule NOUN cana-727 35 5 of of ADP cana-727 35 6 module module NOUN cana-727 35 7 m. m. NOUN cana-727 35 8 if if SCONJ cana-727 35 9 aΞ±b aΞ±b NOUN cana-727 35 10 , , PUNCT cana-727 35 11 then then ADV cana-727 35 12 aΞ²b aΞ²b VERB cana-727 35 13 . . PUNCT cana-727 36 1 proposition proposition NOUN cana-727 36 2 2.1 2.1 NUM cana-727 36 3 : : PUNCT cana-727 37 1 a a DET cana-727 37 2 module module NOUN cana-727 37 3 m m NOUN cana-727 37 4 is be AUX cana-727 37 5 p p NOUN cana-727 37 6 - - PUNCT cana-727 37 7 extending extend VERB cana-727 37 8 if if SCONJ cana-727 38 1 and and CCONJ cana-727 38 2 only only ADV cana-727 38 3 if if SCONJ cana-727 38 4 for for ADP cana-727 38 5 each each DET cana-727 38 6 cyclic cyclic ADJ cana-727 38 7 submodule submodule NOUN cana-727 38 8 a a PRON cana-727 38 9 of of ADP cana-727 38 10 m m PROPN cana-727 38 11 , , PUNCT cana-727 38 12 there there PRON cana-727 38 13 is be VERB cana-727 38 14 a a DET cana-727 38 15 direct direct ADJ cana-727 38 16 summand summand NOUN cana-727 38 17 d d PROPN cana-727 38 18 of of ADP cana-727 38 19 m m PRON cana-727 38 20 such such ADJ cana-727 38 21 that that SCONJ cana-727 38 22 a𝛼 a𝛼 ADP cana-727 38 23 d. d. PROPN cana-727 38 24 proof proof PROPN cana-727 38 25 : : PUNCT cana-727 38 26 the the DET cana-727 38 27 proof proof NOUN cana-727 38 28 is be AUX cana-727 38 29 routine routine ADJ cana-727 38 30 . . PUNCT cana-727 38 31 " " PUNCT cana-727 39 1 " " PUNCT cana-727 39 2 motivated motivate VERB cana-727 39 3 by by ADP cana-727 39 4 proposition proposition NOUN cana-727 39 5 2.1 2.1 NUM cana-727 39 6 and and CCONJ cana-727 39 7 akalan akalan NOUN cana-727 39 8 , , PUNCT cana-727 39 9 birkenmeier birkenmeier NOUN cana-727 39 10 , , PUNCT cana-727 39 11 tercan tercan PROPN cana-727 39 12 's 's PART cana-727 39 13 use use NOUN cana-727 39 14 of of ADP cana-727 39 15 the the DET cana-727 39 16 Ξ² Ξ² NOUN cana-727 39 17 equivalence equivalence NOUN cana-727 39 18 relation relation NOUN cana-727 39 19 in in ADP cana-727 39 20 [ [ X cana-727 39 21 1 1 NUM cana-727 39 22 ] ] PUNCT cana-727 39 23 . . PUNCT cana-727 40 1 as as ADP cana-727 40 2 a a DET cana-727 40 3 generalization generalization NOUN cana-727 40 4 of of ADP cana-727 40 5 goldie goldie PROPN cana-727 40 6 extending extend VERB cana-727 40 7 modules module NOUN cana-727 40 8 , , PUNCT cana-727 40 9 we we PRON cana-727 40 10 introduce introduce VERB cana-727 40 11 a a DET cana-727 40 12 class class NOUN cana-727 40 13 of of ADP cana-727 40 14 modules module NOUN cana-727 40 15 which which PRON cana-727 40 16 is be AUX cana-727 40 17 analogous analogous ADJ cana-727 40 18 to to ADP cana-727 40 19 that that PRON cana-727 40 20 of of ADP cana-727 40 21 𝐺𝑧-extending 𝐺𝑧-extende VERB cana-727 40 22 and and CCONJ cana-727 40 23 πΊπ‘Ÿ-extending πΊπ‘Ÿ-extende VERB cana-727 40 24 modules module NOUN cana-727 40 25 which which PRON cana-727 40 26 are be AUX cana-727 40 27 introduced introduce VERB cana-727 40 28 in in ADP cana-727 40 29 [ [ X cana-727 40 30 6 6 NUM cana-727 40 31 ] ] PUNCT cana-727 40 32 and and CCONJ cana-727 40 33 [ [ X cana-727 40 34 7 7 X cana-727 40 35 ] ] PUNCT cana-727 40 36 respectively respectively ADV cana-727 40 37 . . PUNCT cana-727 40 38 " " PUNCT cana-727 41 1 definition definition NOUN cana-727 41 2 2.2 2.2 NUM cana-727 41 3 : : PUNCT cana-727 41 4 we we PRON cana-727 41 5 call call VERB cana-727 41 6 a a DET cana-727 41 7 module module NOUN cana-727 41 8 m m VERB cana-727 41 9 is be AUX cana-727 41 10 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 41 11 module module NOUN cana-727 41 12 if if SCONJ cana-727 41 13 for for ADP cana-727 41 14 each each DET cana-727 41 15 cyclic cyclic ADJ cana-727 41 16 submodule submodule NOUN cana-727 41 17 a a PRON cana-727 41 18 of of ADP cana-727 41 19 m m PROPN cana-727 41 20 , , PUNCT cana-727 41 21 there there PRON cana-727 41 22 is be VERB cana-727 41 23 a a DET cana-727 41 24 direct direct ADJ cana-727 41 25 summand summand NOUN cana-727 41 26 d d PROPN cana-727 41 27 of of ADP cana-727 41 28 m m PRON cana-727 41 29 such such ADJ cana-727 41 30 that that SCONJ cana-727 41 31 𝐴𝛽𝐷. 𝐴𝛽𝐷. PROPN cana-727 41 32 " " PUNCT cana-727 41 33 note note NOUN cana-727 41 34 that that SCONJ cana-727 41 35 m m NOUN cana-727 41 36 is be AUX cana-727 41 37 g g NOUN cana-727 41 38 - - PUNCT cana-727 41 39 extending extend VERB cana-727 41 40 if if SCONJ cana-727 42 1 and and CCONJ cana-727 42 2 only only ADV cana-727 42 3 if if SCONJ cana-727 42 4 for for ADP cana-727 42 5 each each DET cana-727 42 6 submodule submodule NOUN cana-727 42 7 a a PRON cana-727 42 8 of of ADP cana-727 42 9 m m PRON cana-727 42 10 there there PRON cana-727 42 11 is be VERB cana-727 42 12 a a DET cana-727 42 13 direct direct ADJ cana-727 42 14 summand summand NOUN cana-727 42 15 d d PROPN cana-727 42 16 of of ADP cana-727 42 17 m m PRON cana-727 42 18 such such ADJ cana-727 42 19 that that SCONJ cana-727 42 20 𝐴𝛽𝐷. 𝐴𝛽𝐷. PROPN cana-727 42 21 it it PRON cana-727 42 22 is be AUX cana-727 42 23 clear clear ADJ cana-727 42 24 that that SCONJ cana-727 42 25 the the DET cana-727 42 26 class class NOUN cana-727 42 27 of of ADP cana-727 42 28 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 42 29 contains contain VERB cana-727 42 30 both both PRON cana-727 42 31 of of ADP cana-727 42 32 the the DET cana-727 42 33 classes class NOUN cana-727 42 34 of of ADP cana-727 42 35 gextending gextende VERB cana-727 42 36 and and CCONJ cana-727 42 37 p p NOUN cana-727 42 38 - - PUNCT cana-727 42 39 extending extend VERB cana-727 42 40 modules module NOUN cana-727 42 41 . . PUNCT cana-727 43 1 now now ADV cana-727 43 2 , , PUNCT cana-727 43 3 we we PRON cana-727 43 4 locate locate VERB cana-727 43 5 the the DET cana-727 43 6 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 43 7 condition condition NOUN cana-727 43 8 with with ADP cana-727 43 9 respect respect NOUN cana-727 43 10 to to ADP cana-727 43 11 several several ADJ cana-727 43 12 known know VERB cana-727 43 13 generalizations generalization NOUN cana-727 43 14 of of ADP cana-727 43 15 the the DET cana-727 43 16 extending extend VERB cana-727 43 17 property property NOUN cana-727 43 18 . . PUNCT cana-727 43 19 " " PUNCT cana-727 44 1 proposition proposition NOUN cana-727 44 2 2.3 2.3 NUM cana-727 44 3 : : PUNCT cana-727 44 4 make make VERB cana-727 44 5 m m PRON cana-727 44 6 a a DET cana-727 44 7 module module NOUN cana-727 44 8 . . PUNCT cana-727 45 1 let let VERB cana-727 45 2 's us PRON cana-727 45 3 think think VERB cana-727 45 4 about about ADP cana-727 45 5 the the DET cana-727 45 6 aforementioned aforementioned ADJ cana-727 45 7 circumstances circumstance NOUN cana-727 45 8 . . PUNCT cana-727 46 1 ( ( PUNCT cana-727 46 2 i i NOUN cana-727 46 3 ) ) PUNCT cana-727 46 4 m m AUX cana-727 46 5 is be AUX cana-727 46 6 cs cs PROPN cana-727 46 7 . . PROPN cana-727 46 8 ( ( PUNCT cana-727 46 9 ii ii NOUN cana-727 46 10 ) ) PUNCT cana-727 46 11 m m VERB cana-727 46 12 is be AUX cana-727 46 13 g g NOUN cana-727 46 14 - - PUNCT cana-727 46 15 extending extending ADJ cana-727 46 16 . . PUNCT cana-727 47 1 ( ( PUNCT cana-727 47 2 iii iii X cana-727 47 3 ) ) PUNCT cana-727 47 4 m m VERB cana-727 47 5 is be AUX cana-727 47 6 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 47 7 . . PUNCT cana-727 48 1 ( ( PUNCT cana-727 48 2 iv iv X cana-727 48 3 ) ) PUNCT cana-727 48 4 m m VERB cana-727 48 5 is be AUX cana-727 48 6 p p ADJ cana-727 48 7 - - PUNCT cana-727 48 8 extending extending ADJ cana-727 48 9 . . PUNCT cana-727 49 1 communications communication NOUN cana-727 49 2 on on ADP cana-727 49 3 applied apply VERB cana-727 49 4 nonlinear nonlinear ADJ cana-727 49 5 analysis analysis NOUN cana-727 49 6 issn issn NOUN cana-727 49 7 : : PUNCT cana-727 49 8 1074 1074 NUM cana-727 49 9 - - PUNCT cana-727 49 10 133x 133x NUM cana-727 49 11 vol vol NOUN cana-727 49 12 31 31 NUM cana-727 49 13 no no NOUN cana-727 49 14 . . PUNCT cana-727 50 1 3s 3s NUM cana-727 50 2 ( ( PUNCT cana-727 50 3 2024 2024 NUM cana-727 50 4 ) ) PUNCT cana-727 50 5 11 11 NUM cana-727 50 6 https://internationalpubls.com https://internationalpubls.com X cana-727 50 7 " " PUNCT cana-727 50 8 then then ADV cana-727 50 9 ( ( PUNCT cana-727 50 10 i i NOUN cana-727 50 11 ) ) PUNCT cana-727 50 12 ⟹(ii ⟹(ii NUM cana-727 50 13 ) ) PUNCT cana-727 50 14 ⟹(iii ⟹(iii NUM cana-727 50 15 ) ) PUNCT cana-727 50 16 and and CCONJ cana-727 50 17 ( ( PUNCT cana-727 50 18 i i NOUN cana-727 50 19 ) ) PUNCT cana-727 50 20 ⟹ ⟹ PROPN cana-727 50 21 ( ( PUNCT cana-727 50 22 iv)⟹(iii iv)⟹(iii NOUN cana-727 50 23 ) ) PUNCT cana-727 50 24 . . PUNCT cana-727 51 1 in in ADP cana-727 51 2 general general ADJ cana-727 51 3 , , PUNCT cana-727 51 4 the the DET cana-727 51 5 converse converse NOUN cana-727 51 6 implications implication NOUN cana-727 51 7 do do AUX cana-727 51 8 not not PART cana-727 51 9 hold hold VERB cana-727 51 10 . . PUNCT cana-727 51 11 " " PUNCT cana-727 52 1 proof proof NOUN cana-727 52 2 : : PUNCT cana-727 52 3 ( ( PUNCT cana-727 52 4 i i NOUN cana-727 52 5 ) ) PUNCT cana-727 52 6 ⟹(ii ⟹(ii NUM cana-727 52 7 ) ) PUNCT cana-727 52 8 ⟹(iii ⟹(iii NUM cana-727 52 9 ) ) PUNCT cana-727 52 10 and and CCONJ cana-727 52 11 ( ( PUNCT cana-727 52 12 i i NOUN cana-727 52 13 ) ) PUNCT cana-727 52 14 ⟹ ⟹ PROPN cana-727 52 15 ( ( PUNCT cana-727 52 16 iv)⟹(iii iv)⟹(iii NOUN cana-727 52 17 ) ) PUNCT cana-727 52 18 are be AUX cana-727 52 19 clear clear ADJ cana-727 52 20 . . PUNCT cana-727 53 1 ( ( PUNCT cana-727 53 2 ii)⇏(i)"let ii)⇏(i)"let NOUN cana-727 53 3 m m VERB cana-727 53 4 be be AUX cana-727 53 5 the the DET cana-727 53 6 𝕫-module 𝕫-module NOUN cana-727 53 7 π•«π‘β¨β„š π•«π‘β¨β„š PROPN cana-727 53 8 , , PUNCT cana-727 53 9 where where SCONJ cana-727 53 10 p p NOUN cana-727 53 11 is be AUX cana-727 53 12 any any DET cana-727 53 13 prime prime ADJ cana-727 53 14 integer integer NOUN cana-727 53 15 . . PUNCT cana-727 54 1 then then ADV cana-727 54 2 𝑀℀ 𝑀℀ PROPN cana-727 54 3 is be AUX cana-727 54 4 g g NOUN cana-727 54 5 - - PUNCT cana-727 54 6 extending extend VERB cana-727 54 7 by by ADP cana-727 54 8 [ [ X cana-727 54 9 1 1 NUM cana-727 54 10 , , PUNCT cana-727 54 11 corollary corollary ADJ cana-727 54 12 ( ( PUNCT cana-727 54 13 3.3 3.3 NUM cana-727 54 14 ) ) PUNCT cana-727 54 15 ] ] PUNCT cana-727 54 16 . . PUNCT cana-727 55 1 however however ADV cana-727 55 2 𝑀℀ 𝑀℀ PROPN cana-727 55 3 is be AUX cana-727 55 4 not not PART cana-727 55 5 extending extend VERB cana-727 55 6 [ [ X cana-727 55 7 8 8 NUM cana-727 55 8 , , PUNCT cana-727 55 9 example example NOUN cana-727 55 10 10 10 NUM cana-727 55 11 ] ] PUNCT cana-727 55 12 . . PUNCT cana-727 55 13 " " PUNCT cana-727 56 1 ( ( PUNCT cana-727 56 2 iii)⇏(ii iii)⇏(ii PROPN cana-727 56 3 ) ) PUNCT cana-727 56 4 let let VERB cana-727 56 5 𝑀2(𝑅)be 𝑀2(𝑅)be PUNCT cana-727 56 6 the the DET cana-727 56 7 ring ring NOUN cana-727 56 8 as as ADP cana-727 56 9 in in ADP cana-727 56 10 [ [ X cana-727 56 11 9 9 NUM cana-727 56 12 , , PUNCT cana-727 56 13 example example NOUN cana-727 56 14 13.8 13.8 NUM cana-727 56 15 ] ] PUNCT cana-727 56 16 . . PUNCT cana-727 57 1 then then ADV cana-727 57 2 𝑀2(𝑅 𝑀2(𝑅 NOUN cana-727 57 3 ) ) PUNCT cana-727 57 4 is be AUX cana-727 57 5 a a DET cana-727 57 6 von von PROPN cana-727 57 7 neumann neumann PROPN cana-727 57 8 regular regular PROPN cana-727 57 9 ring ring NOUN cana-727 57 10 which which PRON cana-727 57 11 is be AUX cana-727 57 12 not not PART cana-727 57 13 a a DET cana-727 57 14 baer baer PROPN cana-727 57 15 ring ring NOUN cana-727 57 16 . . PUNCT cana-727 58 1 hence hence ADV cana-727 58 2 it it PRON cana-727 58 3 is be AUX cana-727 58 4 neither neither CCONJ cana-727 58 5 right right ADJ cana-727 58 6 nor nor CCONJ cana-727 58 7 left leave VERB cana-727 58 8 cs c NOUN cana-727 58 9 , , PUNCT cana-727 58 10 by by ADP cana-727 58 11 [ [ X cana-727 58 12 10 10 NUM cana-727 58 13 , , PUNCT cana-727 58 14 example example NOUN cana-727 58 15 2.7 2.7 NUM cana-727 58 16 ] ] PUNCT cana-727 58 17 , , PUNCT cana-727 58 18 however however ADV cana-727 58 19 it it PRON cana-727 58 20 's be AUX cana-727 58 21 well well ADV cana-727 58 22 acknowledged acknowledge VERB cana-727 58 23 that that SCONJ cana-727 58 24 each each DET cana-727 58 25 von von PROPN cana-727 58 26 neumann neumann PROPN cana-727 58 27 regular regular PROPN cana-727 58 28 ring ring NOUN cana-727 58 29 is be AUX cana-727 58 30 nonsingular nonsingular ADJ cana-727 58 31 , , PUNCT cana-727 58 32 therefore therefore ADV cana-727 58 33 𝑀2(𝑅 𝑀2(𝑅 NOUN cana-727 58 34 ) ) PUNCT cana-727 58 35 is be AUX cana-727 58 36 not not PART cana-727 58 37 is be AUX cana-727 58 38 gextending gextende VERB cana-727 58 39 , , PUNCT cana-727 58 40 see see VERB cana-727 58 41 [ [ X cana-727 58 42 1 1 NUM cana-727 58 43 , , PUNCT cana-727 58 44 proposition proposition NOUN cana-727 58 45 1.8 1.8 NUM cana-727 58 46 ] ] PUNCT cana-727 58 47 . . PUNCT cana-727 59 1 also also ADV cana-727 59 2 , , PUNCT cana-727 59 3 this this PRON cana-727 59 4 is be AUX cana-727 59 5 an an DET cana-727 59 6 example example NOUN cana-727 59 7 to to PART cana-727 59 8 show show VERB cana-727 59 9 that that SCONJ cana-727 59 10 ( ( PUNCT cana-727 59 11 iv)⇏(i iv)⇏(i NOUN cana-727 59 12 ) ) PUNCT cana-727 59 13 . . PUNCT cana-727 60 1 ( ( PUNCT cana-727 60 2 iii)⇏(iv iii)⇏(iv NOUN cana-727 60 3 ) ) PUNCT cana-727 60 4 let let VERB cana-727 60 5 m m PRON cana-727 60 6 be be AUX cana-727 60 7 the the DET cana-727 60 8 𝕫-module 𝕫-module NOUN cana-727 60 9 𝕫2⨁𝕫8 𝕫2⨁𝕫8 NOUN cana-727 60 10 . . PUNCT cana-727 61 1 then then ADV cana-727 61 2 𝑀℀ 𝑀℀ PROPN cana-727 61 3 is be AUX cana-727 61 4 𝐺𝑝-extending 𝐺𝑝-extending ADJ cana-727 61 5 but but CCONJ cana-727 61 6 not not PART cana-727 61 7 p p NOUN cana-727 61 8 - - PUNCT cana-727 61 9 extending extending ADJ cana-727 61 10 , , PUNCT cana-727 61 11 see see VERB cana-727 61 12 [ [ X cana-727 61 13 1 1 NUM cana-727 61 14 , , PUNCT cana-727 61 15 corollary corollary ADJ cana-727 61 16 3.3 3.3 NUM cana-727 61 17 ] ] PUNCT cana-727 61 18 . . PUNCT cana-727 62 1 the the DET cana-727 62 2 condition condition NOUN cana-727 62 3 under under ADP cana-727 62 4 which which PRON cana-727 62 5 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 62 6 and and CCONJ cana-727 62 7 p p NOUN cana-727 62 8 - - PUNCT cana-727 62 9 extending extend VERB cana-727 62 10 modules module NOUN cana-727 62 11 are be AUX cana-727 62 12 equivalent equivalent ADJ cana-727 62 13 is be AUX cana-727 62 14 stated state VERB cana-727 62 15 in in ADP cana-727 62 16 the the DET cana-727 62 17 following follow VERB cana-727 62 18 proposal proposal NOUN cana-727 62 19 . . PUNCT cana-727 63 1 proposition proposition NOUN cana-727 63 2 2.4 2.4 NUM cana-727 63 3 : : PUNCT cana-727 63 4 " " PUNCT cana-727 63 5 let let VERB cana-727 63 6 m m PRON cana-727 63 7 be be AUX cana-727 63 8 a a DET cana-727 63 9 module module NOUN cana-727 63 10 . . PUNCT cana-727 64 1 ( ( PUNCT cana-727 64 2 i i NOUN cana-727 64 3 ) ) PUNCT cana-727 64 4 if if SCONJ cana-727 64 5 m m NOUN cana-727 64 6 is be AUX cana-727 64 7 a a DET cana-727 64 8 ucmodule ucmodule NOUN cana-727 64 9 . . PUNCT cana-727 65 1 then then ADV cana-727 65 2 m m VERB cana-727 65 3 is be AUX cana-727 65 4 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 65 5 if if SCONJ cana-727 65 6 and and CCONJ cana-727 65 7 only only ADV cana-727 65 8 if if SCONJ cana-727 65 9 m m PROPN cana-727 65 10 is be AUX cana-727 65 11 p p ADJ cana-727 65 12 - - PUNCT cana-727 65 13 extending extending ADJ cana-727 65 14 . . PUNCT cana-727 66 1 ( ( PUNCT cana-727 66 2 ii ii NOUN cana-727 66 3 ) ) PUNCT cana-727 66 4 if if SCONJ cana-727 66 5 m m NOUN cana-727 66 6 is be AUX cana-727 66 7 a a DET cana-727 66 8 nonsingular nonsingular ADJ cana-727 66 9 module module NOUN cana-727 66 10 . . PUNCT cana-727 67 1 then then ADV cana-727 67 2 m m VERB cana-727 67 3 is be AUX cana-727 67 4 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 67 5 if if SCONJ cana-727 67 6 and and CCONJ cana-727 67 7 only only ADV cana-727 67 8 if if SCONJ cana-727 67 9 m m PROPN cana-727 67 10 is be AUX cana-727 67 11 p p ADJ cana-727 67 12 - - PUNCT cana-727 67 13 extending extending ADJ cana-727 67 14 . . PUNCT cana-727 68 1 ( ( PUNCT cana-727 68 2 iii iii X cana-727 68 3 ) ) PUNCT cana-727 68 4 if if SCONJ cana-727 68 5 m m NOUN cana-727 68 6 is be AUX cana-727 68 7 an an DET cana-727 68 8 indecomposable indecomposable ADJ cana-727 68 9 module module NOUN cana-727 68 10 . . PUNCT cana-727 69 1 then then ADV cana-727 69 2 m m VERB cana-727 69 3 is be AUX cana-727 69 4 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 69 5 if if SCONJ cana-727 69 6 and and CCONJ cana-727 69 7 only only ADV cana-727 69 8 if if SCONJ cana-727 69 9 m m NOUN cana-727 69 10 is be AUX cana-727 69 11 pextending pextende VERB cana-727 69 12 . . PUNCT cana-727 69 13 " " PUNCT cana-727 70 1 proof proof NOUN cana-727 70 2 : : PUNCT cana-727 70 3 ( ( PUNCT cana-727 70 4 i i NOUN cana-727 70 5 ) ) PUNCT cana-727 70 6 " " PUNCT cana-727 70 7 assume assume VERB cana-727 70 8 that that SCONJ cana-727 70 9 m m PROPN cana-727 70 10 is be AUX cana-727 70 11 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 70 12 and and CCONJ cana-727 70 13 let let VERB cana-727 70 14 a a PRON cana-727 70 15 be be AUX cana-727 70 16 a a DET cana-727 70 17 cyclic cyclic ADJ cana-727 70 18 submodule submodule NOUN cana-727 70 19 of of ADP cana-727 70 20 m m PROPN cana-727 70 21 , , PUNCT cana-727 70 22 then then ADV cana-727 70 23 there there PRON cana-727 70 24 exists exist VERB cana-727 70 25 a a DET cana-727 70 26 direct direct ADJ cana-727 70 27 d d NOUN cana-727 70 28 of of ADP cana-727 70 29 m m PRON cana-727 70 30 such such ADJ cana-727 70 31 that that SCONJ cana-727 70 32 𝐴𝛽𝐷. 𝐴𝛽𝐷. PROPN cana-727 70 33 one one PRON cana-727 70 34 can can AUX cana-727 70 35 easily easily ADV cana-727 70 36 show show VERB cana-727 70 37 that that SCONJ cana-727 70 38 ( ( PUNCT cana-727 70 39 𝐴 𝐴 PROPN cana-727 70 40 ∩ ∩ ADJ cana-727 70 41 𝐷)𝛼𝐴 𝐷)𝛼𝐴 PROPN cana-727 70 42 and and CCONJ cana-727 70 43 ( ( PUNCT cana-727 70 44 𝐴 𝐴 PROPN cana-727 70 45 ∩ ∩ NOUN cana-727 70 46 𝐷)𝛼𝐷. 𝐷)𝛼𝐷. NOUN cana-727 70 47 but but CCONJ cana-727 70 48 m m PROPN cana-727 70 49 is be AUX cana-727 70 50 uc uc PROPN cana-727 70 51 module module NOUN cana-727 70 52 , , PUNCT cana-727 70 53 therefore therefore ADV cana-727 70 54 Ξ± Ξ± PROPN cana-727 70 55 is be AUX cana-727 70 56 transitive transitive ADJ cana-727 70 57 , , PUNCT cana-727 70 58 hence hence ADV cana-727 70 59 𝐴𝛼𝐷. 𝐴𝛼𝐷. PROPN cana-727 70 60 thus thus ADV cana-727 70 61 m m VERB cana-727 70 62 is be AUX cana-727 70 63 p p ADJ cana-727 70 64 - - PUNCT cana-727 70 65 extending extending NOUN cana-727 70 66 . . PUNCT cana-727 71 1 the the DET cana-727 71 2 converse converse NOUN cana-727 71 3 is be AUX cana-727 71 4 clear clear ADJ cana-727 71 5 . . PUNCT cana-727 72 1 ( ( PUNCT cana-727 72 2 ii ii NOUN cana-727 72 3 ) ) PUNCT cana-727 72 4 let let VERB cana-727 72 5 m m PRON cana-727 72 6 be be AUX cana-727 72 7 a a DET cana-727 72 8 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 72 9 and and CCONJ cana-727 72 10 let let VERB cana-727 72 11 a a PRON cana-727 72 12 be be AUX cana-727 72 13 a a DET cana-727 72 14 cyclic cyclic ADJ cana-727 72 15 submodule submodule NOUN cana-727 72 16 of of ADP cana-727 72 17 m m PROPN cana-727 72 18 , , PUNCT cana-727 72 19 then then ADV cana-727 72 20 there there PRON cana-727 72 21 is be VERB cana-727 72 22 a a DET cana-727 72 23 direct direct ADJ cana-727 72 24 d d NOUN cana-727 72 25 of of ADP cana-727 72 26 m m PRON cana-727 72 27 such such ADJ cana-727 72 28 that that SCONJ cana-727 72 29 𝐴𝛽𝐷. 𝐴𝛽𝐷. PROPN cana-727 72 30 it it PRON cana-727 72 31 is be AUX cana-727 72 32 sufficient sufficient ADJ cana-727 72 33 to to PART cana-727 72 34 show show VERB cana-727 72 35 that that SCONJ cana-727 72 36 𝐴 𝐴 PROPN cana-727 72 37 ≀ ≀ NOUN cana-727 72 38 𝐷. 𝐷. PROPN cana-727 72 39 since since SCONJ cana-727 72 40 𝐴+𝐷 𝐴+𝐷 PROPN cana-727 72 41 𝐷 𝐷 PROPN cana-727 72 42 β‰… β‰… PROPN cana-727 72 43 𝐴 𝐴 PROPN cana-727 72 44 𝐴∩𝐷 𝐴∩𝐷 PROPN cana-727 72 45 is be AUX cana-727 72 46 singular singular ADJ cana-727 72 47 and and CCONJ cana-727 72 48 𝐴+𝐷 𝐴+𝐷 PROPN cana-727 72 49 𝐷 𝐷 PROPN cana-727 72 50 ≀ ≀ PROPN cana-727 72 51 𝑀 𝑀 PROPN cana-727 72 52 𝐷 𝐷 PROPN cana-727 72 53 β‰… β‰… PROPN cana-727 72 54 𝐷′ 𝐷′ PROPN cana-727 72 55 is be AUX cana-727 72 56 nonsingular nonsingular ADJ cana-727 72 57 , , PUNCT cana-727 72 58 hence hence ADV cana-727 72 59 a+d a+d VERB cana-727 72 60 = = SYM cana-727 72 61 d d X cana-727 72 62 which which PRON cana-727 72 63 implies imply VERB cana-727 72 64 that that SCONJ cana-727 72 65 𝐴 𝐴 PROPN cana-727 72 66 ≀ ≀ NOUN cana-727 72 67 𝐷. 𝐷. PROPN cana-727 72 68 the the DET cana-727 72 69 converse converse NOUN cana-727 72 70 is be AUX cana-727 72 71 obvious obvious ADJ cana-727 72 72 . . PUNCT cana-727 73 1 ( ( PUNCT cana-727 73 2 iii iii X cana-727 73 3 ) ) PUNCT cana-727 73 4 let let VERB cana-727 73 5 m m PRON cana-727 73 6 be be AUX cana-727 73 7 a a DET cana-727 73 8 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 73 9 and and CCONJ cana-727 73 10 let let VERB cana-727 73 11 a a PRON cana-727 73 12 be be AUX cana-727 73 13 a a DET cana-727 73 14 cyclic cyclic ADJ cana-727 73 15 submodule submodule NOUN cana-727 73 16 of of ADP cana-727 73 17 m m PROPN cana-727 73 18 , , PUNCT cana-727 73 19 then then ADV cana-727 73 20 there there PRON cana-727 73 21 is be VERB cana-727 73 22 a a DET cana-727 73 23 direct direct ADJ cana-727 73 24 d d NOUN cana-727 73 25 of of ADP cana-727 73 26 m m PRON cana-727 73 27 such such ADJ cana-727 73 28 that that SCONJ cana-727 73 29 𝐴𝛽𝐷. 𝐴𝛽𝐷. PROPN cana-727 73 30 since since SCONJ cana-727 73 31 m m PROPN cana-727 73 32 is be AUX cana-727 73 33 indecomposable indecomposable ADJ cana-727 73 34 , , PUNCT cana-727 73 35 then then ADV cana-727 73 36 d d X cana-727 73 37 = = NOUN cana-727 73 38 m. m. NOUN cana-727 73 39 thus thus ADV cana-727 73 40 m m VERB cana-727 73 41 is be AUX cana-727 73 42 p p ADJ cana-727 73 43 - - PUNCT cana-727 73 44 extending extend VERB cana-727 73 45 module module NOUN cana-727 73 46 . . PUNCT cana-727 74 1 the the DET cana-727 74 2 converse converse NOUN cana-727 74 3 is be AUX cana-727 74 4 clear clear ADJ cana-727 74 5 . . PUNCT cana-727 74 6 " " PUNCT cana-727 75 1 corollary corollary ADJ cana-727 75 2 2.5 2.5 NUM cana-727 75 3 : : PUNCT cana-727 75 4 " " PUNCT cana-727 75 5 let let VERB cana-727 75 6 m m PRON cana-727 75 7 be be AUX cana-727 75 8 an an DET cana-727 75 9 indecomposable indecomposable ADJ cana-727 75 10 module module NOUN cana-727 75 11 . . PUNCT cana-727 76 1 then then ADV cana-727 76 2 the the DET cana-727 76 3 following follow VERB cana-727 76 4 statements statement NOUN cana-727 76 5 are be AUX cana-727 76 6 equivalent equivalent ADJ cana-727 76 7 : : PUNCT cana-727 76 8 ( ( PUNCT cana-727 76 9 i i NOUN cana-727 76 10 ) ) PUNCT cana-727 76 11 m m VERB cana-727 76 12 is be AUX cana-727 76 13 uniform uniform ADJ cana-727 76 14 . . PUNCT cana-727 77 1 ( ( PUNCT cana-727 77 2 ii ii NOUN cana-727 77 3 ) ) PUNCT cana-727 77 4 m m VERB cana-727 77 5 is be AUX cana-727 77 6 cs cs PROPN cana-727 77 7 . . PUNCT cana-727 77 8 ( ( PUNCT cana-727 77 9 iii iii X cana-727 77 10 ) ) PUNCT cana-727 77 11 m m VERB cana-727 77 12 is be AUX cana-727 77 13 g g NOUN cana-727 77 14 - - PUNCT cana-727 77 15 extending extending ADJ cana-727 77 16 . . PUNCT cana-727 78 1 ( ( PUNCT cana-727 78 2 iv iv X cana-727 78 3 ) ) PUNCT cana-727 78 4 m m VERB cana-727 78 5 is be AUX cana-727 78 6 p p ADJ cana-727 78 7 - - PUNCT cana-727 78 8 extending extending ADJ cana-727 78 9 . . PUNCT cana-727 79 1 communications communication NOUN cana-727 79 2 on on ADP cana-727 79 3 applied apply VERB cana-727 79 4 nonlinear nonlinear ADJ cana-727 79 5 analysis analysis NOUN cana-727 79 6 issn issn NOUN cana-727 79 7 : : PUNCT cana-727 79 8 1074 1074 NUM cana-727 79 9 - - PUNCT cana-727 79 10 133x 133x NUM cana-727 79 11 vol vol NOUN cana-727 79 12 31 31 NUM cana-727 79 13 no no NOUN cana-727 79 14 . . PUNCT cana-727 80 1 3s 3s NUM cana-727 80 2 ( ( PUNCT cana-727 80 3 2024 2024 NUM cana-727 80 4 ) ) PUNCT cana-727 80 5 12 12 NUM cana-727 80 6 https://internationalpubls.com https://internationalpubls.com X cana-727 80 7 ( ( PUNCT cana-727 80 8 v v NOUN cana-727 80 9 ) ) PUNCT cana-727 80 10 m m VERB cana-727 80 11 is be AUX cana-727 80 12 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 80 13 . . PUNCT cana-727 80 14 " " PUNCT cana-727 81 1 example example NOUN cana-727 81 2 2.6 2.6 NUM cana-727 81 3 : : PUNCT cana-727 81 4 " " PUNCT cana-727 81 5 let let VERB cana-727 81 6 f f PRON cana-727 81 7 be be AUX cana-727 81 8 a a DET cana-727 81 9 field field NOUN cana-727 81 10 and and CCONJ cana-727 81 11 v v AUX cana-727 81 12 be be AUX cana-727 81 13 a a DET cana-727 81 14 vector vector NOUN cana-727 81 15 space space NOUN cana-727 81 16 over over ADP cana-727 81 17 f f PROPN cana-727 81 18 with with ADP cana-727 81 19 dim dim PROPN cana-727 81 20 ( ( PUNCT cana-727 81 21 𝐹𝑉)=2 𝐹𝑉)=2 NOUN cana-727 81 22 . . PUNCT cana-727 82 1 let let VERB cana-727 82 2 r r NOUN cana-727 82 3 be be AUX cana-727 82 4 the the DET cana-727 82 5 trivial trivial ADJ cana-727 82 6 extension extension NOUN cana-727 82 7 of of ADP cana-727 82 8 f f PROPN cana-727 82 9 with with ADP cana-727 82 10 v v PROPN cana-727 82 11 , , PUNCT cana-727 82 12 i.e i.e PROPN cana-727 82 13 , , PUNCT cana-727 82 14 𝑅 𝑅 PROPN cana-727 82 15 = = PROPN cana-727 82 16 [ [ PUNCT cana-727 82 17 𝐹 𝐹 PROPN cana-727 82 18 𝑉 𝑉 PROPN cana-727 82 19 0 0 PUNCT cana-727 82 20 𝐹 𝐹 PROPN cana-727 82 21 ] ] PUNCT cana-727 83 1 = = PUNCT cana-727 83 2 { { PUNCT cana-727 83 3 [ [ PUNCT cana-727 83 4 𝑓 𝑓 PROPN cana-727 83 5 𝑣 𝑣 ADP cana-727 83 6 0 0 PUNCT cana-727 83 7 𝑓 𝑓 NOUN cana-727 83 8 ] ] PUNCT cana-727 83 9 : : PUNCT cana-727 83 10 𝑓 𝑓 X cana-727 83 11 ∈ ∈ PROPN cana-727 83 12 𝐹 𝐹 PROPN cana-727 83 13 , , PUNCT cana-727 83 14 𝑣 𝑣 PROPN cana-727 83 15 ∈ ∈ PROPN cana-727 83 16 𝑉 𝑉 PROPN cana-727 83 17 } } PUNCT cana-727 83 18 . . PUNCT cana-727 84 1 since since SCONJ cana-727 84 2 𝑅𝑅 𝑅𝑅 PROPN cana-727 84 3 is be AUX cana-727 84 4 indecomposable indecomposable ADJ cana-727 84 5 which which PRON cana-727 84 6 is be AUX cana-727 84 7 not not PART cana-727 84 8 cs cs ADJ cana-727 84 9 , , PUNCT cana-727 84 10 then then ADV cana-727 84 11 𝑅𝑅 𝑅𝑅 PROPN cana-727 84 12 is be AUX cana-727 84 13 not not PART cana-727 84 14 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 84 15 . . PUNCT cana-727 84 16 " " PUNCT cana-727 85 1 3 3 X cana-727 85 2 . . PUNCT cana-727 85 3 characterizations characterization NOUN cana-727 85 4 of of ADP cana-727 85 5 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 85 6 . . PUNCT cana-727 86 1 " " PUNCT cana-727 86 2 in in ADP cana-727 86 3 this this DET cana-727 86 4 section section NOUN cana-727 86 5 , , PUNCT cana-727 86 6 we we PRON cana-727 86 7 give give VERB cana-727 86 8 equivalent equivalent ADJ cana-727 86 9 conditions condition NOUN cana-727 86 10 to to ADP cana-727 86 11 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 86 12 property property NOUN cana-727 86 13 . . PUNCT cana-727 87 1 we we PRON cana-727 87 2 start start VERB cana-727 87 3 by by ADP cana-727 87 4 the the DET cana-727 87 5 following follow VERB cana-727 87 6 theorem theorem PROPN cana-727 87 7 . . PUNCT cana-727 87 8 theorem theorem VERB cana-727 87 9 3.1 3.1 NUM cana-727 87 10 : : PUNCT cana-727 87 11 an an DET cana-727 87 12 rmodule rmodule NOUN cana-727 87 13 m m VERB cana-727 87 14 is be AUX cana-727 87 15 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 87 16 if if SCONJ cana-727 87 17 and and CCONJ cana-727 87 18 only only ADV cana-727 87 19 if if SCONJ cana-727 87 20 for for ADP cana-727 87 21 each each DET cana-727 87 22 cyclic cyclic ADJ cana-727 87 23 submodule submodule NOUN cana-727 87 24 a a PRON cana-727 87 25 of of ADP cana-727 87 26 m m PROPN cana-727 87 27 , , PUNCT cana-727 87 28 there there PRON cana-727 87 29 is be VERB cana-727 87 30 a a DET cana-727 87 31 direct direct ADJ cana-727 87 32 summand summand NOUN cana-727 87 33 d d PROPN cana-727 87 34 of of ADP cana-727 87 35 m m PRON cana-727 87 36 such such ADJ cana-727 87 37 that that SCONJ cana-727 87 38 𝐴𝛽𝐷 𝐴𝛽𝐷 PROPN cana-727 87 39 and and CCONJ cana-727 87 40 d d X cana-727 87 41 ' ' PUNCT cana-727 87 42 is be AUX cana-727 87 43 a a DET cana-727 87 44 complement complement NOUN cana-727 87 45 of of ADP cana-727 87 46 a a PRON cana-727 87 47 , , PUNCT cana-727 87 48 where where SCONJ cana-727 87 49 𝑀 𝑀 PROPN cana-727 87 50 = = PUNCT cana-727 87 51 𝐷⨁𝐷′. 𝐷⨁𝐷′. NUM cana-727 87 52 proof proof NOUN cana-727 87 53 : : PUNCT cana-727 87 54 suppose suppose VERB cana-727 87 55 that that SCONJ cana-727 87 56 m m PROPN cana-727 87 57 is be AUX cana-727 87 58 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 87 59 , , PUNCT cana-727 87 60 let let VERB cana-727 87 61 a a PRON cana-727 87 62 be be AUX cana-727 87 63 a a DET cana-727 87 64 cyclic cyclic ADJ cana-727 87 65 submodule submodule NOUN cana-727 87 66 of of ADP cana-727 87 67 m m PROPN cana-727 87 68 , , PUNCT cana-727 87 69 there there PRON cana-727 87 70 is be VERB cana-727 87 71 a a DET cana-727 87 72 direct direct ADJ cana-727 87 73 summand summand NOUN cana-727 87 74 d d PROPN cana-727 87 75 of of ADP cana-727 87 76 m m PRON cana-727 87 77 such such ADJ cana-727 87 78 that that SCONJ cana-727 87 79 𝐴𝛽𝐷. 𝐴𝛽𝐷. PROPN cana-727 87 80 let let VERB cana-727 87 81 𝑀 𝑀 PROPN cana-727 87 82 = = SYM cana-727 87 83 𝐷⨁𝐷′ 𝐷⨁𝐷′ PROPN cana-727 87 84 , , PUNCT cana-727 87 85 for for ADP cana-727 87 86 some some DET cana-727 87 87 submodule submodule NOUN cana-727 87 88 d d NOUN cana-727 87 89 ' ' PUNCT cana-727 87 90 of of ADP cana-727 87 91 m. m. NOUN cana-727 87 92 since since SCONJ cana-727 87 93 𝐴 𝐴 PROPN cana-727 87 94 ∩ ∩ NOUN cana-727 87 95 𝐷 𝐷 PROPN cana-727 87 96 ≀e ≀e VERB cana-727 87 97 a a DET cana-727 87 98 , , PUNCT cana-727 87 99 then then ADV cana-727 87 100 𝐴 𝐴 PROPN cana-727 87 101 ∩ ∩ ADJ cana-727 87 102 𝐷′ 𝐷′ NOUN cana-727 87 103 = = NOUN cana-727 87 104 0 0 X cana-727 87 105 . . PUNCT cana-727 88 1 now now ADV cana-727 88 2 , , PUNCT cana-727 88 3 let let VERB cana-727 88 4 b b X cana-727 88 5 be be AUX cana-727 88 6 a a DET cana-727 88 7 submodule submodule NOUN cana-727 88 8 of of ADP cana-727 88 9 m m PRON cana-727 88 10 such such ADJ cana-727 88 11 that that DET cana-727 88 12 𝐷′ 𝐷′ PROPN cana-727 88 13 ≀ ≀ ADJ cana-727 88 14 𝐡 𝐡 PROPN cana-727 88 15 and and CCONJ cana-727 88 16 𝐴 𝐴 PROPN cana-727 88 17 ∩ ∩ NOUN cana-727 88 18 𝐡 𝐡 NOUN cana-727 88 19 = = NOUN cana-727 88 20 0 0 NUM cana-727 88 21 . . PUNCT cana-727 89 1 since since SCONJ cana-727 89 2 𝐴 𝐴 PROPN cana-727 89 3 ∩ ∩ NOUN cana-727 89 4 𝐷 𝐷 PROPN cana-727 89 5 ≀e ≀e NOUN cana-727 89 6 d d PROPN cana-727 89 7 , , PUNCT cana-727 89 8 then then ADV cana-727 89 9 𝐡 𝐡 NOUN cana-727 89 10 ∩ ∩ NOUN cana-727 89 11 𝐷 𝐷 NOUN cana-727 89 12 = = NOUN cana-727 89 13 0 0 PROPN cana-727 89 14 . . PUNCT cana-727 90 1 but but CCONJ cana-727 90 2 d d X cana-727 90 3 ' ' PUNCT cana-727 90 4 is be AUX cana-727 90 5 a a DET cana-727 90 6 complement complement NOUN cana-727 90 7 of of ADP cana-727 90 8 d d NOUN cana-727 90 9 , , PUNCT cana-727 90 10 therefore therefore ADV cana-727 90 11 b b X cana-727 90 12 = = SYM cana-727 90 13 d d NOUN cana-727 90 14 ' ' PUNCT cana-727 90 15 . . PUNCT cana-727 91 1 thus thus ADV cana-727 91 2 , , PUNCT cana-727 91 3 d d X cana-727 91 4 ' ' PUNCT cana-727 91 5 is be AUX cana-727 91 6 a a DET cana-727 91 7 complement complement NOUN cana-727 91 8 of of ADP cana-727 91 9 a. a. NOUN cana-727 91 10 the the DET cana-727 91 11 converse converse NOUN cana-727 91 12 is be AUX cana-727 91 13 clear clear ADJ cana-727 91 14 . . PUNCT cana-727 91 15 " " PUNCT cana-727 92 1 " " PUNCT cana-727 92 2 the the DET cana-727 92 3 next next ADJ cana-727 92 4 result result NOUN cana-727 92 5 gives give VERB cana-727 92 6 another another DET cana-727 92 7 characterization characterization NOUN cana-727 92 8 to to ADP cana-727 92 9 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 92 10 modules module NOUN cana-727 92 11 . . PUNCT cana-727 93 1 proposition proposition NOUN cana-727 93 2 3.2 3.2 NUM cana-727 93 3 : : PUNCT cana-727 93 4 let let VERB cana-727 93 5 m m PRON cana-727 93 6 be be AUX cana-727 93 7 an an DET cana-727 93 8 rmodule rmodule NOUN cana-727 93 9 , , PUNCT cana-727 93 10 the the DET cana-727 93 11 following follow VERB cana-727 93 12 conditions condition NOUN cana-727 93 13 are be AUX cana-727 93 14 equivalent equivalent ADJ cana-727 93 15 : : PUNCT cana-727 93 16 ( ( PUNCT cana-727 93 17 i i NOUN cana-727 93 18 ) ) PUNCT cana-727 94 1 m m AUX cana-727 94 2 is be AUX cana-727 94 3 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 94 4 . . PUNCT cana-727 94 5 ( ( PUNCT cana-727 94 6 ii ii NOUN cana-727 94 7 ) ) PUNCT cana-727 94 8 for for ADP cana-727 94 9 all all DET cana-727 94 10 cyclic cyclic PROPN cana-727 94 11 submodule submodule NOUN cana-727 94 12 a a PRON cana-727 94 13 of of ADP cana-727 94 14 m m PROPN cana-727 94 15 , , PUNCT cana-727 94 16 there there PRON cana-727 94 17 exists exist VERB cana-727 94 18 a a DET cana-727 94 19 submodule submodule NOUN cana-727 94 20 x x PUNCT cana-727 94 21 of of ADP cana-727 94 22 m m PROPN cana-727 94 23 and and CCONJ cana-727 94 24 a a DET cana-727 94 25 direct direct ADJ cana-727 94 26 summand summand NOUN cana-727 94 27 d d PROPN cana-727 94 28 of of ADP cana-727 94 29 m m PRON cana-727 94 30 such such ADJ cana-727 94 31 that that SCONJ cana-727 94 32 𝑋 𝑋 PROPN cana-727 94 33 ≀e ≀e VERB cana-727 94 34 a a PRON cana-727 94 35 and and CCONJ cana-727 94 36 𝑋 𝑋 PROPN cana-727 94 37 ≀e ≀e PROPN cana-727 94 38 d. d. PROPN cana-727 94 39 ( ( PUNCT cana-727 94 40 iii iii NOUN cana-727 94 41 ) ) PUNCT cana-727 94 42 for for ADP cana-727 94 43 every every DET cana-727 94 44 cyclic cyclic ADJ cana-727 94 45 submodule submodule NOUN cana-727 94 46 a a PRON cana-727 94 47 of of ADP cana-727 94 48 m m PRON cana-727 94 49 there there PRON cana-727 94 50 exists exist VERB cana-727 94 51 a a DET cana-727 94 52 complement complement NOUN cana-727 94 53 b b NOUN cana-727 94 54 of of ADP cana-727 94 55 a a PRON cana-727 94 56 and and CCONJ cana-727 94 57 a a DET cana-727 94 58 complement complement NOUN cana-727 94 59 c c NOUN cana-727 94 60 of of ADP cana-727 94 61 b b NOUN cana-727 94 62 such such ADJ cana-727 94 63 that that DET cana-727 94 64 aΞ²c aΞ²c NOUN cana-727 94 65 and and CCONJ cana-727 94 66 each each DET cana-727 94 67 homomorphism homomorphism NOUN cana-727 94 68 𝑓 𝑓 X cana-727 94 69 : : PUNCT cana-727 94 70 𝐢⨁𝐡 𝐢⨁𝐡 NOUN cana-727 94 71 β†’ β†’ SYM cana-727 94 72 𝑀 𝑀 PROPN cana-727 94 73 extends extend VERB cana-727 94 74 to to ADP cana-727 94 75 a a DET cana-727 94 76 homomorphism homomorphism NOUN cana-727 94 77 𝑔 𝑔 NOUN cana-727 94 78 : : PUNCT cana-727 94 79 𝑀 𝑀 PROPN cana-727 94 80 β†’ β†’ SYM cana-727 94 81 𝑀. 𝑀. NOUN cana-727 94 82 proof proof NOUN cana-727 94 83 : : PUNCT cana-727 94 84 ( ( PUNCT cana-727 94 85 i i NOUN cana-727 94 86 ) ) PUNCT cana-727 94 87 ⟹(ii ⟹(ii NUM cana-727 94 88 ) ) PUNCT cana-727 94 89 assume assume VERB cana-727 94 90 that that SCONJ cana-727 94 91 m m PROPN cana-727 94 92 is be AUX cana-727 94 93 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 94 94 and and CCONJ cana-727 94 95 let let VERB cana-727 94 96 a a PRON cana-727 94 97 be be AUX cana-727 94 98 a a DET cana-727 94 99 cyclic cyclic ADJ cana-727 94 100 submodule submodule NOUN cana-727 94 101 of of ADP cana-727 94 102 m m PROPN cana-727 94 103 , , PUNCT cana-727 94 104 there there PRON cana-727 94 105 is be VERB cana-727 94 106 a a DET cana-727 94 107 direct direct ADJ cana-727 94 108 summand summand NOUN cana-727 94 109 d d PROPN cana-727 94 110 of of ADP cana-727 94 111 m m PRON cana-727 94 112 such such ADJ cana-727 94 113 that that DET cana-727 94 114 𝐴𝛽𝐷 𝐴𝛽𝐷 PROPN cana-727 94 115 , , PUNCT cana-727 94 116 hence hence ADV cana-727 94 117 𝐴 𝐴 PROPN cana-727 94 118 ∩ ∩ NOUN cana-727 94 119 𝐷 𝐷 PROPN cana-727 94 120 ≀e ≀e VERB cana-727 94 121 a a DET cana-727 94 122 and and CCONJ cana-727 94 123 𝐴 𝐴 PROPN cana-727 94 124 ∩ ∩ NOUN cana-727 94 125 𝐷 𝐷 PROPN cana-727 94 126 ≀e ≀e NOUN cana-727 94 127 d. d. PROPN cana-727 94 128 take take VERB cana-727 94 129 x x PUNCT cana-727 94 130 = = SYM cana-727 94 131 𝐴 𝐴 PROPN cana-727 94 132 ∩ ∩ PROPN cana-727 94 133 𝐷 𝐷 PROPN cana-727 94 134 , , PUNCT cana-727 94 135 we we PRON cana-727 94 136 get get VERB cana-727 94 137 the the DET cana-727 94 138 result result NOUN cana-727 94 139 . . PUNCT cana-727 95 1 ( ( PUNCT cana-727 95 2 ii)⟹(iii ii)⟹(iii NOUN cana-727 95 3 ) ) PUNCT cana-727 95 4 let let VERB cana-727 95 5 a a PRON cana-727 95 6 be be AUX cana-727 95 7 a a DET cana-727 95 8 cyclic cyclic ADJ cana-727 95 9 submodule submodule NOUN cana-727 95 10 of of ADP cana-727 95 11 m. m. NOUN cana-727 95 12 by by ADP cana-727 95 13 ( ( PUNCT cana-727 95 14 ii ii NOUN cana-727 95 15 ) ) PUNCT cana-727 95 16 , , PUNCT cana-727 95 17 there there PRON cana-727 95 18 exists exist VERB cana-727 95 19 a a DET cana-727 95 20 submodule submodule NOUN cana-727 95 21 x x PUNCT cana-727 95 22 of of ADP cana-727 95 23 m m PROPN cana-727 95 24 and and CCONJ cana-727 95 25 a a DET cana-727 95 26 direct direct ADJ cana-727 95 27 summand summand NOUN cana-727 95 28 d d PROPN cana-727 95 29 of of ADP cana-727 95 30 m m PRON cana-727 95 31 such such ADJ cana-727 95 32 that that SCONJ cana-727 95 33 𝑀 𝑀 PROPN cana-727 95 34 = = SYM cana-727 95 35 𝐷⨁𝐷′ 𝐷⨁𝐷′ PROPN cana-727 95 36 , , PUNCT cana-727 95 37 𝑋 𝑋 PROPN cana-727 95 38 ≀e ≀e VERB cana-727 95 39 a a PRON cana-727 95 40 and and CCONJ cana-727 95 41 𝑋 𝑋 PROPN cana-727 95 42 ≀e ≀e PROPN cana-727 95 43 d. d. PROPN cana-727 95 44 take take VERB cana-727 95 45 d d NOUN cana-727 95 46 = = SYM cana-727 95 47 c c PROPN cana-727 95 48 and and CCONJ cana-727 95 49 d d NOUN cana-727 95 50 ' ' PUNCT cana-727 95 51 = = ADJ cana-727 95 52 b. b. PROPN cana-727 95 53 ( ( PUNCT cana-727 95 54 iii)⟹(i iii)⟹(i PROPN cana-727 95 55 ) ) PUNCT cana-727 95 56 let let VERB cana-727 95 57 a a PRON cana-727 95 58 be be AUX cana-727 95 59 a a DET cana-727 95 60 cyclic cyclic ADJ cana-727 95 61 submodule submodule NOUN cana-727 95 62 of of ADP cana-727 95 63 m. m. NOUN cana-727 95 64 from from ADP cana-727 95 65 ( ( PUNCT cana-727 95 66 iii iii NOUN cana-727 95 67 ) ) PUNCT cana-727 95 68 , , PUNCT cana-727 95 69 there there PRON cana-727 95 70 exists exist VERB cana-727 95 71 a a DET cana-727 95 72 complement complement NOUN cana-727 95 73 b b NOUN cana-727 95 74 of of ADP cana-727 95 75 a a PRON cana-727 95 76 and and CCONJ cana-727 95 77 a a DET cana-727 95 78 complement complement NOUN cana-727 95 79 c c NOUN cana-727 95 80 of of ADP cana-727 95 81 b b NOUN cana-727 95 82 such such ADJ cana-727 95 83 that that DET cana-727 95 84 aΞ²c aΞ²c NOUN cana-727 95 85 and and CCONJ cana-727 95 86 every every DET cana-727 95 87 homomorphism homomorphism NOUN cana-727 95 88 𝑓 𝑓 X cana-727 95 89 : : PUNCT cana-727 95 90 𝐢⨁𝐡 𝐢⨁𝐡 NOUN cana-727 95 91 β†’ β†’ SYM cana-727 95 92 𝑀 𝑀 PROPN cana-727 95 93 extends extend VERB cana-727 95 94 to to ADP cana-727 95 95 a a DET cana-727 95 96 homomorphism homomorphism NOUN cana-727 95 97 𝑔 𝑔 NOUN cana-727 95 98 : : PUNCT cana-727 95 99 𝑀 𝑀 PROPN cana-727 95 100 β†’ β†’ SYM cana-727 95 101 𝑀and 𝑀and PROPN cana-727 95 102 by by ADP cana-727 95 103 [ [ PUNCT cana-727 95 104 11 11 NUM cana-727 95 105 , , PUNCT cana-727 95 106 lemma lemma PROPN cana-727 95 107 3.97 3.97 NUM cana-727 95 108 ] ] PUNCT cana-727 95 109 , , PUNCT cana-727 95 110 d d X cana-727 95 111 is be AUX cana-727 95 112 a a DET cana-727 95 113 direct direct ADJ cana-727 95 114 summand summand NOUN cana-727 95 115 of of ADP cana-727 95 116 m m PROPN cana-727 95 117 , , PUNCT cana-727 95 118 hence hence ADV cana-727 95 119 m m VERB cana-727 95 120 is be AUX cana-727 95 121 𝐺𝑝extending 𝐺𝑝extende VERB cana-727 95 122 . . PUNCT cana-727 95 123 " " PUNCT cana-727 96 1 communications communication NOUN cana-727 96 2 on on ADP cana-727 96 3 applied apply VERB cana-727 96 4 nonlinear nonlinear ADJ cana-727 96 5 analysis analysis NOUN cana-727 96 6 issn issn NOUN cana-727 96 7 : : PUNCT cana-727 96 8 1074 1074 NUM cana-727 96 9 - - PUNCT cana-727 96 10 133x 133x NUM cana-727 96 11 vol vol NOUN cana-727 96 12 31 31 NUM cana-727 96 13 no no NOUN cana-727 96 14 . . PUNCT cana-727 97 1 3s 3s NUM cana-727 97 2 ( ( PUNCT cana-727 97 3 2024 2024 NUM cana-727 97 4 ) ) PUNCT cana-727 97 5 13 13 NUM cana-727 97 6 https://internationalpubls.com https://internationalpubls.com X cana-727 97 7 theorem theorem VERB cana-727 97 8 3.3 3.3 NUM cana-727 97 9 : : PUNCT cana-727 97 10 a a DET cana-727 97 11 module module NOUN cana-727 97 12 m m VERB cana-727 97 13 is be AUX cana-727 97 14 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 97 15 if if SCONJ cana-727 97 16 and and CCONJ cana-727 97 17 only only ADV cana-727 97 18 if if SCONJ cana-727 97 19 for for ADP cana-727 97 20 every every DET cana-727 97 21 direct direct ADJ cana-727 97 22 summand summand NOUN cana-727 97 23 a a PRON cana-727 97 24 of of ADP cana-727 97 25 the the DET cana-727 97 26 injective injective ADJ cana-727 97 27 hull hull PROPN cana-727 97 28 e(m e(m PROPN cana-727 97 29 ) ) PUNCT cana-727 97 30 of of ADP cana-727 97 31 m m PROPN cana-727 97 32 with with ADP cana-727 97 33 𝐴 𝐴 PROPN cana-727 97 34 ∩ ∩ NOUN cana-727 97 35 𝑀 𝑀 PROPN cana-727 97 36 is be AUX cana-727 97 37 cyclic cyclic ADJ cana-727 97 38 submodule submodule NOUN cana-727 97 39 of of ADP cana-727 97 40 m m PROPN cana-727 97 41 , , PUNCT cana-727 97 42 there there PRON cana-727 97 43 is be VERB cana-727 97 44 a a DET cana-727 97 45 direct direct ADJ cana-727 97 46 summand summand NOUN cana-727 97 47 d d PROPN cana-727 97 48 of of ADP cana-727 97 49 m m PRON cana-727 97 50 such such ADJ cana-727 97 51 that that SCONJ cana-727 97 52 ( ( PUNCT cana-727 97 53 𝐴 𝐴 PROPN cana-727 97 54 ∩ ∩ NOUN cana-727 97 55 𝑀)𝛽𝐷. 𝑀)𝛽𝐷. PROPN cana-727 97 56 proof proof NOUN cana-727 97 57 : : PUNCT cana-727 97 58 " " PUNCT cana-727 97 59 let let VERB cana-727 97 60 a a PRON cana-727 97 61 be be AUX cana-727 97 62 a a DET cana-727 97 63 cyclic cyclic ADJ cana-727 97 64 submodule submodule NOUN cana-727 97 65 of of ADP cana-727 97 66 m m PRON cana-727 97 67 and and CCONJ cana-727 97 68 let let VERB cana-727 97 69 b b X cana-727 97 70 be be AUX cana-727 97 71 a a DET cana-727 97 72 complement complement NOUN cana-727 97 73 of of ADP cana-727 97 74 a a PRON cana-727 97 75 , , PUNCT cana-727 97 76 then then ADV cana-727 97 77 𝐴⨁𝐡 𝐴⨁𝐡 VERB cana-727 97 78 ≀e ≀e NOUN cana-727 97 79 m. m. NOUN cana-727 97 80 since since SCONJ cana-727 97 81 𝑀 𝑀 PROPN cana-727 97 82 ≀e ≀e VERB cana-727 97 83 e(m e(m PROPN cana-727 97 84 ) ) PUNCT cana-727 97 85 , , PUNCT cana-727 97 86 then then ADV cana-727 97 87 𝐴⨁𝐡 𝐴⨁𝐡 VERB cana-727 97 88 ≀e ≀e PROPN cana-727 97 89 e(m e(m PROPN cana-727 97 90 ) ) PUNCT cana-727 97 91 implies imply VERB cana-727 97 92 𝐸(𝑀 𝐸(𝑀 NOUN cana-727 97 93 ) ) PUNCT cana-727 97 94 = = PRON cana-727 97 95 𝐸(𝐴)⨁𝐸(𝐡 𝐸(𝐴)⨁𝐸(𝐡 NOUN cana-727 97 96 ) ) PUNCT cana-727 97 97 . . PUNCT cana-727 98 1 it it PRON cana-727 98 2 can can AUX cana-727 98 3 be be AUX cana-727 98 4 seen see VERB cana-727 98 5 that that SCONJ cana-727 98 6 𝐸(𝐴 𝐸(𝐴 NOUN cana-727 98 7 ) ) PUNCT cana-727 98 8 ∩ ∩ NOUN cana-727 98 9 𝑀 𝑀 PROPN cana-727 98 10 is be AUX cana-727 98 11 cyclic cyclic ADJ cana-727 98 12 submodule submodule NOUN cana-727 98 13 in in ADP cana-727 98 14 m. m. NOUN cana-727 98 15 by by ADP cana-727 98 16 our our PRON cana-727 98 17 assumption assumption NOUN cana-727 98 18 , , PUNCT cana-727 98 19 there there PRON cana-727 98 20 is be VERB cana-727 98 21 a a DET cana-727 98 22 direct direct ADJ cana-727 98 23 summand summand NOUN cana-727 98 24 d d PROPN cana-727 98 25 of of ADP cana-727 98 26 m m PRON cana-727 98 27 such such ADJ cana-727 98 28 that that SCONJ cana-727 98 29 ( ( PUNCT cana-727 98 30 𝐸(𝐴 𝐸(𝐴 NOUN cana-727 98 31 ) ) PUNCT cana-727 98 32 ∩ ∩ NOUN cana-727 98 33 𝑀)𝛽𝐷. 𝑀)𝛽𝐷. PROPN cana-727 98 34 but but CCONJ cana-727 98 35 we we PRON cana-727 98 36 have(𝐴 have(𝐴 ADJ cana-727 98 37 ∩ ∩ NOUN cana-727 98 38 𝑀)𝛽(𝐸(𝐴 𝑀)𝛽(𝐸(𝐴 NOUN cana-727 98 39 ) ) PUNCT cana-727 98 40 ∩ ∩ PROPN cana-727 98 41 𝑀 𝑀 PROPN cana-727 98 42 ) ) PUNCT cana-727 98 43 , , PUNCT cana-727 98 44 hence hence ADV cana-727 98 45 𝐴𝛽𝐷. 𝐴𝛽𝐷. PROPN cana-727 98 46 the the DET cana-727 98 47 converse converse NOUN cana-727 98 48 implication implication NOUN cana-727 98 49 is be AUX cana-727 98 50 clear clear ADJ cana-727 98 51 . . PUNCT cana-727 98 52 " " PUNCT cana-727 98 53 theorem theorem VERB cana-727 98 54 3.4 3.4 NUM cana-727 98 55 : : PUNCT cana-727 98 56 " " PUNCT cana-727 98 57 suppose suppose VERB cana-727 98 58 m m PRON cana-727 98 59 is be AUX cana-727 98 60 an an DET cana-727 98 61 r r NOUN cana-727 98 62 - - PUNCT cana-727 98 63 module module NOUN cana-727 98 64 . . PUNCT cana-727 99 1 the the DET cana-727 99 2 assertions assertion NOUN cana-727 99 3 that that PRON cana-727 99 4 follow follow VERB cana-727 99 5 are be AUX cana-727 99 6 identical identical ADJ cana-727 99 7 . . PUNCT cana-727 100 1 ( ( PUNCT cana-727 100 2 i i NOUN cana-727 100 3 ) ) PUNCT cana-727 100 4 m m AUX cana-727 100 5 is be AUX cana-727 100 6 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 100 7 module module NOUN cana-727 100 8 . . PUNCT cana-727 101 1 ( ( PUNCT cana-727 101 2 ii ii NOUN cana-727 101 3 ) ) PUNCT cana-727 101 4 a a DET cana-727 101 5 decomposition decomposition NOUN cana-727 101 6 exists exist VERB cana-727 101 7 for for ADP cana-727 101 8 each each DET cana-727 101 9 cyclic cyclic ADJ cana-727 101 10 submodule submodule NOUN cana-727 101 11 a a PRON cana-727 101 12 of of ADP cana-727 101 13 the the DET cana-727 101 14 module module NOUN cana-727 101 15 m. m. NOUN cana-727 101 16 m m PROPN cana-727 101 17 = = PUNCT cana-727 101 18 dοƒ…d dοƒ…d PROPN cana-727 101 19 ' ' PUNCT cana-727 101 20 , , PUNCT cana-727 101 21 such such ADJ cana-727 101 22 that that SCONJ cana-727 101 23 ( ( PUNCT cana-727 101 24 d'+a d'+a NOUN cana-727 101 25 ) ) PUNCT cana-727 101 26 Ξ²m Ξ²m NOUN cana-727 101 27 . . PUNCT cana-727 102 1 ( ( PUNCT cana-727 102 2 ii ii NOUN cana-727 102 3 ) ) PUNCT cana-727 102 4 fore fore NOUN cana-727 102 5 very very ADV cana-727 102 6 cyclic cyclic ADJ cana-727 102 7 submodule submodule NOUN cana-727 102 8 a a PRON cana-727 102 9 of of ADP cana-727 102 10 m m PROPN cana-727 102 11 , , PUNCT cana-727 102 12 there there PRON cana-727 102 13 is be VERB cana-727 102 14 a a DET cana-727 102 15 decomposition decomposition NOUN cana-727 102 16 a a DET cana-727 102 17 m m NOUN cana-727 102 18 = = SYM cana-727 102 19 a a DET cana-727 102 20 l l NOUN cana-727 102 21 οƒ… οƒ… VERB cana-727 102 22 a a DET cana-727 102 23 k k PROPN cana-727 102 24 such such ADJ cana-727 102 25 that that SCONJ cana-727 102 26 l l NOUN cana-727 102 27 is be AUX cana-727 102 28 a a DET cana-727 102 29 direct direct ADJ cana-727 102 30 summand summand NOUN cana-727 102 31 of of ADP cana-727 102 32 m m PROPN cana-727 102 33 and and CCONJ cana-727 102 34 kΞ²m kΞ²m PROPN cana-727 102 35 . . PUNCT cana-727 102 36 " " PUNCT cana-727 103 1 proof proof NOUN cana-727 103 2 : : PUNCT cana-727 103 3 ( ( PUNCT cana-727 103 4 i)οƒž(ii)"let i)οƒž(ii)"let AUX cana-727 103 5 m m AUX cana-727 103 6 be be AUX cana-727 103 7 a a DET cana-727 103 8 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 103 9 and and CCONJ cana-727 103 10 let let VERB cana-727 103 11 a a PRON cana-727 103 12 be be AUX cana-727 103 13 a a DET cana-727 103 14 cyclic cyclic ADJ cana-727 103 15 submodule submodule NOUN cana-727 103 16 of of ADP cana-727 103 17 m m PROPN cana-727 103 18 , , PUNCT cana-727 103 19 there there PRON cana-727 103 20 exists exist VERB cana-727 103 21 direct direct ADJ cana-727 103 22 summand summand NOUN cana-727 103 23 d d PROPN cana-727 103 24 of of ADP cana-727 103 25 m m PRON cana-727 103 26 such such ADJ cana-727 103 27 that that SCONJ cana-727 103 28 a a DET cana-727 103 29 Ξ²d Ξ²d NOUN cana-727 103 30 , , PUNCT cana-727 103 31 then then ADV cana-727 103 32 m m VERB cana-727 103 33 = = PUNCT cana-727 103 34 dοƒ…d dοƒ…d PROPN cana-727 103 35 ' ' PUNCT cana-727 103 36 , , PUNCT cana-727 103 37 d d X cana-727 103 38 ' ' PUNCT cana-727 103 39 ≀ ≀ NUM cana-727 103 40 m. m. NOUN cana-727 103 41 since since SCONJ cana-727 103 42 { { PUNCT cana-727 103 43 a a PRON cana-727 103 44 , , PUNCT cana-727 103 45 d d NOUN cana-727 103 46 ' ' PUNCT cana-727 103 47 } } PUNCT cana-727 103 48 is be AUX cana-727 103 49 an an DET cana-727 103 50 independent independent ADJ cana-727 103 51 family family NOUN cana-727 103 52 , , PUNCT cana-727 103 53 then then ADV cana-727 103 54 ( ( PUNCT cana-727 103 55 a+d a+d NOUN cana-727 103 56 ' ' PUNCT cana-727 103 57 ) ) PUNCT cana-727 103 58 Ξ²m Ξ²m VERB cana-727 103 59 , , PUNCT cana-727 103 60 see see VERB cana-727 103 61 [ [ X cana-727 103 62 12 12 NUM cana-727 103 63 , , PUNCT cana-727 103 64 proposition proposition NOUN cana-727 103 65 1.4 1.4 NUM cana-727 103 66 ] ] PUNCT cana-727 103 67 . . PUNCT cana-727 103 68 " " PUNCT cana-727 104 1 ( ( PUNCT cana-727 104 2 ii)οƒž(iii ii)οƒž(iii PROPN cana-727 104 3 ) ) PUNCT cana-727 104 4 " " PUNCT cana-727 104 5 let let VERB cana-727 104 6 a a PRON cana-727 104 7 be be AUX cana-727 104 8 a a DET cana-727 104 9 cyclic cyclic ADJ cana-727 104 10 submodule submodule NOUN cana-727 104 11 of of ADP cana-727 104 12 m. m. NOUN cana-727 104 13 by by ADP cana-727 104 14 ( ( PUNCT cana-727 104 15 ii ii NOUN cana-727 104 16 ) ) PUNCT cana-727 104 17 , , PUNCT cana-727 104 18 there there PRON cana-727 104 19 is be VERB cana-727 104 20 a a DET cana-727 104 21 decomposition decomposition NOUN cana-727 104 22 m m NOUN cana-727 104 23 = = PUNCT cana-727 104 24 dοƒ…d dοƒ…d PROPN cana-727 104 25 ' ' PUNCT cana-727 104 26 , , PUNCT cana-727 104 27 such such ADJ cana-727 104 28 that that SCONJ cana-727 104 29 ( ( PUNCT cana-727 104 30 d'+a d'+a NOUN cana-727 104 31 ) ) PUNCT cana-727 104 32 Ξ²m Ξ²m NOUN cana-727 104 33 . . PROPN cana-727 105 1 claim claim VERB cana-727 105 2 that that SCONJ cana-727 105 3 a a DET cana-727 105 4 m m NOUN cana-727 105 5 = = PUNCT cana-727 105 6 a a DET cana-727 105 7 ad ad NOUN cana-727 105 8 + + CCONJ cana-727 105 9 οƒ… οƒ… VERB cana-727 105 10 a a DET cana-727 105 11 ad ad NOUN cana-727 105 12 + + NOUN cana-727 105 13 ' ' PUNCT cana-727 105 14 . . PUNCT cana-727 106 1 since since SCONJ cana-727 106 2 m m PROPN cana-727 106 3 = = SYM cana-727 106 4 dοƒ…d dοƒ…d PROPN cana-727 106 5 ' ' NUM cana-727 106 6 , , PUNCT cana-727 106 7 then then ADV cana-727 106 8 a a DET cana-727 106 9 m m NOUN cana-727 106 10 = = SYM cana-727 106 11 a a DET cana-727 106 12 dd dd NOUN cana-727 106 13 ' ' X cana-727 106 14 + + PROPN cana-727 106 15 = = PUNCT cana-727 106 16 a a DET cana-727 106 17 d d NOUN cana-727 106 18 + + CCONJ cana-727 106 19 a a DET cana-727 106 20 ad ad NOUN cana-727 106 21 + + NOUN cana-727 106 22 ' ' PUNCT cana-727 106 23 and and CCONJ cana-727 106 24 a a DET cana-727 106 25 ad ad NOUN cana-727 106 26 + + X cana-727 106 27   PUNCT cana-727 106 28 a a DET cana-727 106 29 ad ad NOUN cana-727 106 30 + + NOUN cana-727 106 31 ' ' PUNCT cana-727 106 32 = = NOUN cana-727 106 33 a a DET cana-727 106 34 add add NOUN cana-727 106 35 ) ) PUNCT cana-727 106 36 ' ' PUNCT cana-727 106 37 ( ( PUNCT cana-727 106 38 + + ADJ cana-727 106 39   NOUN cana-727 106 40 = = PUNCT cana-727 106 41 a a DET cana-727 106 42 dda dda ADJ cana-727 106 43 ) ) PUNCT cana-727 106 44 ' ' PUNCT cana-727 106 45 ( ( PUNCT cana-727 106 46 + + NOUN cana-727 106 47 = = PRON cana-727 106 48 a a X cana-727 106 49 , , PUNCT cana-727 106 50 hence hence ADV cana-727 106 51 a a DET cana-727 106 52 m m NOUN cana-727 106 53 = = SYM cana-727 106 54 a a DET cana-727 106 55 ad ad NOUN cana-727 106 56 + + CCONJ cana-727 106 57 οƒ… οƒ… VERB cana-727 106 58 a a DET cana-727 106 59 ad ad NOUN cana-727 106 60 + + NOUN cana-727 106 61 ' ' PUNCT cana-727 106 62 . . PUNCT cana-727 107 1 take take VERB cana-727 107 2 k k NOUN cana-727 107 3 = = PUNCT cana-727 107 4 d'+a d'+a NOUN cana-727 107 5 and and CCONJ cana-727 107 6 l l NOUN cana-727 107 7 = = PUNCT cana-727 107 8 d+a d+a PROPN cana-727 107 9 , , PUNCT cana-727 107 10 so so SCONJ cana-727 107 11 we we PRON cana-727 107 12 get get VERB cana-727 107 13 the the DET cana-727 107 14 result result NOUN cana-727 107 15 . . PUNCT cana-727 107 16 " " PUNCT cana-727 108 1 ( ( PUNCT cana-727 108 2 iii)οƒž(i)"to iii)οƒž(i)"to PROPN cana-727 108 3 show show VERB cana-727 108 4 that that SCONJ cana-727 108 5 m m NOUN cana-727 108 6 is be AUX cana-727 108 7 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 108 8 , , PUNCT cana-727 108 9 let let VERB cana-727 108 10 a a PRON cana-727 108 11 be be AUX cana-727 108 12 a a DET cana-727 108 13 cyclic cyclic ADJ cana-727 108 14 submodule submodule NOUN cana-727 108 15 of of ADP cana-727 108 16 m. m. NOUN cana-727 108 17 by by ADP cana-727 108 18 ( ( PUNCT cana-727 108 19 iii iii NOUN cana-727 108 20 ) ) PUNCT cana-727 108 21 , , PUNCT cana-727 108 22 there there PRON cana-727 108 23 is be VERB cana-727 108 24 a a DET cana-727 108 25 decomposition decomposition NOUN cana-727 108 26 a a DET cana-727 108 27 m m NOUN cana-727 108 28 = = SYM cana-727 108 29 a a DET cana-727 108 30 l l NOUN cana-727 108 31 οƒ… οƒ… VERB cana-727 108 32 a a DET cana-727 108 33 k k PROPN cana-727 108 34 such such ADJ cana-727 108 35 that that SCONJ cana-727 108 36 l l NOUN cana-727 108 37 is be AUX cana-727 108 38 a a DET cana-727 108 39 direct direct ADJ cana-727 108 40 summand summand NOUN cana-727 108 41 of of ADP cana-727 108 42 m m PROPN cana-727 108 43 and and CCONJ cana-727 108 44 kΞ²m kΞ²m PROPN cana-727 108 45 . . PUNCT cana-727 109 1 it it PRON cana-727 109 2 is be AUX cana-727 109 3 enough enough ADJ cana-727 109 4 to to PART cana-727 109 5 show show VERB cana-727 109 6 that that SCONJ cana-727 109 7 a a DET cana-727 109 8 Ξ²l Ξ²l NOUN cana-727 109 9 . . PUNCT cana-727 110 1 let let VERB cana-727 110 2 i i PRON cana-727 110 3 : : PUNCT cana-727 110 4 lβ†’ lβ†’ NOUN cana-727 110 5 m m AUX cana-727 110 6 be be AUX cana-727 110 7 the the DET cana-727 110 8 injection injection NOUN cana-727 110 9 map map NOUN cana-727 110 10 . . PUNCT cana-727 111 1 since since SCONJ cana-727 111 2 kΞ²m kΞ²m PROPN cana-727 111 3 , , PUNCT cana-727 111 4 then then ADV cana-727 111 5 i i PRON cana-727 111 6 -1 -1 PUNCT cana-727 111 7 ( ( PUNCT cana-727 111 8 k k X cana-727 111 9 ) ) PUNCT cana-727 111 10 Ξ²i Ξ²i PRON cana-727 111 11 -1 -1 PUNCT cana-727 111 12 ( ( PUNCT cana-727 111 13 m m NOUN cana-727 111 14 ) ) PUNCT cana-727 111 15 , , PUNCT cana-727 111 16 that that ADV cana-727 111 17 is is ADV cana-727 111 18 ( ( PUNCT cana-727 111 19 lk)Ξ²d lk)Ξ²d PROPN cana-727 111 20 . . PUNCT cana-727 112 1 one one PRON cana-727 112 2 can can AUX cana-727 112 3 easily easily ADV cana-727 112 4 show show VERB cana-727 112 5 that that SCONJ cana-727 112 6 lk lk NOUN cana-727 112 7 = = SYM cana-727 112 8 a a NOUN cana-727 112 9 , , PUNCT cana-727 112 10 so so ADV cana-727 112 11 m m VERB cana-727 112 12 is be AUX cana-727 112 13 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 112 14 module module NOUN cana-727 112 15 . . PUNCT cana-727 112 16 " " PUNCT cana-727 113 1 " " PUNCT cana-727 113 2 proposition proposition NOUN cana-727 113 3 3.5 3.5 NUM cana-727 113 4 : : PUNCT cana-727 113 5 let let VERB cana-727 113 6 m m PRON cana-727 113 7 be be AUX cana-727 113 8 an an DET cana-727 113 9 r r NOUN cana-727 113 10 - - PUNCT cana-727 113 11 module module NOUN cana-727 113 12 . . PUNCT cana-727 114 1 then then ADV cana-727 114 2 m m VERB cana-727 114 3 is be AUX cana-727 114 4 𝐺𝑃-extending 𝐺𝑃-extende VERB cana-727 114 5 module module NOUN cana-727 114 6 if if SCONJ cana-727 114 7 and and CCONJ cana-727 114 8 only only ADV cana-727 114 9 if if SCONJ cana-727 114 10 for for ADP cana-727 114 11 every every DET cana-727 114 12 cyclic cyclic ADJ cana-727 114 13 submodule submodule NOUN cana-727 114 14 a a PRON cana-727 114 15 of of ADP cana-727 114 16 m m PROPN cana-727 114 17 , , PUNCT cana-727 114 18 there there PRON cana-727 114 19 exists exist VERB cana-727 114 20 an an DET cana-727 114 21 idempotent idempotent ADJ cana-727 114 22 f f PROPN cana-727 114 23 οƒŽend οƒŽend NOUN cana-727 114 24 ( ( PUNCT cana-727 114 25 m m NOUN cana-727 114 26 ) ) PUNCT cana-727 114 27 such such ADJ cana-727 114 28 that that SCONJ cana-727 114 29 a a DET cana-727 114 30 Ξ²f Ξ²f INTJ cana-727 114 31 ( ( PUNCT cana-727 114 32 m m NOUN cana-727 114 33 ) ) PUNCT cana-727 114 34 . . PUNCT cana-727 114 35 " " PUNCT cana-727 115 1 4 4 NUM cana-727 115 2 . . PUNCT cana-727 115 3 decompositions decomposition NOUN cana-727 115 4 . . PUNCT cana-727 116 1 " " PUNCT cana-727 116 2 there there PRON cana-727 116 3 are be VERB cana-727 116 4 nonsingular nonsingular ADJ cana-727 116 5 modules module NOUN cana-727 116 6 𝑀 𝑀 NOUN cana-727 116 7 = = SYM cana-727 116 8 𝑀1⨁𝑀2 𝑀1⨁𝑀2 PROPN cana-727 116 9 in in ADP cana-727 116 10 which which PRON cana-727 116 11 𝑀1 𝑀1 NOUN cana-727 116 12 and and CCONJ cana-727 116 13 𝑀2 𝑀2 PROPN cana-727 116 14 are be AUX cana-727 116 15 p p NOUN cana-727 116 16 - - PUNCT cana-727 116 17 extending extend VERB cana-727 116 18 , , PUNCT cana-727 116 19 but but CCONJ cana-727 116 20 m m NOUN cana-727 116 21 is be AUX cana-727 116 22 not not PART cana-727 116 23 pextending pextende VERB cana-727 116 24 ( ( PUNCT cana-727 116 25 e.g e.g NOUN cana-727 116 26 , , PUNCT cana-727 116 27 let let VERB cana-727 116 28 r r NOUN cana-727 116 29 = = SYM cana-727 116 30 𝕫[π‘₯ 𝕫[π‘₯ PROPN cana-727 116 31 ] ] PUNCT cana-727 116 32 be be VERB cana-727 116 33 a a DET cana-727 116 34 polynomial polynomial ADJ cana-727 116 35 ring ring NOUN cana-727 116 36 of of ADP cana-727 116 37 integers integer NOUN cana-727 116 38 and and CCONJ cana-727 116 39 let let VERB cana-727 116 40 m m VERB cana-727 116 41 = = ADJ cana-727 116 42 𝕫[π‘₯]⨁𝕫[π‘₯ 𝕫[π‘₯]⨁𝕫[π‘₯ PROPN cana-727 116 43 ] ] PUNCT cana-727 116 44 ) ) PUNCT cana-727 116 45 . . PUNCT cana-727 117 1 note note VERB cana-727 117 2 that that SCONJ cana-727 117 3 𝕫[π‘₯ 𝕫[π‘₯ PROPN cana-727 117 4 ] ] X cana-727 117 5 is be AUX cana-727 117 6 𝐺 𝐺 PROPN cana-727 117 7 βˆ’extending βˆ’extending NOUN cana-727 117 8 , , PUNCT cana-727 117 9 by by ADP cana-727 117 10 [ [ X cana-727 117 11 1 1 NUM cana-727 117 12 ] ] PUNCT cana-727 117 13 and and CCONJ cana-727 117 14 hence hence ADV cana-727 117 15 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 117 16 but but CCONJ cana-727 117 17 m m NOUN cana-727 117 18 is be AUX cana-727 117 19 not not PART cana-727 117 20 p p NOUN cana-727 117 21 - - PUNCT cana-727 117 22 extending extending NOUN cana-727 117 23 which which PRON cana-727 117 24 is be AUX cana-727 117 25 nonsingular nonsingular ADJ cana-727 117 26 , , PUNCT cana-727 117 27 thus thus ADV cana-727 117 28 by by ADP cana-727 117 29 proposition proposition NOUN cana-727 117 30 2.4 2.4 NUM cana-727 117 31 m m NOUN cana-727 117 32 is be AUX cana-727 117 33 not not PART cana-727 117 34 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 117 35 . . PUNCT cana-727 118 1 next next ADV cana-727 118 2 , , PUNCT cana-727 118 3 we we PRON cana-727 118 4 give give VERB cana-727 118 5 various various ADJ cana-727 118 6 conditions condition NOUN cana-727 118 7 under under ADP cana-727 118 8 which which PRON cana-727 118 9 the the DET cana-727 118 10 direct direct ADJ cana-727 118 11 sum sum NOUN cana-727 118 12 of of ADP cana-727 118 13 𝐺𝑝-extending 𝐺𝑝-extending PROPN cana-727 118 14 is be AUX cana-727 118 15 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 118 16 . . PUNCT cana-727 118 17 " " PUNCT cana-727 119 1 communications communication NOUN cana-727 119 2 on on ADP cana-727 119 3 applied apply VERB cana-727 119 4 nonlinear nonlinear ADJ cana-727 119 5 analysis analysis NOUN cana-727 119 6 issn issn NOUN cana-727 119 7 : : PUNCT cana-727 119 8 1074 1074 NUM cana-727 119 9 - - PUNCT cana-727 119 10 133x 133x NUM cana-727 119 11 vol vol NOUN cana-727 119 12 31 31 NUM cana-727 119 13 no no NOUN cana-727 119 14 . . PUNCT cana-727 120 1 3s 3s NUM cana-727 120 2 ( ( PUNCT cana-727 120 3 2024 2024 NUM cana-727 120 4 ) ) PUNCT cana-727 120 5 14 14 NUM cana-727 120 6 https://internationalpubls.com https://internationalpubls.com X cana-727 120 7 proposition proposition NOUN cana-727 120 8 4.1 4.1 NUM cana-727 120 9 : : PUNCT cana-727 120 10 let let VERB cana-727 120 11 m m PRON cana-727 120 12 = = VERB cana-727 120 13 m1οƒ…m2 m1οƒ…m2 PROPN cana-727 120 14 be be AUX cana-727 120 15 a a DET cana-727 120 16 distributive distributive ADJ cana-727 120 17 module module NOUN cana-727 120 18 if if SCONJ cana-727 120 19 m1 m1 PROPN cana-727 120 20 and and CCONJ cana-727 120 21 m2 m2 PROPN cana-727 120 22 are be AUX cana-727 120 23 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 120 24 modules module NOUN cana-727 120 25 , , PUNCT cana-727 120 26 then then ADV cana-727 120 27 m m VERB cana-727 120 28 is be AUX cana-727 120 29 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 120 30 . . PUNCT cana-727 121 1 proof proof NOUN cana-727 121 2 : : PUNCT cana-727 121 3 " " PUNCT cana-727 121 4 let let VERB cana-727 121 5 a a PRON cana-727 121 6 be be AUX cana-727 121 7 a a DET cana-727 121 8 cyclic cyclic ADJ cana-727 121 9 submodule submodule NOUN cana-727 121 10 of of ADP cana-727 121 11 m. m. NOUN cana-727 121 12 since since SCONJ cana-727 121 13 m m PROPN cana-727 121 14 is be AUX cana-727 121 15 distributive distributive ADJ cana-727 121 16 , , PUNCT cana-727 121 17 then then ADV cana-727 121 18 a a DET cana-727 121 19 = = PUNCT cana-727 121 20 a∩m a∩m ADJ cana-727 121 21 = = PUNCT cana-727 121 22 a∩(m1οƒ…m2 a∩(m1οƒ…m2 PROPN cana-727 121 23 ) ) PUNCT cana-727 121 24 = = SYM cana-727 121 25 ( ( PUNCT cana-727 121 26 a∩m1)οƒ… a∩m1)οƒ… X cana-727 121 27 ( ( PUNCT cana-727 121 28 a∩m2 a∩m2 PROPN cana-727 121 29 ) ) PUNCT cana-727 121 30 . . PUNCT cana-727 122 1 since since SCONJ cana-727 122 2 a a PRON cana-727 122 3 is be AUX cana-727 122 4 cyclic cyclic ADJ cana-727 122 5 in in ADP cana-727 122 6 m m PROPN cana-727 122 7 , , PUNCT cana-727 122 8 then then ADV cana-727 122 9 a∩m1 a∩m1 PROPN cana-727 122 10 and and CCONJ cana-727 122 11 a∩m2 a∩m2 PROPN cana-727 122 12 are be AUX cana-727 122 13 cyclic cyclic ADJ cana-727 122 14 in in ADP cana-727 122 15 m1 m1 PROPN cana-727 122 16 and and CCONJ cana-727 122 17 m2 m2 PROPN cana-727 122 18 respectively respectively ADV cana-727 122 19 . . PUNCT cana-727 123 1 but but CCONJ cana-727 123 2 m1 m1 PROPN cana-727 123 3 and and CCONJ cana-727 123 4 m2 m2 PROPN cana-727 123 5 are be AUX cana-727 123 6 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 123 7 modules module NOUN cana-727 123 8 , , PUNCT cana-727 123 9 therefore therefore ADV cana-727 123 10 there there PRON cana-727 123 11 are be VERB cana-727 123 12 direct direct ADJ cana-727 123 13 summand summand NOUN cana-727 123 14 𝐷1 𝐷1 NOUN cana-727 123 15 of of ADP cana-727 123 16 𝑀1 𝑀1 PROPN cana-727 123 17 and and CCONJ cana-727 123 18 𝐷2 𝐷2 NOUN cana-727 123 19 of of ADP cana-727 123 20 𝑀2 𝑀2 PROPN cana-727 123 21 such such ADJ cana-727 123 22 that that PRON cana-727 123 23 ( ( PUNCT cana-727 123 24 𝐴⋂𝐷1)𝛽𝐷1 𝐴⋂𝐷1)𝛽𝐷1 PROPN cana-727 123 25 and and CCONJ cana-727 123 26 ( ( PUNCT cana-727 123 27 𝐴⋂𝐷2)𝛽𝐷2 𝐴⋂𝐷2)𝛽𝐷2 VERB cana-727 123 28 hence hence ADV cana-727 123 29 a a DET cana-727 123 30 Ξ²(𝐷1⨁𝐷2 Ξ²(𝐷1⨁𝐷2 NOUN cana-727 123 31 ) ) PUNCT cana-727 123 32 , , PUNCT cana-727 123 33 by by ADP cana-727 123 34 [ [ X cana-727 123 35 12 12 NUM cana-727 123 36 , , PUNCT cana-727 123 37 proposition proposition NOUN cana-727 123 38 1.4 1.4 NUM cana-727 123 39 ] ] PUNCT cana-727 123 40 . . PUNCT cana-727 124 1 thus thus ADV cana-727 124 2 , , PUNCT cana-727 124 3 m m VERB cana-727 124 4 is be AUX cana-727 124 5 𝐺𝑝extending 𝐺𝑝extende VERB cana-727 124 6 module module NOUN cana-727 124 7 . . PUNCT cana-727 124 8 " " PUNCT cana-727 125 1 the the DET cana-727 125 2 following following ADJ cana-727 125 3 statements statement NOUN cana-727 125 4 are be AUX cana-727 125 5 also also ADV cana-727 125 6 easily easily ADV cana-727 125 7 proved prove VERB cana-727 125 8 by by ADP cana-727 125 9 using use VERB cana-727 125 10 a a DET cana-727 125 11 similar similar ADJ cana-727 125 12 argument argument NOUN cana-727 125 13 . . PUNCT cana-727 126 1 proposition proposition NOUN cana-727 126 2 4.2 4.2 NUM cana-727 126 3 : : PUNCT cana-727 126 4 " " PUNCT cana-727 126 5 let let VERB cana-727 126 6 m m PRON cana-727 126 7 = = VERB cana-727 126 8 m1οƒ…m2 m1οƒ…m2 PROPN cana-727 126 9 be be AUX cana-727 126 10 a a DET cana-727 126 11 duo duo NOUN cana-727 126 12 module module NOUN cana-727 126 13 if if SCONJ cana-727 126 14 m1 m1 PROPN cana-727 126 15 and and CCONJ cana-727 126 16 m2 m2 PROPN cana-727 126 17 are be AUX cana-727 126 18 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 126 19 modules module NOUN cana-727 126 20 , , PUNCT cana-727 126 21 then then ADV cana-727 126 22 m m VERB cana-727 126 23 is be AUX cana-727 126 24 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 126 25 . . PUNCT cana-727 126 26 " " PUNCT cana-727 127 1 proposition proposition NOUN cana-727 127 2 4.3 4.3 NUM cana-727 127 3 : : PUNCT cana-727 127 4 let let VERB cana-727 127 5 m1 m1 PROPN cana-727 127 6 and and CCONJ cana-727 127 7 m2 m2 PROPN cana-727 127 8 be be AUX cana-727 127 9 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 127 10 modules module NOUN cana-727 127 11 such such ADJ cana-727 127 12 that that DET cana-727 127 13 annm1+annm2 annm1+annm2 PROPN cana-727 127 14 = = SYM cana-727 127 15 r r NOUN cana-727 127 16 , , PUNCT cana-727 127 17 then then ADV cana-727 127 18 m1οƒ…m2 m1οƒ…m2 PROPN cana-727 127 19 is be AUX cana-727 127 20 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 127 21 module module NOUN cana-727 127 22 . . PUNCT cana-727 128 1 proposition proposition NOUN cana-727 128 2 4.4 4.4 NUM cana-727 128 3 : : PUNCT cana-727 128 4 " " PUNCT cana-727 128 5 let let VERB cana-727 128 6 m m PRON cana-727 128 7 = = VERB cana-727 128 8 m1οƒ…m2 m1οƒ…m2 PROPN cana-727 128 9 be be AUX cana-727 128 10 an an DET cana-727 128 11 rmodule rmodule NOUN cana-727 128 12 with with ADP cana-727 128 13 m1 m1 PROPN cana-727 128 14 being be AUX cana-727 128 15 𝐺𝑝-extending 𝐺𝑝-extending PROPN cana-727 128 16 and and CCONJ cana-727 128 17 m2 m2 PROPN cana-727 128 18 is be AUX cana-727 128 19 semisimple semisimple ADJ cana-727 128 20 . . PUNCT cana-727 128 21 suppose suppose VERB cana-727 128 22 that that SCONJ cana-727 128 23 for for ADP cana-727 128 24 any any DET cana-727 128 25 cyclic cyclic ADJ cana-727 128 26 submodule submodule NOUN cana-727 128 27 a a PRON cana-727 128 28 of of ADP cana-727 128 29 m m PROPN cana-727 128 30 , , PUNCT cana-727 128 31 a∩m1 a∩m1 PROPN cana-727 128 32 is be AUX cana-727 128 33 a a DET cana-727 128 34 direct direct ADJ cana-727 128 35 summand summand NOUN cana-727 128 36 of of ADP cana-727 128 37 a a PRON cana-727 128 38 , , PUNCT cana-727 128 39 then then ADV cana-727 128 40 m m VERB cana-727 128 41 is be AUX cana-727 128 42 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 128 43 . . PUNCT cana-727 129 1 proof proof NOUN cana-727 129 2 : : PUNCT cana-727 129 3 let let VERB cana-727 129 4 a a PRON cana-727 129 5 be be AUX cana-727 129 6 a a DET cana-727 129 7 cyclic cyclic ADJ cana-727 129 8 submodule submodule NOUN cana-727 129 9 of of ADP cana-727 129 10 m m PROPN cana-727 129 11 , , PUNCT cana-727 129 12 then then ADV cana-727 129 13 it it PRON cana-727 129 14 is be AUX cana-727 129 15 easy easy ADJ cana-727 129 16 to to PART cana-727 129 17 see see VERB cana-727 129 18 that that DET cana-727 129 19 a+m1 a+m1 NOUN cana-727 129 20 = = SYM cana-727 129 21 m1οƒ… m1οƒ… PROPN cana-727 129 22 [ [ X cana-727 129 23 ( ( PUNCT cana-727 129 24 a+m1)∩m2 a+m1)∩m2 NOUN cana-727 129 25 ] ] PUNCT cana-727 129 26 . . PUNCT cana-727 130 1 since since SCONJ cana-727 130 2 m2 m2 PROPN cana-727 130 3 is be AUX cana-727 130 4 semisimple semisimple ADJ cana-727 130 5 , , PUNCT cana-727 130 6 then then ADV cana-727 130 7 ( ( PUNCT cana-727 130 8 a+m1)∩m2 a+m1)∩m2 NOUN cana-727 130 9 is be AUX cana-727 130 10 a a DET cana-727 130 11 direct direct ADJ cana-727 130 12 summand summand NOUN cana-727 130 13 of of ADP cana-727 130 14 m2 m2 PROPN cana-727 130 15 and and CCONJ cana-727 130 16 therefore therefore ADV cana-727 130 17 a+m1 a+m1 PROPN cana-727 130 18 is be AUX cana-727 130 19 a a DET cana-727 130 20 direct direct ADJ cana-727 130 21 summand summand NOUN cana-727 130 22 of of ADP cana-727 130 23 m. m. NOUN cana-727 130 24 by by ADP cana-727 130 25 our our PRON cana-727 130 26 assumption assumption NOUN cana-727 130 27 , , PUNCT cana-727 130 28 a∩m1 a∩m1 PROPN cana-727 130 29 is be AUX cana-727 130 30 a a DET cana-727 130 31 direct direct ADJ cana-727 130 32 summand summand NOUN cana-727 130 33 of of ADP cana-727 130 34 a a PRON cana-727 130 35 , , PUNCT cana-727 130 36 then then ADV cana-727 130 37 a a DET cana-727 130 38 = = X cana-727 130 39 ( ( PUNCT cana-727 130 40 a∩m1)οƒ…a a∩m1)οƒ…a NOUN cana-727 130 41 ' ' PUNCT cana-727 130 42 , , PUNCT cana-727 130 43 for for ADP cana-727 130 44 some some DET cana-727 130 45 submodule submodule NOUN cana-727 130 46 a a PRON cana-727 130 47 ' ' PUNCT cana-727 130 48 of of ADP cana-727 130 49 a. a. NOUN cana-727 130 50 one one NOUN cana-727 130 51 can can AUX cana-727 130 52 easily easily ADV cana-727 130 53 show show VERB cana-727 130 54 that that SCONJ cana-727 130 55 a∩m1 a∩m1 PROPN cana-727 130 56 is be AUX cana-727 130 57 cyclic cyclic ADJ cana-727 130 58 in in ADP cana-727 130 59 m1 m1 PROPN cana-727 130 60 . . PUNCT cana-727 131 1 but but CCONJ cana-727 131 2 m1 m1 PROPN cana-727 131 3 is be AUX cana-727 131 4 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 131 5 , , PUNCT cana-727 131 6 then then ADV cana-727 131 7 there there PRON cana-727 131 8 is be VERB cana-727 131 9 a a DET cana-727 131 10 direct direct ADJ cana-727 131 11 summand summand NOUN cana-727 131 12 d d PROPN cana-727 131 13 of of ADP cana-727 131 14 m1 m1 PROPN cana-727 131 15 such such ADJ cana-727 131 16 that that SCONJ cana-727 131 17 ( ( PUNCT cana-727 131 18 a∩m1 a∩m1 PROPN cana-727 131 19 ) ) PUNCT cana-727 131 20 Ξ² Ξ² PROPN cana-727 131 21 d d NOUN cana-727 131 22 is be AUX cana-727 131 23 hence hence ADV cana-727 131 24 a a PRON cana-727 131 25 = = PUNCT cana-727 131 26 ( ( PUNCT cana-727 131 27 ( ( PUNCT cana-727 131 28 a∩m1)οƒ…a')Ξ²(m1+a a∩m1)οƒ…a')Ξ²(m1+a NOUN cana-727 131 29 ) ) PUNCT cana-727 131 30 . . PUNCT cana-727 132 1 thus thus ADV cana-727 132 2 , , PUNCT cana-727 132 3 m m VERB cana-727 132 4 is be AUX cana-727 132 5 𝐺𝑝extending 𝐺𝑝extende VERB cana-727 132 6 . . PUNCT cana-727 132 7 " " PUNCT cana-727 133 1 proposition proposition NOUN cana-727 133 2 4.5 4.5 NUM cana-727 133 3 : : PUNCT cana-727 133 4 " " PUNCT cana-727 133 5 let let VERB cana-727 133 6 m m PRON cana-727 133 7 = = VERB cana-727 133 8 m1οƒ…m2 m1οƒ…m2 NOUN cana-727 133 9 such such ADJ cana-727 133 10 that that DET cana-727 133 11 m1 m1 PROPN cana-727 133 12 is be AUX cana-727 133 13 𝐺𝑝-extending 𝐺𝑝-extending PROPN cana-727 133 14 and and CCONJ cana-727 133 15 m2 m2 PROPN cana-727 133 16 is be AUX cana-727 133 17 injective injective ADJ cana-727 133 18 module module NOUN cana-727 133 19 . . PUNCT cana-727 134 1 then then ADV cana-727 134 2 m m VERB cana-727 134 3 is be AUX cana-727 134 4 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 134 5 if if SCONJ cana-727 134 6 and and CCONJ cana-727 134 7 only only ADV cana-727 134 8 if if SCONJ cana-727 134 9 for for ADP cana-727 134 10 every every DET cana-727 134 11 cyclic cyclic ADJ cana-727 134 12 submodule submodule NOUN cana-727 134 13 a a PRON cana-727 134 14 of of ADP cana-727 134 15 m m PRON cana-727 134 16 such such ADJ cana-727 134 17 that that SCONJ cana-727 134 18 a∩m2β‰ 0 a∩m2β‰ 0 PROPN cana-727 134 19 there there PRON cana-727 134 20 is be VERB cana-727 134 21 a a DET cana-727 134 22 direct direct ADJ cana-727 134 23 summand summand NOUN cana-727 134 24 d d PROPN cana-727 134 25 of of ADP cana-727 134 26 m m PRON cana-727 134 27 such such ADJ cana-727 134 28 that that SCONJ cana-727 134 29 𝐴𝛽𝐷. 𝐴𝛽𝐷. ADJ cana-727 134 30 proof proof NOUN cana-727 134 31 : : PUNCT cana-727 134 32 suppose suppose VERB cana-727 134 33 that that SCONJ cana-727 134 34 for for SCONJ cana-727 134 35 every every DET cana-727 134 36 cyclic cyclic ADJ cana-727 134 37 submodule submodule NOUN cana-727 134 38 a a PRON cana-727 134 39 of of ADP cana-727 134 40 m m PRON cana-727 134 41 such such ADJ cana-727 134 42 that that SCONJ cana-727 134 43 a∩m2β‰ 0 a∩m2β‰ 0 PROPN cana-727 134 44 there there ADV cana-727 134 45 exists exist VERB cana-727 134 46 direct direct ADJ cana-727 134 47 summand summand NOUN cana-727 134 48 d d PROPN cana-727 134 49 of of ADP cana-727 134 50 m m PRON cana-727 134 51 such such ADJ cana-727 134 52 that that SCONJ cana-727 134 53 𝐴𝛽𝐷. 𝐴𝛽𝐷. PROPN cana-727 134 54 let let VERB cana-727 134 55 a a PRON cana-727 134 56 be be AUX cana-727 134 57 a a DET cana-727 134 58 cyclic cyclic ADJ cana-727 134 59 submodule submodule NOUN cana-727 134 60 of of ADP cana-727 134 61 m m PRON cana-727 134 62 such such ADJ cana-727 134 63 that that SCONJ cana-727 134 64 a∩m2 a∩m2 PROPN cana-727 134 65 = = SYM cana-727 134 66 0 0 NUM cana-727 134 67 . . PUNCT cana-727 135 1 by by ADP cana-727 135 2 [ [ X cana-727 135 3 2 2 NUM cana-727 135 4 ] ] PUNCT cana-727 135 5 , , PUNCT cana-727 135 6 there there PRON cana-727 135 7 is be VERB cana-727 135 8 a a DET cana-727 135 9 submodule submodule NOUN cana-727 135 10 m m NOUN cana-727 135 11 ' ' PUNCT cana-727 135 12 of of ADP cana-727 135 13 m m AUX cana-727 135 14 containing contain VERB cana-727 135 15 a a DET cana-727 135 16 such such ADJ cana-727 135 17 that that SCONJ cana-727 135 18 m m VERB cana-727 135 19 = = ADJ cana-727 135 20 m'οƒ…m2 m'οƒ…m2 X cana-727 135 21 . . PUNCT cana-727 136 1 since since SCONJ cana-727 136 2 m'β‰… m'β‰… PROPN cana-727 136 3 𝑀 𝑀 PROPN cana-727 136 4 𝑀2 𝑀2 PROPN cana-727 136 5 β‰…m1 β‰…m1 NOUN cana-727 136 6 is be AUX cana-727 136 7 𝐺𝑝-extending 𝐺𝑝-extending PROPN cana-727 136 8 and and CCONJ cana-727 136 9 a a PRON cana-727 136 10 is be AUX cana-727 136 11 cyclic cyclic ADJ cana-727 136 12 submodule submodule NOUN cana-727 136 13 of of ADP cana-727 136 14 m m PROPN cana-727 136 15 ' ' PUNCT cana-727 136 16 , , PUNCT cana-727 136 17 then then ADV cana-727 136 18 there there PRON cana-727 136 19 is be VERB cana-727 136 20 a a DET cana-727 136 21 direct direct ADJ cana-727 136 22 summand summand NOUN cana-727 136 23 k k PROPN cana-727 136 24 of of ADP cana-727 136 25 m m PROPN cana-727 136 26 ' ' PART cana-727 136 27 such such ADJ cana-727 136 28 that that DET cana-727 136 29 aΞ²k aΞ²k NOUN cana-727 136 30 . . PUNCT cana-727 137 1 thus thus ADV cana-727 137 2 , , PUNCT cana-727 137 3 m m VERB cana-727 137 4 is be AUX cana-727 137 5 𝐺𝑝extending 𝐺𝑝extending PROPN cana-727 137 6 . . PUNCT cana-727 138 1 the the DET cana-727 138 2 converse converse NOUN cana-727 138 3 is be AUX cana-727 138 4 obvious obvious ADJ cana-727 138 5 . . PUNCT cana-727 138 6 " " PUNCT cana-727 139 1 we we PRON cana-727 139 2 now now ADV cana-727 139 3 list list VERB cana-727 139 4 several several ADJ cana-727 139 5 circumstances circumstance NOUN cana-727 139 6 in in ADP cana-727 139 7 which which PRON cana-727 139 8 a a DET cana-727 139 9 direct direct ADJ cana-727 139 10 summand summand NOUN cana-727 139 11 of of ADP cana-727 139 12 a a DET cana-727 139 13 module module NOUN cana-727 139 14 that that PRON cana-727 139 15 extends extend VERB cana-727 139 16 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 139 17 is be AUX cana-727 139 18 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 139 19 . . PUNCT cana-727 139 20 proposition proposition NOUN cana-727 139 21 4.6:"let 4.6:"let NOUN cana-727 139 22 a a PRON cana-727 139 23 be be AUX cana-727 139 24 a a DET cana-727 139 25 direct direct ADJ cana-727 139 26 summand summand NOUN cana-727 139 27 of of ADP cana-727 139 28 a a DET cana-727 139 29 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 139 30 module module NOUN cana-727 139 31 m m NOUN cana-727 139 32 , , PUNCT cana-727 139 33 if if SCONJ cana-727 139 34 the the DET cana-727 139 35 intersection intersection NOUN cana-727 139 36 of of ADP cana-727 139 37 a a PRON cana-727 139 38 with with ADP cana-727 139 39 any any DET cana-727 139 40 direct direct ADJ cana-727 139 41 summand summand NOUN cana-727 139 42 of of ADP cana-727 139 43 m m PROPN cana-727 139 44 is be AUX cana-727 139 45 a a DET cana-727 139 46 direct direct ADJ cana-727 139 47 summand summand NOUN cana-727 139 48 of of ADP cana-727 139 49 a a PRON cana-727 139 50 , , PUNCT cana-727 139 51 then then ADV cana-727 139 52 a a DET cana-727 139 53 is be AUX cana-727 139 54 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 139 55 module module NOUN cana-727 139 56 . . PUNCT cana-727 139 57 " " PUNCT cana-727 140 1 proof proof NOUN cana-727 140 2 : : PUNCT cana-727 140 3 " " PUNCT cana-727 140 4 let let VERB cana-727 140 5 x x PRON cana-727 140 6 be be AUX cana-727 140 7 a a DET cana-727 140 8 cyclic cyclic NOUN cana-727 140 9 in in ADP cana-727 140 10 a a PRON cana-727 140 11 , , PUNCT cana-727 140 12 then then ADV cana-727 140 13 x x PUNCT cana-727 140 14 is be AUX cana-727 140 15 cyclic cyclic ADJ cana-727 140 16 in in ADP cana-727 140 17 m. m. NOUN cana-727 140 18 but but CCONJ cana-727 140 19 m m PROPN cana-727 140 20 is be AUX cana-727 140 21 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 140 22 , , PUNCT cana-727 140 23 therefore therefore ADV cana-727 140 24 there there PRON cana-727 140 25 exists exist VERB cana-727 140 26 a a DET cana-727 140 27 direct direct ADJ cana-727 140 28 summand summand NOUN cana-727 140 29 d d PROPN cana-727 140 30 of of ADP cana-727 140 31 m m PRON cana-727 140 32 such such ADJ cana-727 140 33 that that DET cana-727 140 34 xΞ²d xΞ²d NOUN cana-727 140 35 . . PUNCT cana-727 141 1 it it PRON cana-727 141 2 can can AUX cana-727 141 3 be be AUX cana-727 141 4 seen see VERB cana-727 141 5 that that SCONJ cana-727 141 6 xΞ² xΞ² PROPN cana-727 141 7 ( ( PUNCT cana-727 141 8 ad ad PROPN cana-727 141 9 ) ) PUNCT cana-727 141 10 . . PUNCT cana-727 142 1 by by ADP cana-727 142 2 our our PRON cana-727 142 3 assumption assumption NOUN cana-727 142 4 ad ad NOUN cana-727 142 5 is be AUX cana-727 142 6 a a DET cana-727 142 7 direct direct ADJ cana-727 142 8 summand summand NOUN cana-727 142 9 of of ADP cana-727 142 10 a. a. NOUN cana-727 142 11 thus thus ADV cana-727 142 12 , , PUNCT cana-727 142 13 a a DET cana-727 142 14 is be AUX cana-727 142 15 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 142 16 . . PUNCT cana-727 142 17 " " PUNCT cana-727 143 1 communications communication NOUN cana-727 143 2 on on ADP cana-727 143 3 applied apply VERB cana-727 143 4 nonlinear nonlinear ADJ cana-727 143 5 analysis analysis NOUN cana-727 143 6 issn issn NOUN cana-727 143 7 : : PUNCT cana-727 143 8 1074 1074 NUM cana-727 143 9 - - PUNCT cana-727 143 10 133x 133x NUM cana-727 143 11 vol vol NOUN cana-727 143 12 31 31 NUM cana-727 143 13 no no NOUN cana-727 143 14 . . PUNCT cana-727 144 1 3s 3s NUM cana-727 144 2 ( ( PUNCT cana-727 144 3 2024 2024 NUM cana-727 144 4 ) ) PUNCT cana-727 144 5 15 15 NUM cana-727 144 6 https://internationalpubls.com https://internationalpubls.com X cana-727 144 7 proposition proposition NOUN cana-727 144 8 4.7:"let 4.7:"let NOUN cana-727 144 9 a a DET cana-727 144 10 be be AUX cana-727 144 11 a a DET cana-727 144 12 cyclic cyclic ADJ cana-727 144 13 submodule submodule NOUN cana-727 144 14 of of ADP cana-727 144 15 a a DET cana-727 144 16 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 144 17 module module NOUN cana-727 144 18 m m NOUN cana-727 144 19 . . PUNCT cana-727 144 20 " " PUNCT cana-727 145 1 ( ( PUNCT cana-727 145 2 i i NOUN cana-727 145 3 ) ) PUNCT cana-727 145 4 if if SCONJ cana-727 145 5 for for ADP cana-727 145 6 each each DET cana-727 145 7 𝑒2 𝑒2 NOUN cana-727 145 8 = = PUNCT cana-727 145 9 𝑒 𝑒 PROPN cana-727 145 10 ∈ ∈ PROPN cana-727 145 11 𝐸𝑛𝑑(𝑀𝑅 𝐸𝑛𝑑(𝑀𝑅 NUM cana-727 145 12 ) ) PUNCT cana-727 145 13 , , PUNCT cana-727 145 14 there there PRON cana-727 145 15 exists exist VERB cana-727 145 16 𝑓2 𝑓2 NOUN cana-727 145 17 = = SYM cana-727 145 18 𝑓 𝑓 PRON cana-727 145 19 ∈ ∈ PROPN cana-727 145 20 𝐸𝑛𝑑(𝐴𝑅 𝐸𝑛𝑑(𝐴𝑅 NOUN cana-727 145 21 ) ) PUNCT cana-727 145 22 such such ADJ cana-727 145 23 that that SCONJ cana-727 145 24 𝐴 𝐴 PROPN cana-727 145 25 ∩ ∩ ADJ cana-727 145 26 𝑒𝑀 𝑒𝑀 PROPN cana-727 145 27 ≀𝑒 ≀𝑒 NOUN cana-727 145 28 𝑓𝐴 𝑓𝐴 NOUN cana-727 145 29 , , PUNCT cana-727 145 30 then then ADV cana-727 145 31 a a DET cana-727 145 32 is be AUX cana-727 145 33 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 145 34 . . PUNCT cana-727 146 1 ( ( PUNCT cana-727 146 2 ii ii NOUN cana-727 146 3 ) ) PUNCT cana-727 146 4 if if SCONJ cana-727 146 5 for for ADP cana-727 146 6 each each DET cana-727 146 7 𝑒2 𝑒2 NOUN cana-727 146 8 = = PUNCT cana-727 146 9 𝑒 𝑒 PROPN cana-727 146 10 ∈ ∈ PROPN cana-727 146 11 𝐸𝑛𝑑(𝑀𝑅 𝐸𝑛𝑑(𝑀𝑅 NUM cana-727 146 12 ) ) PUNCT cana-727 146 13 , , PUNCT cana-727 146 14 there there PRON cana-727 146 15 exists exist VERB cana-727 146 16 𝑓2 𝑓2 NOUN cana-727 146 17 = = SYM cana-727 146 18 𝑓 𝑓 PRON cana-727 146 19 ∈ ∈ PROPN cana-727 146 20 𝐸𝑛𝑑(𝐴𝑅 𝐸𝑛𝑑(𝐴𝑅 NOUN cana-727 146 21 ) ) PUNCT cana-727 146 22 such such ADJ cana-727 146 23 that that DET cana-727 146 24 𝑒𝑀𝛽𝑓𝑀 𝑒𝑀𝛽𝑓𝑀 NOUN cana-727 146 25 and and CCONJ cana-727 146 26 𝑓𝐴 𝑓𝐴 PROPN cana-727 146 27 βŠ† βŠ† NUM cana-727 146 28 𝐴 𝐴 PROPN cana-727 146 29 , , PUNCT cana-727 146 30 then then ADV cana-727 146 31 a a DET cana-727 146 32 is be AUX cana-727 146 33 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 146 34 . . PUNCT cana-727 147 1 proof proof NOUN cana-727 147 2 : : PUNCT cana-727 147 3 ( ( PUNCT cana-727 147 4 i i NOUN cana-727 147 5 ) ) PUNCT cana-727 147 6 " " PUNCT cana-727 147 7 let let VERB cana-727 147 8 y y PRON cana-727 147 9 be be AUX cana-727 147 10 a a DET cana-727 147 11 cyclic cyclic ADJ cana-727 147 12 submodule submodule NOUN cana-727 147 13 of of ADP cana-727 147 14 a. a. NOUN cana-727 147 15 hence hence ADV cana-727 147 16 y y PROPN cana-727 147 17 is be AUX cana-727 147 18 a a DET cana-727 147 19 cyclic cyclic ADJ cana-727 147 20 submodule submodule NOUN cana-727 147 21 of of ADP cana-727 147 22 m. m. NOUN cana-727 147 23 by by ADP cana-727 147 24 proposition proposition NOUN cana-727 147 25 3.2 3.2 NUM cana-727 147 26 , , PUNCT cana-727 147 27 there there PRON cana-727 147 28 is be VERB cana-727 147 29 x≀e x≀e PROPN cana-727 147 30 y y PROPN cana-727 147 31 and and CCONJ cana-727 147 32 𝑒2 𝑒2 PROPN cana-727 148 1 = = PUNCT cana-727 148 2 𝑒 𝑒 PROPN cana-727 148 3 ∈ ∈ PROPN cana-727 148 4 𝐸𝑛𝑑(𝑀𝑅 𝐸𝑛𝑑(𝑀𝑅 VERB cana-727 148 5 ) ) PUNCT cana-727 148 6 such such ADJ cana-727 148 7 that that SCONJ cana-727 148 8 x≀e x≀e PROPN cana-727 148 9 em em PRON cana-727 148 10 . . PUNCT cana-727 149 1 then then ADV cana-727 149 2 x≀e x≀e PROPN cana-727 149 3 em∩ em∩ PROPN cana-727 149 4 𝐴 𝐴 PROPN cana-727 149 5 ≀e ≀e VERB cana-727 149 6 f f PROPN cana-727 149 7 a a X cana-727 149 8 , , PUNCT cana-727 149 9 for for ADP cana-727 149 10 some some DET cana-727 149 11 𝑓2 𝑓2 NOUN cana-727 149 12 = = PUNCT cana-727 149 13 𝑓 𝑓 PROPN cana-727 149 14 ∈ ∈ PROPN cana-727 149 15 𝐸𝑛𝑑(𝑀𝑅).thus 𝐸𝑛𝑑(𝑀𝑅).thus PROPN cana-727 149 16 , , PUNCT cana-727 149 17 a a PRON cana-727 149 18 is be AUX cana-727 149 19 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 149 20 . . PUNCT cana-727 149 21 " " PUNCT cana-727 150 1 ( ( PUNCT cana-727 150 2 ii ii NOUN cana-727 150 3 ) ) PUNCT cana-727 150 4 " " PUNCT cana-727 150 5 let let VERB cana-727 150 6 y y PRON cana-727 150 7 be be AUX cana-727 150 8 a a DET cana-727 150 9 cyclic cyclic ADJ cana-727 150 10 submodule submodule NOUN cana-727 150 11 of of ADP cana-727 150 12 a a PRON cana-727 150 13 , , PUNCT cana-727 150 14 then then ADV cana-727 150 15 y y PROPN cana-727 150 16 is be AUX cana-727 150 17 cyclic cyclic ADJ cana-727 150 18 in in ADP cana-727 150 19 m. m. NOUN cana-727 150 20 then then ADV cana-727 150 21 there there PRON cana-727 150 22 exists exist VERB cana-727 150 23 𝑒2 𝑒2 PROPN cana-727 150 24 = = PUNCT cana-727 150 25 𝑒 𝑒 PROPN cana-727 150 26 ∈ ∈ PROPN cana-727 150 27 𝐸𝑛𝑑(𝑀𝑅 𝐸𝑛𝑑(𝑀𝑅 VERB cana-727 150 28 ) ) PUNCT cana-727 150 29 such such ADJ cana-727 150 30 that that DET cana-727 150 31 yΞ²em yΞ²em NOUN cana-727 150 32 . . PUNCT cana-727 151 1 hence hence ADV cana-727 151 2 yΞ²fm yΞ²fm PROPN cana-727 151 3 . . PUNCT cana-727 152 1 since since SCONJ cana-727 152 2 faβŠ† faβŠ† PROPN cana-727 152 3 𝐴 𝐴 PROPN cana-727 152 4 , , PUNCT cana-727 152 5 a a PRON cana-727 152 6 is be AUX cana-727 152 7 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 152 8 . . PUNCT cana-727 152 9 " " PUNCT cana-727 153 1 proposition proposition NOUN cana-727 153 2 4.8:"let 4.8:"let NOUN cana-727 154 1 k k PROPN cana-727 154 2 be be AUX cana-727 154 3 a a DET cana-727 154 4 projection projection ADJ cana-727 154 5 invariant invariant ADJ cana-727 154 6 cyclic cyclic ADJ cana-727 154 7 submodule submodule NOUN cana-727 154 8 of of ADP cana-727 154 9 m. m. NOUN cana-727 154 10 if if SCONJ cana-727 154 11 m m PROPN cana-727 154 12 is be AUX cana-727 154 13 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 154 14 , , PUNCT cana-727 154 15 then then ADV cana-727 154 16 there there PRON cana-727 154 17 exists exist VERB cana-727 154 18 𝑀1 𝑀1 NOUN cana-727 154 19 ≀ ≀ X cana-727 154 20 𝑀 𝑀 PROPN cana-727 154 21 such such ADJ cana-727 154 22 that that DET cana-727 154 23 𝑀 𝑀 PROPN cana-727 154 24 = = PUNCT cana-727 154 25 𝑀1⨁𝐾 𝑀1⨁𝐾 PROPN cana-727 154 26 and and CCONJ cana-727 154 27 k k PROPN cana-727 154 28 is be AUX cana-727 154 29 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 154 30 . . PUNCT cana-727 155 1 proof proof NOUN cana-727 155 2 : : PUNCT cana-727 155 3 there there PRON cana-727 155 4 exists exist VERB cana-727 155 5 𝑒2 𝑒2 PROPN cana-727 155 6 = = PUNCT cana-727 155 7 𝑒 𝑒 PROPN cana-727 155 8 ∈ ∈ PROPN cana-727 155 9 𝐸𝑛𝑑(𝑀𝑅 𝐸𝑛𝑑(𝑀𝑅 VERB cana-727 155 10 ) ) PUNCT cana-727 156 1 such such ADJ cana-727 156 2 that that DET cana-727 156 3 𝐾𝛽𝑒𝑀. 𝐾𝛽𝑒𝑀. PROPN cana-727 156 4 but but CCONJ cana-727 156 5 𝐾 𝐾 PROPN cana-727 156 6 = = NOUN cana-727 156 7 𝑒𝐾⨁(1 𝑒𝐾⨁(1 PROPN cana-727 156 8 βˆ’ βˆ’ PROPN cana-727 156 9 𝑒)𝐾 𝑒)𝐾 NOUN cana-727 156 10 , , PUNCT cana-727 156 11 𝑒𝐾 𝑒𝐾 NOUN cana-727 156 12 = = SYM cana-727 156 13 𝐾 𝐾 PROPN cana-727 156 14 ∩ ∩ ADJ cana-727 156 15 𝑒𝑀 𝑒𝑀 NOUN cana-727 156 16 , , PUNCT cana-727 156 17 and and CCONJ cana-727 156 18 ( ( PUNCT cana-727 156 19 1 1 NUM cana-727 156 20 βˆ’ βˆ’ PROPN cana-727 156 21 𝑒)𝐾 𝑒)𝐾 ADJ cana-727 156 22 = = SYM cana-727 156 23 𝐾 𝐾 PROPN cana-727 156 24 ∩ ∩ NOUN cana-727 156 25 ( ( PUNCT cana-727 156 26 1 1 NUM cana-727 156 27 βˆ’ βˆ’ NOUN cana-727 156 28 𝑒)𝑀 𝑒)𝑀 VERB cana-727 156 29 because because SCONJ cana-727 156 30 k k PROPN cana-727 156 31 is be AUX cana-727 156 32 projection projection NOUN cana-727 156 33 invariant invariant ADJ cana-727 156 34 , , PUNCT cana-727 156 35 then then ADV cana-727 156 36 𝑒𝐾 𝑒𝐾 NOUN cana-727 156 37 ≀𝑒 ≀𝑒 NOUN cana-727 156 38 𝑒𝑀 𝑒𝑀 PROPN cana-727 156 39 and and CCONJ cana-727 156 40 𝑒𝐾 𝑒𝐾 NOUN cana-727 156 41 ≀𝑒 ≀𝑒 NOUN cana-727 156 42 𝐾. 𝐾. PROPN cana-727 156 43 hence hence ADV cana-727 156 44 𝐾 𝐾 PROPN cana-727 156 45 ∩ ∩ NOUN cana-727 156 46 ( ( PUNCT cana-727 156 47 1 1 NUM cana-727 156 48 βˆ’ βˆ’ NOUN cana-727 156 49 𝑒)𝑀 𝑒)𝑀 VERB cana-727 156 50 = = NOUN cana-727 156 51 0 0 X cana-727 156 52 . . PUNCT cana-727 157 1 so so ADV cana-727 157 2 𝐾 𝐾 PROPN cana-727 157 3 = = PUNCT cana-727 157 4 𝑒𝐾 𝑒𝐾 NOUN cana-727 157 5 ≀𝑒 ≀𝑒 NOUN cana-727 157 6 𝑒𝑀. 𝑒𝑀. NOUN cana-727 157 7 since since SCONJ cana-727 157 8 k k PROPN cana-727 157 9 is be AUX cana-727 157 10 cyclic cyclic ADJ cana-727 157 11 in in ADP cana-727 157 12 m m PROPN cana-727 157 13 , , PUNCT cana-727 157 14 then then ADV cana-727 157 15 k k X cana-727 157 16 = = NOUN cana-727 157 17 em em VERB cana-727 157 18 . . PUNCT cana-727 158 1 let let AUX cana-727 158 2 𝑀1 𝑀1 NOUN cana-727 158 3 = = PUNCT cana-727 158 4 ( ( PUNCT cana-727 158 5 1 1 NUM cana-727 158 6 βˆ’ βˆ’ PROPN cana-727 158 7 𝑒)𝑀. 𝑒)𝑀. X cana-727 158 8 therefore therefore ADV cana-727 158 9 𝑀 𝑀 PROPN cana-727 158 10 = = PUNCT cana-727 158 11 𝑀1⨁𝐾. 𝑀1⨁𝐾. PROPN cana-727 158 12 observe observe VERB cana-727 158 13 that that SCONJ cana-727 158 14 , , PUNCT cana-727 158 15 by by ADP cana-727 158 16 proposition proposition NOUN cana-727 158 17 4.7 4.7 NUM cana-727 158 18 ( ( PUNCT cana-727 158 19 ii ii NOUN cana-727 158 20 ) ) PUNCT cana-727 158 21 , , PUNCT cana-727 158 22 k k PROPN cana-727 158 23 is be AUX cana-727 158 24 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 158 25 . . PUNCT cana-727 158 26 " " PUNCT cana-727 158 27 theorem theorem VERB cana-727 158 28 4.9 4.9 NUM cana-727 158 29 : : PUNCT cana-727 158 30 let let VERB cana-727 158 31 m m PRON cana-727 158 32 be be AUX cana-727 158 33 a a DET cana-727 158 34 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 158 35 module module NOUN cana-727 158 36 . . PUNCT cana-727 159 1 if if SCONJ cana-727 159 2 m m PROPN cana-727 159 3 has have AUX cana-727 159 4 sip sip NOUN cana-727 159 5 or or CCONJ cana-727 159 6 satisfies satisfy VERB cana-727 159 7 the the DET cana-727 159 8 𝐢3 𝐢3 ADJ cana-727 159 9 condition condition NOUN cana-727 159 10 , , PUNCT cana-727 159 11 then then ADV cana-727 159 12 any any DET cana-727 159 13 cyclic cyclic ADJ cana-727 159 14 direct direct ADJ cana-727 159 15 summand summand NOUN cana-727 159 16 of of ADP cana-727 159 17 m m PROPN cana-727 159 18 is be AUX cana-727 159 19 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 159 20 . . PUNCT cana-727 160 1 proof proof NOUN cana-727 160 2 : : PUNCT cana-727 160 3 " " PUNCT cana-727 160 4 let let VERB cana-727 160 5 𝑀 𝑀 PROPN cana-727 160 6 = = PUNCT cana-727 160 7 𝑁⨁𝑁′ 𝑁⨁𝑁′ PROPN cana-727 160 8 for for ADP cana-727 160 9 some some DET cana-727 160 10 submodules submodule NOUN cana-727 160 11 n n CCONJ cana-727 160 12 , , PUNCT cana-727 160 13 n n CCONJ cana-727 160 14 ' ' PUNCT cana-727 160 15 of of ADP cana-727 160 16 m m PRON cana-727 160 17 where where SCONJ cana-727 160 18 n n PRON cana-727 160 19 is be AUX cana-727 160 20 cyclic cyclic ADJ cana-727 160 21 in in ADP cana-727 160 22 m. m. NOUN cana-727 160 23 using use VERB cana-727 160 24 proposition proposition NOUN cana-727 160 25 4.8(i 4.8(i NUM cana-727 160 26 ) ) PUNCT cana-727 160 27 , , PUNCT cana-727 160 28 where where SCONJ cana-727 160 29 n n PRON cana-727 160 30 is be AUX cana-727 160 31 taken take VERB cana-727 160 32 to to PART cana-727 160 33 be be AUX cana-727 160 34 cyclic cyclic ADJ cana-727 160 35 in in ADP cana-727 160 36 m m PROPN cana-727 160 37 and and CCONJ cana-727 160 38 applying apply VERB cana-727 160 39 the the DET cana-727 160 40 sip sip NOUN cana-727 160 41 gives give VERB cana-727 160 42 that that PRON cana-727 160 43 n n NOUN cana-727 160 44 is be AUX cana-727 160 45 a a DET cana-727 160 46 𝐺𝑝-extending 𝐺𝑝-extending PROPN cana-727 160 47 . . PUNCT cana-727 161 1 now now ADV cana-727 161 2 assume assume VERB cana-727 161 3 that that SCONJ cana-727 161 4 m m NOUN cana-727 161 5 satisfies satisfy VERB cana-727 161 6 the the DET cana-727 161 7 𝐢3 𝐢3 ADJ cana-727 161 8 condition condition NOUN cana-727 161 9 . . PUNCT cana-727 162 1 let let VERB cana-727 162 2 πœ‹ πœ‹ NOUN cana-727 162 3 : : PUNCT cana-727 162 4 𝑀 𝑀 PROPN cana-727 162 5 β†’ β†’ SYM cana-727 162 6 𝑁 𝑁 PROPN cana-727 162 7 be be AUX cana-727 162 8 the the DET cana-727 162 9 canonical canonical ADJ cana-727 162 10 projection projection NOUN cana-727 162 11 . . PUNCT cana-727 163 1 let let VERB cana-727 163 2 k k PRON cana-727 163 3 be be AUX cana-727 163 4 any any DET cana-727 163 5 cyclic cyclic ADJ cana-727 163 6 submodule submodule NOUN cana-727 163 7 of of ADP cana-727 163 8 n n CCONJ cana-727 163 9 , , PUNCT cana-727 163 10 then then ADV cana-727 163 11 k k PROPN cana-727 163 12 is be AUX cana-727 163 13 cyclic cyclic ADJ cana-727 163 14 in in ADP cana-727 163 15 m. m. NOUN cana-727 163 16 by by ADP cana-727 163 17 hypothesis hypothesis NOUN cana-727 163 18 , , PUNCT cana-727 163 19 there there PRON cana-727 163 20 exists exist VERB cana-727 163 21 a a DET cana-727 163 22 direct direct ADJ cana-727 163 23 summand summand NOUN cana-727 163 24 l l NOUN cana-727 163 25 of of ADP cana-727 163 26 m m PRON cana-727 163 27 such such ADJ cana-727 163 28 that that SCONJ cana-727 163 29 𝐾 𝐾 PROPN cana-727 163 30 ∩ ∩ ADJ cana-727 163 31 𝐿 𝐿 PROPN cana-727 163 32 ≀𝑒 ≀𝑒 NOUN cana-727 163 33 𝐾 𝐾 PROPN cana-727 163 34 and and CCONJ cana-727 163 35 ∩ ∩ ADJ cana-727 163 36 𝐿 𝐿 PROPN cana-727 163 37 ≀𝑒 ≀𝑒 PROPN cana-727 163 38 𝐿 𝐿 PROPN cana-727 163 39 . . PUNCT cana-727 164 1 since since SCONJ cana-727 164 2 m m PROPN cana-727 164 3 satisfies satisfy VERB cana-727 164 4 𝐢3 𝐢3 ADJ cana-727 164 5 condition condition NOUN cana-727 164 6 , , PUNCT cana-727 164 7 𝑁′⨁𝐿 𝑁′⨁𝐿 INTJ cana-727 164 8 is be AUX cana-727 164 9 a a DET cana-727 164 10 direct direct ADJ cana-727 164 11 summand summand NOUN cana-727 164 12 of of ADP cana-727 164 13 m. m. NOUN cana-727 164 14 it it PRON cana-727 164 15 can can AUX cana-727 164 16 be be AUX cana-727 164 17 seen see VERB cana-727 164 18 that that SCONJ cana-727 164 19 𝑁′⨁𝐿 𝑁′⨁𝐿 ADP cana-727 164 20 = = PUNCT cana-727 164 21 π‘β€²β¨πœ‹(𝐿)(see π‘β€²β¨πœ‹(𝐿)(see PROPN cana-727 165 1 [ [ X cana-727 165 2 11 11 NUM cana-727 165 3 , , PUNCT cana-727 165 4 lemma lemma PROPN cana-727 165 5 2.71 2.71 NUM cana-727 165 6 ] ] PUNCT cana-727 165 7 ) ) PUNCT cana-727 165 8 . . PUNCT cana-727 166 1 hence hence ADV cana-727 166 2 πœ‹(𝐿 πœ‹(𝐿 NUM cana-727 166 3 ) ) PUNCT cana-727 166 4 is be AUX cana-727 166 5 a a DET cana-727 166 6 direct direct ADJ cana-727 166 7 summand summand NOUN cana-727 166 8 of of ADP cana-727 166 9 n. n. NOUN cana-727 166 10 for for ADP cana-727 166 11 any any DET cana-727 166 12 0 0 NUM cana-727 166 13 β‰  β‰  PROPN cana-727 166 14 𝑦 𝑦 PROPN cana-727 166 15 ∈ ∈ PROPN cana-727 166 16 πœ‹(𝐿 πœ‹(𝐿 NOUN cana-727 166 17 ) ) PUNCT cana-727 166 18 , , PUNCT cana-727 166 19 𝑦 𝑦 NOUN cana-727 166 20 = = PUNCT cana-727 166 21 πœ‹(π‘₯ πœ‹(π‘₯ ADV cana-727 166 22 ) ) PUNCT cana-727 166 23 for for ADP cana-727 166 24 some some PRON cana-727 166 25 0 0 NUM cana-727 166 26 β‰  β‰  PROPN cana-727 166 27 π‘₯ π‘₯ DET cana-727 166 28 ∈ ∈ NOUN cana-727 166 29 𝐿. 𝐿. NOUN cana-727 166 30 there there PRON cana-727 166 31 exists exist VERB cana-727 166 32 an an DET cana-727 166 33 π‘Ÿ π‘Ÿ NOUN cana-727 166 34 ∈ ∈ PROPN cana-727 166 35 𝑅 𝑅 PROPN cana-727 166 36 such such ADJ cana-727 166 37 that that DET cana-727 166 38 0 0 NUM cana-727 166 39 β‰  β‰  NOUN cana-727 166 40 π‘₯π‘Ÿ π‘₯π‘Ÿ ADP cana-727 166 41 ∈ ∈ PROPN cana-727 166 42 𝐾 𝐾 PROPN cana-727 166 43 ∩ ∩ NOUN cana-727 166 44 𝐿. 𝐿. VERB cana-727 166 45 so so ADV cana-727 166 46 π‘₯π‘Ÿ π‘₯π‘Ÿ ADP cana-727 166 47 = = PUNCT cana-727 166 48 π‘˜ π‘˜ X cana-727 166 49 = = SYM cana-727 166 50 π‘₯1 π‘₯1 PROPN cana-727 166 51 , , PUNCT cana-727 166 52 where where SCONJ cana-727 166 53 π‘˜ π‘˜ PRON cana-727 166 54 ∈ ∈ PROPN cana-727 166 55 𝐾 𝐾 PROPN cana-727 166 56 and and CCONJ cana-727 166 57 π‘₯1 π‘₯1 PROPN cana-727 166 58 ∈ ∈ PROPN cana-727 166 59 𝐿. 𝐿. VERB cana-727 166 60 now now ADV cana-727 166 61 0 0 NUM cana-727 166 62 β‰  β‰  PROPN cana-727 166 63 π‘₯π‘Ÿ π‘₯π‘Ÿ NOUN cana-727 166 64 = = PUNCT cana-727 166 65 πœ‹(π‘₯)π‘Ÿ πœ‹(π‘₯)π‘Ÿ PROPN cana-727 166 66 = = PUNCT cana-727 166 67 π‘˜ π‘˜ X cana-727 166 68 = = SYM cana-727 166 69 πœ‹(π‘₯1 πœ‹(π‘₯1 X cana-727 166 70 ) ) PUNCT cana-727 166 71 ∈ ∈ PROPN cana-727 166 72 𝐾 𝐾 PROPN cana-727 166 73 ∩ ∩ NOUN cana-727 166 74 πœ‹(𝐿 πœ‹(𝐿 NUM cana-727 166 75 ) ) PUNCT cana-727 166 76 . . PUNCT cana-727 167 1 it it PRON cana-727 167 2 follows follow VERB cana-727 167 3 that that SCONJ cana-727 167 4 𝐾 𝐾 PROPN cana-727 167 5 ∩ ∩ NOUN cana-727 167 6 πœ‹(𝐿 πœ‹(𝐿 NOUN cana-727 167 7 ) ) PUNCT cana-727 167 8 ≀𝑒 ≀𝑒 NOUN cana-727 167 9 πœ‹(𝐿 πœ‹(𝐿 NUM cana-727 167 10 ) ) PUNCT cana-727 167 11 . . PUNCT cana-727 168 1 it it PRON cana-727 168 2 is be AUX cana-727 168 3 clear clear ADJ cana-727 168 4 that that SCONJ cana-727 168 5 πœ‹(𝐿 πœ‹(𝐿 NOUN cana-727 168 6 ) ) PUNCT cana-727 169 1 = = SYM cana-727 169 2 𝑁 𝑁 PROPN cana-727 169 3 ∩ ∩ NOUN cana-727 169 4 ( ( PUNCT cana-727 169 5 π‘β€²β¨πœ‹(𝐿 π‘β€²β¨πœ‹(𝐿 ADJ cana-727 169 6 ) ) PUNCT cana-727 169 7 ) ) PUNCT cana-727 170 1 = = SYM cana-727 170 2 𝑁 𝑁 PROPN cana-727 170 3 ∩ ∩ NOUN cana-727 170 4 ( ( PUNCT cana-727 170 5 𝑁′⨁𝐿 𝑁′⨁𝐿 NOUN cana-727 170 6 ) ) PUNCT cana-727 170 7 . . PUNCT cana-727 171 1 hence hence ADV cana-727 171 2 𝐾 𝐾 PROPN cana-727 171 3 ∩ ∩ NOUN cana-727 171 4 πœ‹(𝐿 πœ‹(𝐿 NUM cana-727 171 5 ) ) PUNCT cana-727 171 6 = = SYM cana-727 171 7 𝐾 𝐾 PROPN cana-727 171 8 ∩ ∩ NOUN cana-727 171 9 ( ( PUNCT cana-727 171 10 𝑁′⨁𝐿 𝑁′⨁𝐿 X cana-727 171 11 ) ) PUNCT cana-727 171 12 ≀𝑒 ≀𝑒 NOUN cana-727 171 13 𝐾. 𝐾. PROPN cana-727 171 14 thus thus ADV cana-727 171 15 , , PUNCT cana-727 171 16 n n PROPN cana-727 171 17 is be AUX cana-727 171 18 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 171 19 . . PUNCT cana-727 171 20 " " PUNCT cana-727 172 1 " " PUNCT cana-727 172 2 next next ADV cana-727 172 3 , , PUNCT cana-727 172 4 we we PRON cana-727 172 5 investigate investigate VERB cana-727 172 6 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 172 7 essential essential ADJ cana-727 172 8 extensions extension NOUN cana-727 172 9 of of ADP cana-727 172 10 a a DET cana-727 172 11 module module NOUN cana-727 172 12 or or CCONJ cana-727 172 13 ring ring NOUN cana-727 172 14 . . PUNCT cana-727 173 1 let let VERB cana-727 173 2 us we PRON cana-727 173 3 begin begin VERB cana-727 173 4 with with ADP cana-727 173 5 the the DET cana-727 173 6 following follow VERB cana-727 173 7 useful useful ADJ cana-727 173 8 result result NOUN cana-727 173 9 which which PRON cana-727 173 10 provides provide VERB cana-727 173 11 relative relative ADJ cana-727 173 12 injectivity injectivity NOUN cana-727 173 13 or or CCONJ cana-727 173 14 certain certain ADJ cana-727 173 15 direct direct ADJ cana-727 173 16 summands summand NOUN cana-727 173 17 of of ADP cana-727 173 18 a a DET cana-727 173 19 goldie goldie NOUN cana-727 173 20 extending extend VERB cana-727 173 21 module module NOUN cana-727 173 22 ( ( PUNCT cana-727 173 23 or or CCONJ cana-727 173 24 nonsingular nonsingular ADJ cana-727 173 25 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 173 26 module module NOUN cana-727 173 27 ) ) PUNCT cana-727 173 28 . . PUNCT cana-727 173 29 " " PUNCT cana-727 174 1 " " PUNCT cana-727 174 2 let let VERB cana-727 174 3 n n PRON cana-727 174 4 , , PUNCT cana-727 174 5 m m VERB cana-727 174 6 be be VERB cana-727 174 7 modules module NOUN cana-727 174 8 . . PUNCT cana-727 175 1 n n PRON cana-727 175 2 is be AUX cana-727 175 3 said say VERB cana-727 175 4 to to PART cana-727 175 5 be be AUX cana-727 175 6 m m NOUN cana-727 175 7 - - NOUN cana-727 175 8 ejective ejective ADJ cana-727 175 9 if if SCONJ cana-727 175 10 , , PUNCT cana-727 175 11 for for ADP cana-727 175 12 each each DET cana-727 175 13 𝐾 𝐾 PROPN cana-727 175 14 ≀ ≀ NOUN cana-727 175 15 𝑀 𝑀 PROPN cana-727 175 16 and and CCONJ cana-727 175 17 each each DET cana-727 175 18 homomorphism homomorphism NOUN cana-727 175 19 𝑓 𝑓 X cana-727 175 20 : : PUNCT cana-727 175 21 𝐾 𝐾 PROPN cana-727 175 22 ⟢ ⟢ NOUN cana-727 175 23 𝑁 𝑁 PROPN cana-727 175 24 , , PUNCT cana-727 175 25 there there PRON cana-727 175 26 exists exist VERB cana-727 175 27 a a DET cana-727 175 28 homomorphism homomorphism NOUN cana-727 175 29 𝑔 𝑔 NOUN cana-727 175 30 : : PUNCT cana-727 175 31 𝑀 𝑀 PROPN cana-727 175 32 β†’ β†’ SYM cana-727 175 33 𝑁 𝑁 PROPN cana-727 175 34 and and CCONJ cana-727 175 35 𝑋 𝑋 NOUN cana-727 175 36 ≀𝑒 ≀𝑒 NOUN cana-727 175 37 𝐾 𝐾 PROPN cana-727 175 38 such such ADJ cana-727 175 39 that that DET cana-727 175 40 g(x g(x NOUN cana-727 175 41 ) ) PUNCT cana-727 176 1 = = SYM cana-727 176 2 f f X cana-727 176 3 ( ( PUNCT cana-727 176 4 x x NOUN cana-727 176 5 ) ) PUNCT cana-727 176 6 , , PUNCT cana-727 176 7 for for ADP cana-727 176 8 all all PRON cana-727 176 9 π‘₯ π‘₯ DET cana-727 176 10 ∈ ∈ PROPN cana-727 176 11 𝑋 𝑋 NOUN cana-727 176 12 , , PUNCT cana-727 176 13 see see VERB cana-727 176 14 [ [ X cana-727 176 15 1 1 NUM cana-727 176 16 ] ] PUNCT cana-727 176 17 . . PUNCT cana-727 176 18 " " PUNCT cana-727 177 1 proposition proposition NOUN cana-727 177 2 4.10 4.10 NUM cana-727 177 3 : : PUNCT cana-727 177 4 " " PUNCT cana-727 177 5 let let VERB cana-727 177 6 r r NOUN cana-727 177 7 be be AUX cana-727 177 8 any any DET cana-727 177 9 ring ring NOUN cana-727 177 10 , , PUNCT cana-727 177 11 𝑀1 𝑀1 ADV cana-727 177 12 a a DET cana-727 177 13 semisimple semisimple ADJ cana-727 177 14 right right ADJ cana-727 177 15 rmodule rmodule NOUN cana-727 177 16 , , PUNCT cana-727 177 17 and and CCONJ cana-727 177 18 𝑀2 𝑀2 PROPN cana-727 177 19 a a DET cana-727 177 20 right right ADJ cana-727 177 21 rmodule rmodule NOUN cana-727 177 22 with with ADP cana-727 177 23 zero zero NUM cana-727 177 24 socle socle NOUN cana-727 177 25 such such ADJ cana-727 177 26 that that PRON cana-727 177 27 𝑀 𝑀 PROPN cana-727 177 28 = = SYM cana-727 177 29 𝑀1⨁𝑀2 𝑀1⨁𝑀2 X cana-727 177 30 is be AUX cana-727 177 31 a a DET cana-727 177 32 goldie goldie NOUN cana-727 177 33 extending extend VERB cana-727 177 34 ucmodule ucmodule NOUN cana-727 177 35 . . PUNCT cana-727 178 1 then then ADV cana-727 178 2 𝑀1 𝑀1 PROPN cana-727 178 3 is be AUX cana-727 178 4 𝑀2 𝑀2 NOUN cana-727 178 5 ejective ejective ADJ cana-727 178 6 . . PUNCT cana-727 178 7 " " PUNCT cana-727 179 1 communications communication NOUN cana-727 179 2 on on ADP cana-727 179 3 applied apply VERB cana-727 179 4 nonlinear nonlinear ADJ cana-727 179 5 analysis analysis NOUN cana-727 179 6 issn issn NOUN cana-727 179 7 : : PUNCT cana-727 179 8 1074 1074 NUM cana-727 179 9 - - PUNCT cana-727 179 10 133x 133x NUM cana-727 179 11 vol vol NOUN cana-727 179 12 31 31 NUM cana-727 179 13 no no NOUN cana-727 179 14 . . PUNCT cana-727 180 1 3s 3s NUM cana-727 180 2 ( ( PUNCT cana-727 180 3 2024 2024 NUM cana-727 180 4 ) ) PUNCT cana-727 180 5 16 16 NUM cana-727 181 1 https://internationalpubls.com https://internationalpubls.com X cana-727 181 2 proof proof NOUN cana-727 181 3 : : PUNCT cana-727 181 4 " " PUNCT cana-727 181 5 obviously obviously ADV cana-727 181 6 , , PUNCT cana-727 181 7 𝑀1 𝑀1 PROPN cana-727 181 8 = = SYM cana-727 181 9 π‘†π‘œπ‘(𝑀 π‘†π‘œπ‘(𝑀 PROPN cana-727 181 10 ) ) PUNCT cana-727 181 11 . . PUNCT cana-727 182 1 let let VERB cana-727 182 2 n n PRON cana-727 182 3 be be AUX cana-727 182 4 any any DET cana-727 182 5 submodule submodule NOUN cana-727 182 6 of of ADP cana-727 182 7 𝑀2 𝑀2 PROPN cana-727 182 8 , , PUNCT cana-727 182 9 and and CCONJ cana-727 182 10 let let VERB cana-727 182 11 πœ‘ πœ‘ PRON cana-727 182 12 : : PUNCT cana-727 182 13 𝑁 𝑁 PROPN cana-727 182 14 β†’ β†’ PUNCT cana-727 182 15 𝑀1 𝑀1 ADV cana-727 182 16 be be AUX cana-727 182 17 a a DET cana-727 182 18 homomorphism homomorphism NOUN cana-727 182 19 . . PUNCT cana-727 183 1 let let VERB cana-727 183 2 𝐿 𝐿 PROPN cana-727 183 3 = = PRON cana-727 183 4 { { PUNCT cana-727 183 5 π‘₯ π‘₯ X cana-727 183 6 βˆ’ βˆ’ NOUN cana-727 183 7 πœ‘(π‘₯ πœ‘(π‘₯ NOUN cana-727 183 8 ): ): PUNCT cana-727 183 9 π‘₯ π‘₯ PRON cana-727 183 10 ∈ ∈ PROPN cana-727 183 11 𝑁 𝑁 PROPN cana-727 183 12 } } PUNCT cana-727 183 13 . . PUNCT cana-727 184 1 then then ADV cana-727 184 2 l l PROPN cana-727 184 3 is be AUX cana-727 184 4 a a DET cana-727 184 5 submodule submodule NOUN cana-727 184 6 of of ADP cana-727 184 7 m m PROPN cana-727 184 8 and and CCONJ cana-727 184 9 𝐿 𝐿 PROPN cana-727 184 10 ∩ ∩ NOUN cana-727 184 11 𝑀1=0 𝑀1=0 PROPN cana-727 184 12 . . PUNCT cana-727 185 1 there there PRON cana-727 185 2 exists exist VERB cana-727 185 3 submodules submodule NOUN cana-727 185 4 k k PROPN cana-727 185 5 , , PUNCT cana-727 185 6 k k X cana-727 185 7 ' ' PUNCT cana-727 185 8 of of ADP cana-727 185 9 m m PRON cana-727 185 10 such such ADJ cana-727 185 11 that that SCONJ cana-727 185 12 𝑀 𝑀 PROPN cana-727 185 13 = = PUNCT cana-727 185 14 𝐾⨁𝐾′ 𝐾⨁𝐾′ PROPN cana-727 185 15 , , PUNCT cana-727 185 16 𝐾 𝐾 PROPN cana-727 185 17 ∩ ∩ ADJ cana-727 185 18 𝐿 𝐿 PROPN cana-727 185 19 ≀𝑒 ≀𝑒 PROPN cana-727 185 20 𝐿 𝐿 PROPN cana-727 185 21 and and CCONJ cana-727 185 22 𝐾 𝐾 PROPN cana-727 185 23 ∩ ∩ ADJ cana-727 185 24 𝐿 𝐿 NOUN cana-727 185 25 ≀𝑒 ≀𝑒 NOUN cana-727 186 1 𝐾. 𝐾. PROPN cana-727 187 1 it it PRON cana-727 187 2 is be AUX cana-727 187 3 clear clear ADJ cana-727 187 4 that that SCONJ cana-727 187 5 k k PROPN cana-727 187 6 is be AUX cana-727 187 7 a a DET cana-727 187 8 closure closure NOUN cana-727 187 9 of of ADP cana-727 187 10 𝐾 𝐾 PROPN cana-727 187 11 ∩ ∩ ADJ cana-727 187 12 𝐿 𝐿 PROPN cana-727 187 13 in in ADP cana-727 187 14 m. m. NOUN cana-727 187 15 by by ADP cana-727 187 16 assumption assumption NOUN cana-727 187 17 , , PUNCT cana-727 187 18 𝐿 𝐿 PROPN cana-727 187 19 ≀ ≀ NOUN cana-727 187 20 𝐾. 𝐾. PROPN cana-727 187 21 since since SCONJ cana-727 187 22 𝐾 𝐾 PROPN cana-727 187 23 ∩ ∩ ADJ cana-727 187 24 𝐿 𝐿 PROPN cana-727 187 25 ∩ ∩ NOUN cana-727 187 26 𝑀1 𝑀1 NOUN cana-727 187 27 = = SYM cana-727 187 28 𝐿 𝐿 PROPN cana-727 187 29 ∩ ∩ NOUN cana-727 187 30 𝑀1 𝑀1 NOUN cana-727 187 31 = = SYM cana-727 187 32 0 0 NUM cana-727 187 33 , , PUNCT cana-727 187 34 𝐾 𝐾 PROPN cana-727 187 35 ∩ ∩ ADJ cana-727 187 36 𝐿 𝐿 PROPN cana-727 187 37 ∩ ∩ NOUN cana-727 187 38 π‘†π‘œπ‘(𝑀 π‘†π‘œπ‘(𝑀 NOUN cana-727 187 39 ) ) PUNCT cana-727 187 40 = = SYM cana-727 187 41 π‘†π‘œπ‘(𝐿 π‘†π‘œπ‘(𝐿 X cana-727 187 42 ) ) PUNCT cana-727 187 43 = = PUNCT cana-727 188 1 0 0 X cana-727 188 2 . . PUNCT cana-727 189 1 it it PRON cana-727 189 2 follows follow VERB cana-727 189 3 that that SCONJ cana-727 189 4 π‘†π‘œπ‘(𝐾 π‘†π‘œπ‘(𝐾 PUNCT cana-727 189 5 ) ) PUNCT cana-727 189 6 = = SYM cana-727 189 7 𝐾 𝐾 PROPN cana-727 189 8 ∩ ∩ NOUN cana-727 189 9 𝑀1 𝑀1 NOUN cana-727 189 10 = = SYM cana-727 189 11 0 0 X cana-727 189 12 . . PUNCT cana-727 190 1 hence hence ADV cana-727 190 2 𝑀1 𝑀1 PROPN cana-727 190 3 = = SYM cana-727 190 4 π‘†π‘œπ‘(𝑀 π‘†π‘œπ‘(𝑀 NOUN cana-727 190 5 ) ) PUNCT cana-727 190 6 βŠ† βŠ† NUM cana-727 190 7 𝐾′. 𝐾′. NUM cana-727 190 8 thus thus ADV cana-727 190 9 , , PUNCT cana-727 190 10 𝐾′ 𝐾′ PROPN cana-727 190 11 = = SYM cana-727 190 12 𝑀1⨁(𝐾′ 𝑀1⨁(𝐾′ PROPN cana-727 190 13 ∩ ∩ ADJ cana-727 190 14 𝑀2 𝑀2 PROPN cana-727 190 15 ) ) PUNCT cana-727 190 16 and and CCONJ cana-727 190 17 𝑀 𝑀 PROPN cana-727 190 18 = = PUNCT cana-727 190 19 𝐾⨁𝑀1(𝐾′⋂𝑀2 𝐾⨁𝑀1(𝐾′⋂𝑀2 ADJ cana-727 190 20 ) ) PUNCT cana-727 190 21 . . PUNCT cana-727 191 1 let let VERB cana-727 191 2 πœ‹ πœ‹ NOUN cana-727 191 3 : : PUNCT cana-727 191 4 𝑀 𝑀 PROPN cana-727 191 5 β†’ β†’ PUNCT cana-727 191 6 𝑀1 𝑀1 ADV cana-727 191 7 denote denote VERB cana-727 191 8 the the DET cana-727 191 9 canonical canonical ADJ cana-727 191 10 projection projection NOUN cana-727 191 11 with with ADP cana-727 191 12 kernel kernel PROPN cana-727 191 13 𝐾⨁(𝐾′⋂𝑀2 𝐾⨁(𝐾′⋂𝑀2 PROPN cana-727 191 14 ) ) PUNCT cana-727 191 15 . . PUNCT cana-727 192 1 let let VERB cana-727 192 2 πœƒ πœƒ PRON cana-727 192 3 be be AUX cana-727 192 4 the the DET cana-727 192 5 restriction restriction NOUN cana-727 192 6 of of ADP cana-727 192 7 πœ‹ πœ‹ PRON cana-727 192 8 to to ADP cana-727 192 9 𝑀2 𝑀2 PROPN cana-727 192 10 . . PUNCT cana-727 193 1 then then ADV cana-727 193 2 πœƒ πœƒ X cana-727 193 3 : : PUNCT cana-727 193 4 𝑀2 𝑀2 PROPN cana-727 193 5 β†’ β†’ SYM cana-727 193 6 𝑀1 𝑀1 PROPN cana-727 193 7 . . PUNCT cana-727 194 1 let let VERB cana-727 194 2 x x PRON cana-727 194 3 be be AUX cana-727 194 4 any any DET cana-727 194 5 element element NOUN cana-727 194 6 of of ADP cana-727 194 7 n. n. NOUN cana-727 194 8 since since SCONJ cana-727 194 9 π‘₯(π‘₯ π‘₯(π‘₯ NOUN cana-727 194 10 βˆ’ βˆ’ NOUN cana-727 194 11 πœ‘(π‘₯ πœ‘(π‘₯ NOUN cana-727 194 12 ) ) PUNCT cana-727 194 13 ) ) PUNCT cana-727 195 1 + + CCONJ cana-727 195 2 πœ‘(π‘₯ πœ‘(π‘₯ NOUN cana-727 195 3 ) ) PUNCT cana-727 195 4 , , PUNCT cana-727 195 5 πœƒ(π‘₯ πœƒ(π‘₯ NOUN cana-727 195 6 ) ) PUNCT cana-727 195 7 = = PUNCT cana-727 195 8 πœ‘(π‘₯ πœ‘(π‘₯ NOUN cana-727 195 9 ) ) PUNCT cana-727 195 10 . . PUNCT cana-727 196 1 it it PRON cana-727 196 2 follows follow VERB cana-727 196 3 that that SCONJ cana-727 196 4 𝑀1 𝑀1 ADV cana-727 196 5 is be AUX cana-727 196 6 𝑀2injective 𝑀2injective ADJ cana-727 196 7 . . PUNCT cana-727 196 8 " " PUNCT cana-727 197 1 corollary corollary ADJ cana-727 197 2 4.11 4.11 NUM cana-727 197 3 : : PUNCT cana-727 197 4 ( ( PUNCT cana-727 197 5 i)"let i)"let ADJ cana-727 197 6 𝑀 𝑀 PROPN cana-727 197 7 = = SYM cana-727 197 8 ⨁𝑖=1 ⨁𝑖=1 X cana-727 197 9 𝑛 𝑛 DET cana-727 197 10 𝑀𝑖 𝑀𝑖 PROPN cana-727 197 11 , , PUNCT cana-727 197 12 where where SCONJ cana-727 197 13 each each DET cana-727 197 14 𝑀𝑖 𝑀𝑖 PROPN cana-727 197 15 is be AUX cana-727 197 16 uniform uniform ADJ cana-727 197 17 . . PUNCT cana-727 198 1 if if SCONJ cana-727 198 2 𝐸(𝑀𝑖 𝐸(𝑀𝑖 NOUN cana-727 198 3 ) ) PUNCT cana-727 198 4 ≇ ≇ PROPN cana-727 198 5 𝐸(𝑀𝑗 𝐸(𝑀𝑗 NUM cana-727 198 6 ) ) PUNCT cana-727 198 7 for for ADP cana-727 198 8 all all PRON cana-727 198 9 𝑖 𝑖 PRON cana-727 198 10 β‰  β‰  PROPN cana-727 198 11 𝑗 𝑗 PROPN cana-727 198 12 , , PUNCT cana-727 198 13 then then ADV cana-727 198 14 m m VERB cana-727 198 15 is be AUX cana-727 198 16 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 198 17 . . PUNCT cana-727 199 1 ( ( PUNCT cana-727 199 2 ii ii NOUN cana-727 199 3 ) ) PUNCT cana-727 199 4 let let VERB cana-727 199 5 s s PRON cana-727 199 6 be be AUX cana-727 199 7 a a DET cana-727 199 8 simple simple ADJ cana-727 199 9 module module NOUN cana-727 199 10 and and CCONJ cana-727 199 11 𝑀1 𝑀1 PROPN cana-727 199 12 , , PUNCT cana-727 199 13 𝑀2 𝑀2 PROPN cana-727 199 14 ≀ ≀ PUNCT cana-727 199 15 𝐸(𝑆 𝐸(𝑆 PROPN cana-727 199 16 ) ) PUNCT cana-727 199 17 . . PUNCT cana-727 200 1 if if SCONJ cana-727 200 2 there there PRON cana-727 200 3 exists exist VERB cana-727 200 4 a a DET cana-727 200 5 homomorphism homomorphism NOUN cana-727 200 6 β„Ž β„Ž PROPN cana-727 200 7 : : PUNCT cana-727 200 8 𝑀2 𝑀2 PROPN cana-727 200 9 β†’ β†’ SYM cana-727 200 10 𝑆 𝑆 PROPN cana-727 200 11 such such ADJ cana-727 200 12 that that DET cana-727 200 13 β„Ž(𝑆 β„Ž(𝑆 NOUN cana-727 200 14 ) ) PUNCT cana-727 200 15 β‰  β‰  PROPN cana-727 200 16 0 0 NUM cana-727 200 17 , , PUNCT cana-727 200 18 then then ADV cana-727 200 19 𝑀 𝑀 PROPN cana-727 200 20 = = SYM cana-727 200 21 𝑀1⨁𝑀2 𝑀1⨁𝑀2 X cana-727 200 22 is be AUX cana-727 200 23 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 200 24 . . PUNCT cana-727 201 1 proof proof NOUN cana-727 201 2 : : PUNCT cana-727 201 3 ( ( PUNCT cana-727 201 4 i i NOUN cana-727 201 5 ) ) PUNCT cana-727 201 6 from from ADP cana-727 201 7 [ [ X cana-727 201 8 1 1 NUM cana-727 201 9 , , PUNCT cana-727 201 10 corollary corollary NOUN cana-727 201 11 4.11 4.11 NUM cana-727 201 12 ] ] PUNCT cana-727 201 13 , , PUNCT cana-727 201 14 m m PROPN cana-727 201 15 is be AUX cana-727 201 16 goldie goldie PROPN cana-727 201 17 extending extending NOUN cana-727 201 18 . . PUNCT cana-727 202 1 thus thus ADV cana-727 202 2 proposition proposition NOUN cana-727 202 3 2.3 2.3 NUM cana-727 202 4 gives give VERB cana-727 202 5 that that PRON cana-727 202 6 m m VERB cana-727 202 7 is be AUX cana-727 202 8 𝐺𝑝extending 𝐺𝑝extende VERB cana-727 202 9 . . PUNCT cana-727 203 1 ( ( PUNCT cana-727 203 2 ii ii NOUN cana-727 203 3 ) ) PUNCT cana-727 203 4 by by ADP cana-727 203 5 [ [ X cana-727 203 6 1 1 NUM cana-727 203 7 , , PUNCT cana-727 203 8 corollary corollary ADJ cana-727 203 9 4.14 4.14 NUM cana-727 203 10 ] ] PUNCT cana-727 203 11 , , PUNCT cana-727 203 12 𝑀1 𝑀1 PROPN cana-727 203 13 is be AUX cana-727 203 14 𝑀2ejective 𝑀2ejective ADJ cana-727 203 15 and and CCONJ cana-727 203 16 so so ADV cana-727 203 17 it it PRON cana-727 203 18 is be AUX cana-727 203 19 g g NOUN cana-727 203 20 - - PUNCT cana-727 203 21 extending extending ADJ cana-727 203 22 . . PUNCT cana-727 204 1 now now ADV cana-727 204 2 , , PUNCT cana-727 204 3 by by ADP cana-727 204 4 proposition proposition NOUN cana-727 204 5 2.3 2.3 NUM cana-727 204 6 m m NOUN cana-727 204 7 is be AUX cana-727 204 8 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 204 9 . . PUNCT cana-727 204 10 " " PUNCT cana-727 204 11 example example NOUN cana-727 205 1 4.12 4.12 NUM cana-727 205 2 : : PUNCT cana-727 205 3 ( ( PUNCT cana-727 205 4 i i NOUN cana-727 205 5 ) ) PUNCT cana-727 205 6 " " PUNCT cana-727 205 7 let let VERB cana-727 205 8 m m PRON cana-727 205 9 be be AUX cana-727 205 10 the the DET cana-727 205 11 𝕫-module 𝕫-module NOUN cana-727 205 12 ( ( PUNCT cana-727 205 13 𝕫/𝕫𝑝)β¨β„š 𝕫/𝕫𝑝)β¨β„š NOUN cana-727 205 14 and and CCONJ cana-727 205 15 let let VERB cana-727 205 16 t t PROPN cana-727 205 17 be be AUX cana-727 205 18 the the DET cana-727 205 19 polynomial polynomial ADJ cana-727 205 20 ring ring NOUN cana-727 205 21 𝕫[π‘₯ 𝕫[π‘₯ PROPN cana-727 205 22 ] ] PUNCT cana-727 205 23 . . PUNCT cana-727 206 1 then then ADV cana-727 206 2 𝑀𝕫 𝑀𝕫 PROPN cana-727 206 3 is be AUX cana-727 206 4 included include VERB cana-727 206 5 in in ADP cana-727 206 6 corollary corollary ADJ cana-727 206 7 4.11(i 4.11(i PROPN cana-727 206 8 ) ) PUNCT cana-727 206 9 . . PUNCT cana-727 207 1 on on ADP cana-727 207 2 the the DET cana-727 207 3 other other ADJ cana-727 207 4 hand hand NOUN cana-727 207 5 , , PUNCT cana-727 207 6 it it PRON cana-727 207 7 is be AUX cana-727 207 8 well well ADV cana-727 207 9 known know VERB cana-727 207 10 that that SCONJ cana-727 207 11 𝑇2 𝑇2 NOUN cana-727 207 12 is be AUX cana-727 207 13 not not PART cana-727 207 14 𝐺𝑝-extendingt 𝐺𝑝-extendingt NOUN cana-727 207 15 module module NOUN cana-727 207 16 . . PUNCT cana-727 208 1 hence hence ADV cana-727 208 2 , , PUNCT cana-727 208 3 we we PRON cana-727 208 4 obtain obtain VERB cana-727 208 5 that that SCONJ cana-727 208 6 the the DET cana-727 208 7 condition condition NOUN cana-727 208 8 𝐸(𝑀𝑖 𝐸(𝑀𝑖 NOUN cana-727 208 9 ) ) PUNCT cana-727 208 10 ≇ ≇ PROPN cana-727 208 11 𝐸(𝑀𝑗 𝐸(𝑀𝑗 NUM cana-727 208 12 ) ) PUNCT cana-727 208 13 for for ADP cana-727 208 14 all all PRON cana-727 208 15 𝑖 𝑖 ADP cana-727 208 16 β‰  β‰  PROPN cana-727 208 17 𝑗 𝑗 PROPN cana-727 208 18 , , PUNCT cana-727 208 19 is be AUX cana-727 208 20 not not PART cana-727 208 21 superfluous superfluous ADJ cana-727 208 22 in in ADP cana-727 208 23 corollary corollary ADJ cana-727 208 24 4.11(i 4.11(i PROPN cana-727 208 25 ) ) PUNCT cana-727 208 26 . . PUNCT cana-727 209 1 ( ( PUNCT cana-727 209 2 ii ii NOUN cana-727 209 3 ) ) PUNCT cana-727 209 4 let let VERB cana-727 209 5 k k PRON cana-727 209 6 be be AUX cana-727 209 7 a a DET cana-727 209 8 field field NOUN cana-727 209 9 and and CCONJ cana-727 209 10 r r NOUN cana-727 209 11 = = SYM cana-727 209 12 k[x k[x PROPN cana-727 209 13 , , PUNCT cana-727 209 14 y y PROPN cana-727 209 15 ] ] PUNCT cana-727 209 16 , , PUNCT cana-727 209 17 the the DET cana-727 209 18 commutative commutative ADJ cana-727 209 19 local local ADJ cana-727 209 20 frobenious frobenious ADJ cana-727 209 21 k k NOUN cana-727 209 22 - - NOUN cana-727 209 23 algebra algebra PROPN cana-727 209 24 ( ( PUNCT cana-727 209 25 see[1 see[1 X cana-727 209 26 , , PUNCT cana-727 209 27 example example NOUN cana-727 209 28 4.15 4.15 NUM cana-727 209 29 ] ] PUNCT cana-727 209 30 ) ) PUNCT cana-727 209 31 defined define VERB cana-727 209 32 by by ADP cana-727 209 33 the the DET cana-727 209 34 relations relation NOUN cana-727 209 35 π‘₯𝑦 π‘₯𝑦 ADJ cana-727 209 36 = = NOUN cana-727 209 37 π‘₯2 π‘₯2 PROPN cana-727 209 38 βˆ’ βˆ’ PROPN cana-727 209 39 𝑦2 𝑦2 PROPN cana-727 209 40 = = PUNCT cana-727 209 41 0 0 X cana-727 209 42 . . PUNCT cana-727 210 1 then then ADV cana-727 210 2 𝑅𝑅 𝑅𝑅 PROPN cana-727 210 3 is be AUX cana-727 210 4 a a DET cana-727 210 5 uniform uniform ADJ cana-727 210 6 injective injective ADJ cana-727 210 7 module module NOUN cana-727 210 8 with with ADP cana-727 210 9 simple simple ADJ cana-727 210 10 submodule submodule NOUN cana-727 210 11 𝐾π‘₯2 𝐾π‘₯2 NOUN cana-727 210 12 . . PUNCT cana-727 211 1 let let VERB cana-727 211 2 𝑀2 𝑀2 PROPN cana-727 211 3 = = SYM cana-727 211 4 π‘₯𝑅 π‘₯𝑅 NOUN cana-727 211 5 = = PUNCT cana-727 211 6 { { PUNCT cana-727 211 7 π‘˜1π‘₯ π‘˜1π‘₯ NOUN cana-727 211 8 + + CCONJ cana-727 211 9 π‘˜.2 π‘˜.2 NOUN cana-727 211 10 π‘₯2 π‘₯2 NOUN cana-727 211 11 : : PUNCT cana-727 211 12 π‘˜π‘– π‘˜π‘– NOUN cana-727 211 13 ∈ ∈ PROPN cana-727 211 14 𝐾 𝐾 PROPN cana-727 211 15 } } PUNCT cana-727 211 16 , , PUNCT cana-727 211 17 and and CCONJ cana-727 211 18 let let VERB cana-727 211 19 h h NOUN cana-727 211 20 be be AUX cana-727 211 21 the the DET cana-727 211 22 r r NOUN cana-727 211 23 - - PUNCT cana-727 211 24 homomorphism homomorphism NOUN cana-727 211 25 , , PUNCT cana-727 211 26 β„Ž β„Ž PROPN cana-727 211 27 : : PUNCT cana-727 211 28 π‘₯𝑅 π‘₯𝑅 NOUN cana-727 211 29 β†’ β†’ SYM cana-727 211 30 𝐾π‘₯2 𝐾π‘₯2 NOUN cana-727 211 31 , , PUNCT cana-727 211 32 defined define VERB cana-727 211 33 by by ADP cana-727 211 34 β„Ž(π‘˜1π‘₯ β„Ž(π‘˜1π‘₯ PRON cana-727 211 35 + + CCONJ cana-727 211 36 π‘˜2π‘₯2 π‘˜2π‘₯2 X cana-727 211 37 ) ) PUNCT cana-727 211 38 = = SYM cana-727 212 1 π‘˜2π‘₯2 π‘˜2π‘₯2 X cana-727 212 2 . . PUNCT cana-727 213 1 then then ADV cana-727 213 2 β„Ž(𝐾π‘₯2 β„Ž(𝐾π‘₯2 NUM cana-727 213 3 ) ) PUNCT cana-727 213 4 β‰  β‰  PROPN cana-727 213 5 0 0 NUM cana-727 213 6 . . PUNCT cana-727 214 1 thus thus ADV cana-727 214 2 , , PUNCT cana-727 214 3 by by ADP cana-727 214 4 corollary corollary ADJ cana-727 214 5 4.11(ii 4.11(ii PROPN cana-727 214 6 ) ) PUNCT cana-727 214 7 , , PUNCT cana-727 214 8 𝑀 𝑀 PROPN cana-727 214 9 = = PUNCT cana-727 214 10 𝑀1⨁π‘₯𝑅 𝑀1⨁π‘₯𝑅 PROPN cana-727 214 11 is be AUX cana-727 214 12 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 214 13 for for ADP cana-727 214 14 any any DET cana-727 214 15 𝑀1 𝑀1 NOUN cana-727 214 16 ≀ ≀ NUM cana-727 214 17 𝑅𝑅. 𝑅𝑅. NOUN cana-727 214 18 " " PUNCT cana-727 214 19 " " PUNCT cana-727 214 20 next next ADJ cana-727 214 21 example example NOUN cana-727 214 22 exhibits exhibit VERB cana-727 214 23 that that SCONJ cana-727 214 24 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 214 25 property property NOUN cana-727 214 26 is be AUX cana-727 214 27 not not PART cana-727 214 28 closed close VERB cana-727 214 29 under under ADP cana-727 214 30 essential essential ADJ cana-727 214 31 extensions extension NOUN cana-727 214 32 of of ADP cana-727 214 33 a a DET cana-727 214 34 module module NOUN cana-727 214 35 . . PUNCT cana-727 214 36 " " PUNCT cana-727 215 1 example example NOUN cana-727 215 2 4.13 4.13 NUM cana-727 215 3 : : PUNCT cana-727 215 4 " " PUNCT cana-727 215 5 let let VERB cana-727 215 6 f f PRON cana-727 215 7 be be AUX cana-727 215 8 any any DET cana-727 215 9 field field NOUN cana-727 215 10 and and CCONJ cana-727 215 11 𝑅 𝑅 NOUN cana-727 216 1 = = SYM cana-727 217 1 [ [ PUNCT cana-727 217 2 𝐹 𝐹 PROPN cana-727 217 3 𝐹 𝐹 PROPN cana-727 217 4 𝐹 𝐹 PROPN cana-727 217 5 0 0 NUM cana-727 217 6 𝐹 𝐹 PROPN cana-727 217 7 0 0 NUM cana-727 217 8 0 0 NUM cana-727 217 9 0 0 NUM cana-727 217 10 𝐹 𝐹 PROPN cana-727 217 11 ] ] PUNCT cana-727 217 12 . . PUNCT cana-727 218 1 then then ADV cana-727 218 2 π‘†π‘œπ‘(𝑅𝑅 π‘†π‘œπ‘(𝑅𝑅 NOUN cana-727 218 3 ) ) PUNCT cana-727 218 4 ≀𝑒 ≀𝑒 NOUN cana-727 218 5 𝑅𝑅. 𝑅𝑅. AUX cana-727 218 6 obviously obviously ADV cana-727 218 7 π‘†π‘œπ‘(𝑅 π‘†π‘œπ‘(𝑅 NUM cana-727 218 8 ) ) PUNCT cana-727 218 9 is be AUX cana-727 218 10 a a DET cana-727 218 11 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 218 12 right right ADJ cana-727 218 13 rmodule rmodule NOUN cana-727 218 14 . . PUNCT cana-727 219 1 however however ADV cana-727 219 2 , , PUNCT cana-727 219 3 it it PRON cana-727 219 4 is be AUX cana-727 219 5 well well ADV cana-727 219 6 known know VERB cana-727 219 7 that that SCONJ cana-727 219 8 𝑅𝑅 𝑅𝑅 PROPN cana-727 219 9 is be AUX cana-727 219 10 not not PART cana-727 219 11 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 219 12 ( ( PUNCT cana-727 219 13 see see VERB cana-727 219 14 [ [ X cana-727 219 15 13 13 NUM cana-727 219 16 , , PUNCT cana-727 219 17 theorem theorem VERB cana-727 219 18 3.4 3.4 NUM cana-727 219 19 ] ] PUNCT cana-727 219 20 ) ) PUNCT cana-727 219 21 . . PUNCT cana-727 219 22 " " PUNCT cana-727 220 1 " " PUNCT cana-727 220 2 in in ADP cana-727 220 3 contrast contrast NOUN cana-727 220 4 to to ADP cana-727 220 5 essential essential ADJ cana-727 220 6 extensions extension NOUN cana-727 220 7 of of ADP cana-727 220 8 a a DET cana-727 220 9 module module NOUN cana-727 220 10 which which PRON cana-727 220 11 satisfies satisfy VERB cana-727 220 12 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 220 13 condition condition NOUN cana-727 220 14 , , PUNCT cana-727 220 15 we we PRON cana-727 220 16 have have VERB cana-727 220 17 the the DET cana-727 220 18 following follow VERB cana-727 220 19 overring overring NOUN cana-727 220 20 of of ADP cana-727 220 21 a a DET cana-727 220 22 ring ring NOUN cana-727 220 23 r r NOUN cana-727 220 24 if if SCONJ cana-727 220 25 sis sis PRON cana-727 220 26 an an DET cana-727 220 27 overring overring NOUN cana-727 220 28 of of ADP cana-727 220 29 r r NOUN cana-727 220 30 such such ADJ cana-727 220 31 that that SCONJ cana-727 220 32 𝑅𝑅 𝑅𝑅 PROPN cana-727 220 33 essential essential ADJ cana-727 220 34 in in ADP cana-727 220 35 𝑆𝑅. 𝑆𝑅. ADJ cana-727 220 36 " " PUNCT cana-727 220 37 theorem theorem VERB cana-727 220 38 4.14:"let 4.14:"let NUM cana-727 220 39 s s AUX cana-727 220 40 be be AUX cana-727 220 41 a a DET cana-727 220 42 right right ADJ cana-727 220 43 essential essential ADJ cana-727 220 44 overring overring NOUN cana-727 220 45 of of ADP cana-727 220 46 r r NOUN cana-727 220 47 ( ( PUNCT cana-727 220 48 i.e. i.e. X cana-727 220 49 , , PUNCT cana-727 220 50 𝑅𝑅 𝑅𝑅 PROPN cana-727 220 51 ≀𝑒 ≀𝑒 NOUN cana-727 220 52 𝑆𝑅 𝑆𝑅 PROPN cana-727 220 53 ) ) PUNCT cana-727 220 54 . . PUNCT cana-727 221 1 if if SCONJ cana-727 221 2 𝑅𝑅 𝑅𝑅 PROPN cana-727 221 3 is be AUX cana-727 221 4 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 221 5 , , PUNCT cana-727 221 6 then then ADV cana-727 221 7 𝑆𝑅 𝑆𝑅 PROPN cana-727 221 8 and and CCONJ cana-727 221 9 𝑆𝑆 𝑆𝑆 PROPN cana-727 221 10 are be AUX cana-727 221 11 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 221 12 . . PUNCT cana-727 222 1 proof proof NOUN cana-727 222 2 : : PUNCT cana-727 222 3 let let VERB cana-727 222 4 π‘Œπ‘… π‘Œπ‘… PROPN cana-727 222 5 be be AUX cana-727 222 6 any any DET cana-727 222 7 cyclic cyclic ADJ cana-727 222 8 submodule submodule NOUN cana-727 222 9 of of ADP cana-727 222 10 𝑆𝑅. 𝑆𝑅. NUM cana-727 222 11 that that ADV cana-727 222 12 much much ADJ cana-727 222 13 is be AUX cana-727 222 14 clear clear ADJ cana-727 222 15 to to PART cana-727 222 16 see.𝑋 see.𝑋 NOUN cana-727 222 17 = = SYM cana-727 222 18 π‘Œ π‘Œ PROPN cana-727 222 19 ∩ ∩ NOUN cana-727 222 20 𝑅 𝑅 PROPN cana-727 222 21 is be AUX cana-727 222 22 cyclic cyclic ADJ cana-727 222 23 submodule submodule NOUN cana-727 222 24 of of ADP cana-727 222 25 𝑅𝑅 𝑅𝑅 PROPN cana-727 222 26 . . PUNCT cana-727 223 1 by by ADP cana-727 223 2 proposition proposition NOUN cana-727 223 3 3.2 3.2 NUM cana-727 223 4 , , PUNCT cana-727 223 5 there there PRON cana-727 223 6 exists exist VERB cana-727 223 7 𝐾𝑅 𝐾𝑅 PROPN cana-727 223 8 ≀ ≀ ADJ cana-727 223 9 𝑅𝑅 𝑅𝑅 PROPN cana-727 223 10 and and CCONJ cana-727 223 11 𝑒2 𝑒2 PROPN cana-727 223 12 = = PUNCT cana-727 223 13 𝑒 𝑒 PROPN cana-727 223 14 ∈ ∈ PROPN cana-727 223 15 𝑅 𝑅 PROPN cana-727 223 16 such such ADJ cana-727 223 17 that that SCONJ cana-727 223 18 𝐾𝑅 𝐾𝑅 PROPN cana-727 223 19 ≀𝑒 ≀𝑒 NOUN cana-727 223 20 𝑋𝑅 𝑋𝑅 PROPN cana-727 223 21 and and CCONJ cana-727 223 22 𝐾𝑅 𝐾𝑅 PROPN cana-727 223 23 ≀𝑒 ≀𝑒 NOUN cana-727 223 24 𝑒𝑅𝑅. 𝑒𝑅𝑅. NOUN cana-727 223 25 communications communication NOUN cana-727 223 26 on on ADP cana-727 223 27 applied apply VERB cana-727 223 28 nonlinear nonlinear ADJ cana-727 223 29 analysis analysis NOUN cana-727 223 30 issn issn NOUN cana-727 223 31 : : PUNCT cana-727 223 32 1074 1074 NUM cana-727 223 33 - - PUNCT cana-727 223 34 133x 133x NUM cana-727 223 35 vol vol NOUN cana-727 223 36 31 31 NUM cana-727 223 37 no no NOUN cana-727 223 38 . . PUNCT cana-727 224 1 3s 3s NUM cana-727 224 2 ( ( PUNCT cana-727 224 3 2024 2024 NUM cana-727 224 4 ) ) PUNCT cana-727 224 5 17 17 NUM cana-727 224 6 https://internationalpubls.com https://internationalpubls.com X cana-727 224 7 notice notice VERB cana-727 224 8 that𝐾𝑅 that𝐾𝑅 NUM cana-727 224 9 ≀𝑒 ≀𝑒 NOUN cana-727 224 10 π‘Œπ‘… π‘Œπ‘… PROPN cana-727 224 11 . . PUNCT cana-727 225 1 now now ADV cana-727 225 2 , , PUNCT cana-727 225 3 let let VERB cana-727 225 4 us we PRON cana-727 225 5 show show VERB cana-727 225 6 that that SCONJ cana-727 225 7 𝐾𝑅 𝐾𝑅 PROPN cana-727 225 8 ≀𝑒 ≀𝑒 NOUN cana-727 225 9 𝑒𝑆𝑅. 𝑒𝑆𝑅. AUX cana-727 225 10 let let VERB cana-727 225 11 0 0 NUM cana-727 225 12 β‰  β‰  PROPN cana-727 225 13 𝑒𝑠 𝑒𝑠 PROPN cana-727 225 14 ∈ ∈ PROPN cana-727 225 15 𝑒𝑆. 𝑒𝑆. NOUN cana-727 225 16 there there PRON cana-727 225 17 exists exist VERB cana-727 225 18 π‘Ÿ1 π‘Ÿ1 PROPN cana-727 225 19 ∈ ∈ PROPN cana-727 225 20 𝑅 𝑅 PROPN cana-727 226 1 such such ADJ cana-727 226 2 that that DET cana-727 226 3 0 0 NUM cana-727 226 4 β‰  β‰  PROPN cana-727 226 5 π‘’π‘ π‘Ÿ1 π‘’π‘ π‘Ÿ1 ADJ cana-727 226 6 ∈ ∈ NOUN cana-727 226 7 𝑅. 𝑅. NOUN cana-727 226 8 hence hence ADV cana-727 226 9 0 0 NUM cana-727 227 1 β‰  β‰  PROPN cana-727 227 2 π‘’π‘ π‘Ÿ1 π‘’π‘ π‘Ÿ1 PROPN cana-727 227 3 ∈ ∈ NOUN cana-727 227 4 𝑒𝑅 𝑒𝑅 VERB cana-727 227 5 , , PUNCT cana-727 227 6 so so SCONJ cana-727 227 7 there there PRON cana-727 227 8 exists exist VERB cana-727 227 9 π‘Ÿ2 π‘Ÿ2 NOUN cana-727 227 10 ∈ ∈ PROPN cana-727 227 11 𝑅 𝑅 PROPN cana-727 227 12 such such ADJ cana-727 227 13 that that DET cana-727 227 14 0 0 NUM cana-727 227 15 β‰  β‰  PROPN cana-727 227 16 π‘’π‘ π‘Ÿ1π‘Ÿ2 π‘’π‘ π‘Ÿ1π‘Ÿ2 NOUN cana-727 227 17 ∈ ∈ NOUN cana-727 227 18 𝐾. 𝐾. PROPN cana-727 227 19 thus thus ADV cana-727 227 20 𝐾𝑅 𝐾𝑅 PROPN cana-727 227 21 ≀𝑒 ≀𝑒 NOUN cana-727 227 22 𝑒𝑆𝑅 𝑒𝑆𝑅 ADP cana-727 227 23 . . PUNCT cana-727 228 1 by by ADP cana-727 228 2 proposition proposition NOUN cana-727 228 3 3.2 3.2 NUM cana-727 228 4 , , PUNCT cana-727 228 5 𝑆𝑅 𝑆𝑅 PROPN cana-727 228 6 is be AUX cana-727 228 7 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 228 8 . . PUNCT cana-727 229 1 a a DET cana-727 229 2 similar similar ADJ cana-727 229 3 demonstration demonstration NOUN cana-727 229 4 illustrates illustrate VERB cana-727 229 5 that that SCONJ cana-727 229 6 𝐾𝑆𝑆 𝐾𝑆𝑆 NUM cana-727 229 7 ≀𝑒 ≀𝑒 NOUN cana-727 229 8 π‘Œπ‘† π‘Œπ‘† PROPN cana-727 229 9 and and CCONJ cana-727 229 10 𝐾𝑆𝑆 𝐾𝑆𝑆 PROPN cana-727 229 11 ≀𝑒 ≀𝑒 NOUN cana-727 229 12 𝑒𝑆𝑆. 𝑒𝑆𝑆. PRON cana-727 229 13 therefore therefore ADV cana-727 229 14 𝑆𝑆 𝑆𝑆 PROPN cana-727 229 15 is be AUX cana-727 229 16 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 229 17 . . PUNCT cana-727 230 1 corollary corollary ADJ cana-727 230 2 4.15 4.15 NUM cana-727 230 3 : : PUNCT cana-727 230 4 let let VERB cana-727 230 5 𝑇 𝑇 PROPN cana-727 230 6 = = SYM cana-727 230 7 π‘‡π‘š(𝑅 π‘‡π‘š(𝑅 PROPN cana-727 230 8 ) ) PUNCT cana-727 230 9 and and CCONJ cana-727 230 10 𝑀 𝑀 PROPN cana-727 230 11 = = PUNCT cana-727 230 12 π‘€π‘š(𝑅 π‘€π‘š(𝑅 PROPN cana-727 230 13 ) ) PUNCT cana-727 230 14 . . PUNCT cana-727 231 1 if if SCONJ cana-727 231 2 𝑇𝑇 𝑇𝑇 PROPN cana-727 231 3 is be AUX cana-727 231 4 𝐺𝑝-extending 𝐺𝑝-extende VERB cana-727 231 5 , , PUNCT cana-727 231 6 then then ADV cana-727 231 7 𝑀𝑇 𝑀𝑇 PROPN cana-727 231 8 and and CCONJ cana-727 231 9 𝑀𝑀 𝑀𝑀 PROPN cana-727 231 10 are be AUX cana-727 231 11 𝐺𝑝extending 𝐺𝑝extende VERB cana-727 231 12 . . PUNCT cana-727 232 1 proof proof NOUN cana-727 232 2 : : PUNCT cana-727 232 3 this this DET cana-727 232 4 outcome outcome NOUN cana-727 232 5 is be AUX cana-727 232 6 a a DET cana-727 232 7 result result NOUN cana-727 232 8 of of ADP cana-727 232 9 theorem theorem ADJ cana-727 232 10 4.14 4.14 NUM cana-727 232 11 and and CCONJ cana-727 232 12 the the DET cana-727 232 13 reality𝑀𝑇 reality𝑀𝑇 PROPN cana-727 232 14 is be AUX cana-727 232 15 a a DET cana-727 232 16 rational rational ADJ cana-727 232 17 extension extension NOUN cana-727 232 18 of of ADP cana-727 232 19 𝑇𝑇. 𝑇𝑇. NUM cana-727 232 20 " " PUNCT cana-727 232 21 " " PUNCT cana-727 232 22 it it PRON cana-727 232 23 is be AUX cana-727 232 24 not not PART cana-727 232 25 known know VERB cana-727 232 26 so so ADV cana-727 232 27 far far ADV cana-727 232 28 whether whether SCONJ cana-727 232 29 direct direct ADJ cana-727 232 30 summands summand NOUN cana-727 232 31 of of ADP cana-727 232 32 goldie goldie PROPN cana-727 232 33 extending extend VERB cana-727 232 34 module module NOUN cana-727 232 35 enjoy enjoy NOUN cana-727 232 36 with with ADP cana-727 232 37 the the DET cana-727 232 38 property property NOUN cana-727 232 39 . . PUNCT cana-727 233 1 like like ADP cana-727 233 2 the the DET cana-727 233 3 former former ADJ cana-727 233 4 case case NOUN cana-727 233 5 the the DET cana-727 233 6 authors author NOUN cana-727 233 7 desire desire VERB cana-727 233 8 to to PART cana-727 233 9 obtain obtain VERB cana-727 233 10 whether whether SCONJ cana-727 233 11 the the DET cana-727 233 12 𝐺𝑝-extending 𝐺𝑝-extending NOUN cana-727 233 13 property property NOUN cana-727 233 14 is be AUX cana-727 233 15 inherited inherit VERB cana-727 233 16 by by ADP cana-727 233 17 its its PRON cana-727 233 18 direct direct ADJ cana-727 233 19 summands summand NOUN cana-727 233 20 or or CCONJ cana-727 233 21 not not PART cana-727 233 22 ? ? PUNCT cana-727 234 1 acknowledgments acknowledgment NOUN cana-727 234 2 the the DET cana-727 234 3 authors author NOUN cana-727 234 4 would would AUX cana-727 234 5 like like VERB cana-727 234 6 to to PART cana-727 234 7 express express VERB cana-727 234 8 their their PRON cana-727 234 9 thanks thank NOUN cana-727 234 10 to to ADP cana-727 234 11 the the DET cana-727 234 12 referee referee NOUN cana-727 234 13 for for ADP cana-727 234 14 her she PRON cana-727 234 15 / / SYM cana-727 234 16 his his PRON cana-727 234 17 careful careful ADJ cana-727 234 18 reading reading NOUN cana-727 234 19 and and CCONJ cana-727 234 20 useful useful ADJ cana-727 234 21 suggestions suggestion NOUN cana-727 234 22 on on ADP cana-727 234 23 this this DET cana-727 234 24 paper paper NOUN cana-727 234 25 . . PUNCT cana-727 234 26 " " PUNCT cana-727 235 1 references reference NOUN cana-727 235 2 [ [ X cana-727 235 3 1 1 NUM cana-727 235 4 ] ] PUNCT cana-727 235 5 e. e. PROPN cana-727 235 6 akalan akalan PROPN cana-727 235 7 , , PUNCT cana-727 235 8 g. g. PROPN cana-727 235 9 f. f. PROPN cana-727 235 10 birkenmeier birkenmeier PROPN cana-727 235 11 , , PUNCT cana-727 235 12 a. a. PROPN cana-727 235 13 tercan tercan PROPN cana-727 235 14 , , PUNCT cana-727 235 15 goldie goldie PROPN cana-727 235 16 extending extend VERB cana-727 235 17 modules module NOUN cana-727 235 18 , , PUNCT cana-727 235 19 comm comm NOUN cana-727 235 20 . . PUNCT cana-727 236 1 algebra algebra PROPN cana-727 236 2 37(2 37(2 PRON cana-727 236 3 ) ) PUNCT cana-727 236 4 ( ( PUNCT cana-727 236 5 2009),:663 2009),:663 NUM cana-727 236 6 - - PUNCT cana-727 236 7 683 683 NUM cana-727 236 8 . . PUNCT cana-727 237 1 [ [ X cana-727 237 2 2 2 NUM cana-727 237 3 ] ] X cana-727 237 4 n. n. NOUN cana-727 237 5 v. v. ADP cana-727 237 6 dung dung PROPN cana-727 237 7 , , PUNCT cana-727 237 8 d. d. PROPN cana-727 237 9 v. v. PROPN cana-727 237 10 huynh huynh PROPN cana-727 237 11 , , PUNCT cana-727 237 12 p. p. PROPN cana-727 237 13 f. f. PROPN cana-727 237 14 smith smith PROPN cana-727 237 15 , , PUNCT cana-727 237 16 wisbauer wisbauer NOUN cana-727 237 17 , , PUNCT cana-727 237 18 r. r. PROPN cana-727 237 19 , , PUNCT cana-727 237 20 extending extend VERB cana-727 237 21 modules module NOUN cana-727 237 22 . . PUNCT cana-727 238 1 harlow harlow PROPN cana-727 238 2 : : PUNCT cana-727 238 3 longman longman NOUN cana-727 238 4 , , PUNCT cana-727 238 5 ( ( PUNCT cana-727 238 6 1994 1994 NUM cana-727 238 7 ) ) PUNCT cana-727 238 8 . . PUNCT cana-727 239 1 [ [ X cana-727 239 2 3 3 X cana-727 239 3 ] ] PUNCT cana-727 239 4 m. m. NOUN cana-727 239 5 a. a. PROPN cana-727 239 6 kamal kamal PROPN cana-727 239 7 , , PUNCT cana-727 239 8 o. o. PROPN cana-727 239 9 a. a. NOUN cana-727 239 10 elmnophy elmnophy NOUN cana-727 239 11 , , PUNCT cana-727 239 12 on on ADP cana-727 239 13 p p NOUN cana-727 239 14 - - PUNCT cana-727 239 15 extending extend VERB cana-727 239 16 modules module NOUN cana-727 239 17 , , PUNCT cana-727 239 18 acta acta PROPN cana-727 239 19 math math PROPN cana-727 239 20 . . PUNCT cana-727 240 1 univ univ PROPN cana-727 240 2 . . PUNCT cana-727 240 3 comenianae comenianae PROPN cana-727 240 4 , , PUNCT cana-727 240 5 ( ( PUNCT cana-727 240 6 2005 2005 NUM cana-727 240 7 ) ) PUNCT cana-727 240 8 , , PUNCT cana-727 240 9 279 279 NUM cana-727 240 10 - - SYM cana-727 240 11 286 286 NUM cana-727 240 12 . . PUNCT cana-727 241 1 [ [ X cana-727 241 2 4 4 NUM cana-727 241 3 ] ] X cana-727 241 4 c. c. PROPN cana-727 241 5 canan canan PROPN cana-727 241 6 yucel yucel PROPN cana-727 241 7 , , PUNCT cana-727 241 8 a. a. NOUN cana-727 241 9 tercan tercan PROPN cana-727 241 10 , , PUNCT cana-727 241 11 modules module NOUN cana-727 241 12 whose whose DET cana-727 241 13 ec ec NOUN cana-727 241 14 - - PUNCT cana-727 241 15 closed close VERB cana-727 241 16 submodules submodule NOUN cana-727 241 17 are be AUX cana-727 241 18 direct direct ADJ cana-727 241 19 summand summand NOUN cana-727 241 20 , , PUNCT cana-727 241 21 taiwanese taiwanese PROPN cana-727 241 22 . . PUNCT cana-727 242 1 j j PROPN cana-727 242 2 , , PUNCT cana-727 242 3 math math NOUN cana-727 242 4 , , PUNCT cana-727 242 5 ( ( PUNCT cana-727 242 6 2009 2009 NUM cana-727 242 7 ) ) PUNCT cana-727 242 8 , , PUNCT cana-727 242 9 1247 1247 NUM cana-727 242 10 - - SYM cana-727 242 11 1256 1256 NUM cana-727 242 12 . . PUNCT cana-727 243 1 [ [ X cana-727 243 2 5 5 X cana-727 243 3 ] ] PUNCT cana-727 243 4 p. p. PROPN cana-727 243 5 f. f. PROPN cana-727 243 6 smith smith PROPN cana-727 243 7 , , PUNCT cana-727 243 8 modules module NOUN cana-727 243 9 for for ADP cana-727 243 10 which which PRON cana-727 243 11 every every DET cana-727 243 12 submodule submodule NOUN cana-727 243 13 has have VERB cana-727 243 14 a a DET cana-727 243 15 unique unique ADJ cana-727 243 16 closure closure NOUN cana-727 243 17 , , PUNCT cana-727 243 18 in in ADP cana-727 243 19 ring ring NOUN cana-727 243 20 theory theory NOUN cana-727 243 21 ( ( PUNCT cana-727 243 22 graville graville NOUN cana-727 243 23 , , PUNCT cana-727 243 24 oh oh INTJ cana-727 243 25 , , PUNCT cana-727 243 26 1992 1992 NUM cana-727 243 27 ) ) PUNCT cana-727 243 28 , , PUNCT cana-727 243 29 302 302 NUM cana-727 243 30 - - SYM cana-727 243 31 313 313 NUM cana-727 243 32 , , PUNCT cana-727 243 33 world world PROPN cana-727 243 34 sci sci PROPN cana-727 243 35 . . PROPN cana-727 243 36 publ publ PROPN cana-727 243 37 . . PUNCT cana-727 243 38 , , PUNCT cana-727 243 39 river river NOUN cana-727 243 40 edge edge NOUN cana-727 243 41 , , PUNCT cana-727 243 42 nj nj PROPN cana-727 243 43 , , PUNCT cana-727 243 44 1993 1993 NUM cana-727 243 45 . . PUNCT cana-727 244 1 [ [ X cana-727 244 2 6 6 NUM cana-727 244 3 ] ] PUNCT cana-727 244 4 a. a. NOUN cana-727 244 5 tercan tercan PROPN cana-727 244 6 , , PUNCT cana-727 244 7 r. r. PROPN cana-727 244 8 yasar yasar PROPN cana-727 244 9 and and CCONJ cana-727 244 10 c. c. PROPN cana-727 244 11 c. c. PROPN cana-727 244 12 yucel yucel PROPN cana-727 244 13 , , PUNCT cana-727 244 14 , , PUNCT cana-727 244 15 goldie goldie PROPN cana-727 244 16 extending extend VERB cana-727 244 17 modules module NOUN cana-727 244 18 on on ADP cana-727 244 19 class class NOUN cana-727 244 20 of of ADP cana-727 244 21 z z NOUN cana-727 244 22 - - PUNCT cana-727 244 23 closed close VERB cana-727 244 24 submodules submodule NOUN cana-727 244 25 . . PUNCT cana-727 245 1 bull bull NOUN cana-727 245 2 . . PUNCT cana-727 246 1 korean korean ADJ cana-727 246 2 math math PROPN cana-727 246 3 . . PUNCT cana-727 247 1 soc soc PROPN cana-727 247 2 . . PUNCT cana-727 248 1 ( ( PUNCT cana-727 248 2 2022 2022 NUM cana-727 248 3 ) ) PUNCT cana-727 248 4 , , PUNCT cana-727 248 5 453 453 NUM cana-727 248 6 - - SYM cana-727 248 7 468 468 NUM cana-727 248 8 . . PUNCT cana-727 249 1 [ [ X cana-727 249 2 7 7 X cana-727 249 3 ] ] PUNCT cana-727 249 4 a. a. NOUN cana-727 249 5 tercan tercan PROPN cana-727 249 6 , , PUNCT cana-727 249 7 r. r. PROPN cana-727 249 8 yasar yasar PROPN cana-727 249 9 and and CCONJ cana-727 249 10 c. c. PROPN cana-727 249 11 c. c. PROPN cana-727 249 12 yucel yucel PROPN cana-727 249 13 , , PUNCT cana-727 249 14 goldie goldie PROPN cana-727 249 15 extending extend VERB cana-727 249 16 modules module NOUN cana-727 249 17 on on ADP cana-727 249 18 class class NOUN cana-727 249 19 of of ADP cana-727 249 20 exact exact ADJ cana-727 249 21 submodules submodule NOUN cana-727 249 22 . . PUNCT cana-727 250 1 comm comm NOUN cana-727 250 2 . . PUNCT cana-727 251 1 algebra algebra PROPN cana-727 251 2 . . PUNCT cana-727 252 1 ( ( PUNCT cana-727 252 2 2022 2022 NUM cana-727 252 3 ) ) PUNCT cana-727 252 4 , , PUNCT cana-727 252 5 1363 1363 NUM cana-727 252 6 - - SYM cana-727 252 7 1371 1371 NUM cana-727 252 8 . . PUNCT cana-727 253 1 [ [ X cana-727 253 2 8 8 NUM cana-727 253 3 ] ] PUNCT cana-727 253 4 p. p. PROPN cana-727 253 5 f. f. PROPN cana-727 253 6 smith smith PROPN cana-727 253 7 and and CCONJ cana-727 253 8 a. a. PROPN cana-727 253 9 tercan tercan PROPN cana-727 253 10 , , PUNCT cana-727 253 11 continuous continuous ADJ cana-727 253 12 and and CCONJ cana-727 253 13 quasicontinuous quasicontinuous ADJ cana-727 253 14 modules module NOUN cana-727 253 15 , , PUNCT cana-727 253 16 houston houston PROPN cana-727 253 17 j. j. PROPN cana-727 253 18 math math PROPN cana-727 253 19 . . PUNCT cana-727 254 1 18 18 NUM cana-727 254 2 ( ( PUNCT cana-727 254 3 1992 1992 NUM cana-727 254 4 ) ) PUNCT cana-727 254 5 , , PUNCT cana-727 254 6 339 339 NUM cana-727 254 7 - - SYM cana-727 254 8 348 348 NUM cana-727 254 9 . . PUNCT cana-727 255 1 [ [ X cana-727 255 2 9 9 NUM cana-727 255 3 ] ] PUNCT cana-727 255 4 k. k. PROPN cana-727 255 5 r. r. PROPN cana-727 255 6 goodearl goodearl PROPN cana-727 255 7 , , PUNCT cana-727 255 8 von von PROPN cana-727 255 9 neumann neumann PROPN cana-727 255 10 regular regular PROPN cana-727 255 11 rings ring NOUN cana-727 255 12 , , PUNCT cana-727 255 13 pitman pitman NOUN cana-727 255 14 , , PUNCT cana-727 255 15 london london PROPN cana-727 255 16 , , PUNCT cana-727 255 17 1976 1976 NUM cana-727 255 18 . . PUNCT cana-727 256 1 [ [ X cana-727 256 2 10 10 NUM cana-727 256 3 ] ] X cana-727 256 4 g. g. PROPN cana-727 256 5 f. f. PROPN cana-727 256 6 birkenmeier birkenmeier PROPN cana-727 256 7 and and CCONJ cana-727 256 8 a. a. PROPN cana-727 256 9 tercan tercan PROPN cana-727 256 10 , , PUNCT cana-727 256 11 when when SCONJ cana-727 256 12 some some DET cana-727 256 13 complement complement NOUN cana-727 256 14 of of ADP cana-727 256 15 a a DET cana-727 256 16 submodule submodule NOUN cana-727 256 17 is be AUX cana-727 256 18 a a DET cana-727 256 19 direct direct ADJ cana-727 256 20 summand summand NOUN cana-727 256 21 , , PUNCT cana-727 256 22 comm comm NOUN cana-727 256 23 . . PUNCT cana-727 257 1 algebra algebra NOUN cana-727 257 2 , , PUNCT cana-727 257 3 35 35 NUM cana-727 257 4 ( ( PUNCT cana-727 257 5 2007 2007 NUM cana-727 257 6 ) ) PUNCT cana-727 257 7 , , PUNCT cana-727 257 8 597 597 NUM cana-727 257 9 - - SYM cana-727 257 10 611 611 NUM cana-727 257 11 . . PUNCT cana-727 258 1 [ [ X cana-727 258 2 11 11 NUM cana-727 258 3 ] ] PUNCT cana-727 258 4 a. a. NOUN cana-727 258 5 tercan tercan PROPN cana-727 258 6 and and CCONJ cana-727 258 7 c.c c.c PROPN cana-727 258 8 . . PROPN cana-727 258 9 yucel yucel PROPN cana-727 258 10 , , PUNCT cana-727 258 11 modules module NOUN cana-727 258 12 theory theory NOUN cana-727 258 13 , , PUNCT cana-727 258 14 extending extend VERB cana-727 258 15 modules module NOUN cana-727 258 16 and and CCONJ cana-727 258 17 generalizations generalization NOUN cana-727 258 18 , , PUNCT cana-727 258 19 frontiers frontier NOUN cana-727 258 20 in in ADP cana-727 258 21 mathematics mathematic NOUN cana-727 258 22 , , PUNCT cana-727 258 23 birkhauser birkhauser NOUN cana-727 258 24 / / SYM cana-727 258 25 springer springer NOUN cana-727 258 26 , , PUNCT cana-727 258 27 2016 2016 NUM cana-727 258 28 . . PUNCT cana-727 259 1 [ [ X cana-727 259 2 12 12 NUM cana-727 259 3 ] ] X cana-727 259 4 e. e. PROPN cana-727 259 5 m. m. PROPN cana-727 259 6 aljumaily aljumaily ADV cana-727 259 7 and and CCONJ cana-727 259 8 b. b. PROPN cana-727 259 9 h. h. PROPN cana-727 259 10 albahraany albahraany PROPN cana-727 259 11 , , PUNCT cana-727 259 12 on on ADP cana-727 259 13 goldie goldie PROPN cana-727 259 14 extending extend VERB cana-727 259 15 modules module NOUN cana-727 259 16 , , PUNCT cana-727 259 17 iraqi iraqi ADJ cana-727 259 18 journal journal NOUN cana-727 259 19 of of ADP cana-727 259 20 science science NOUN cana-727 259 21 , , PUNCT cana-727 259 22 2015 2015 NUM cana-727 259 23 , , PUNCT cana-727 259 24 vol vol NOUN cana-727 259 25 56 56 NUM cana-727 259 26 , , PUNCT cana-727 259 27 no.1b no.1b PROPN cana-727 259 28 , , PUNCT cana-727 259 29 pp pp PRON cana-727 259 30 : : PUNCT cana-727 259 31 492 492 NUM cana-727 259 32 - - SYM cana-727 259 33 498 498 NUM cana-727 259 34 . . PUNCT cana-727 260 1 [ [ X cana-727 260 2 13 13 NUM cana-727 260 3 ] ] PUNCT cana-727 260 4 a. a. NOUN cana-727 260 5 tercan tercan PROPN cana-727 260 6 , , PUNCT cana-727 260 7 on on ADP cana-727 260 8 certain certain ADJ cana-727 260 9 cs cs PROPN cana-727 260 10 - - NOUN cana-727 260 11 rings ring NOUN cana-727 260 12 , , PUNCT cana-727 260 13 comm comm NOUN cana-727 260 14 . . PUNCT cana-727 261 1 algebra algebra PROPN cana-727 261 2 23(1995 23(1995 NUM cana-727 261 3 ) ) PUNCT cana-727 261 4 , , PUNCT cana-727 261 5 no no INTJ cana-727 261 6 . . NOUN cana-727 261 7 2 2 NUM cana-727 261 8 , , PUNCT cana-727 261 9 405 405 NUM cana-727 261 10 - - SYM cana-727 261 11 419 419 NUM cana-727 261 12 . . PUNCT