id sid tid token lemma pos iajs-3089 1 1 ihjpas ihjpas PROPN iajs-3089 1 2 . . PUNCT iajs-3089 2 1 36 36 NUM iajs-3089 2 2 ( ( PUNCT iajs-3089 2 3 3 3 NUM iajs-3089 2 4 ) ) PUNCT iajs-3089 2 5 2023 2023 NUM iajs-3089 2 6 365 365 NUM iajs-3089 2 7 this this DET iajs-3089 2 8 work work NOUN iajs-3089 2 9 is be AUX iajs-3089 2 10 licensed license VERB iajs-3089 2 11 under under ADP iajs-3089 2 12 a a DET iajs-3089 2 13 creative creative ADJ iajs-3089 2 14 commons common NOUN iajs-3089 2 15 attribution attribution NOUN iajs-3089 2 16 4.0 4.0 NUM iajs-3089 2 17 international international ADJ iajs-3089 2 18 license license NOUN iajs-3089 2 19 * * PUNCT iajs-3089 2 20 corresponding correspond VERB iajs-3089 2 21 author author NOUN iajs-3089 2 22 : : PUNCT iajs-3089 2 23 mbhmsc2015110@gmail.com mbhmsc2015110@gmail.com X iajs-3089 2 24 abstract abstract ADV iajs-3089 2 25 let let VERB iajs-3089 2 26 𝑅 𝑅 PROPN iajs-3089 2 27 be be AUX iajs-3089 2 28 a a DET iajs-3089 2 29 commutative commutative ADJ iajs-3089 2 30 ring ring NOUN iajs-3089 2 31 with with ADP iajs-3089 2 32 1 1 NUM iajs-3089 2 33 and and CCONJ iajs-3089 2 34 𝑀 𝑀 PROPN iajs-3089 2 35 be be AUX iajs-3089 2 36 left leave VERB iajs-3089 2 37 unitary unitary ADJ iajs-3089 2 38 𝑅 𝑅 PROPN iajs-3089 2 39 βˆ’ βˆ’ PROPN iajs-3089 2 40 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 2 41 in in ADP iajs-3089 2 42 this this DET iajs-3089 2 43 papers paper NOUN iajs-3089 2 44 we we PRON iajs-3089 2 45 introduced introduce VERB iajs-3089 2 46 and and CCONJ iajs-3089 2 47 studied study VERB iajs-3089 2 48 concept concept NOUN iajs-3089 2 49 p p NOUN iajs-3089 2 50 - - PUNCT iajs-3089 2 51 small small ADJ iajs-3089 2 52 compressible compressible ADJ iajs-3089 2 53 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 2 54 ( ( PUNCT iajs-3089 2 55 an an DET iajs-3089 2 56 𝑅 𝑅 PROPN iajs-3089 2 57 βˆ’ βˆ’ PROPN iajs-3089 2 58 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 2 59 𝑀 𝑀 PROPN iajs-3089 2 60 is be AUX iajs-3089 2 61 said say VERB iajs-3089 2 62 to to PART iajs-3089 2 63 be be AUX iajs-3089 2 64 p p ADJ iajs-3089 2 65 - - PUNCT iajs-3089 2 66 small small ADJ iajs-3089 2 67 compressible compressible NOUN iajs-3089 2 68 if if SCONJ iajs-3089 2 69 𝑀 𝑀 PROPN iajs-3089 2 70 can can AUX iajs-3089 2 71 be be AUX iajs-3089 2 72 embedded embed VERB iajs-3089 2 73 in in ADP iajs-3089 2 74 every every PRON iajs-3089 2 75 of of ADP iajs-3089 2 76 it it PRON iajs-3089 2 77 is be AUX iajs-3089 2 78 nonzero nonzero ADJ iajs-3089 2 79 p p ADJ iajs-3089 2 80 - - PUNCT iajs-3089 2 81 small small ADJ iajs-3089 2 82 submodule submodule NOUN iajs-3089 2 83 of of ADP iajs-3089 2 84 𝑀. 𝑀. PROPN iajs-3089 2 85 equivalently equivalently ADV iajs-3089 2 86 , , PUNCT iajs-3089 2 87 𝑀 𝑀 PROPN iajs-3089 2 88 is be AUX iajs-3089 2 89 p p ADJ iajs-3089 2 90 - - PUNCT iajs-3089 2 91 small small ADJ iajs-3089 2 92 compressible compressible NOUN iajs-3089 2 93 if if SCONJ iajs-3089 2 94 there there PRON iajs-3089 2 95 exists exist VERB iajs-3089 2 96 a a DET iajs-3089 2 97 monomorphism monomorphism NOUN iajs-3089 2 98 : : PUNCT iajs-3089 2 99 𝑀 𝑀 PROPN iajs-3089 2 100 ⟢ ⟢ NOUN iajs-3089 2 101 𝑁 𝑁 PROPN iajs-3089 2 102 , , PUNCT iajs-3089 2 103 0 0 NUM iajs-3089 2 104 β‰  β‰  PROPN iajs-3089 2 105 𝑁 𝑁 PROPN iajs-3089 2 106 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 2 107 𝑀 𝑀 PROPN iajs-3089 2 108 , , PUNCT iajs-3089 2 109 𝑅 𝑅 PROPN iajs-3089 2 110 βˆ’ βˆ’ PROPN iajs-3089 2 111 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ VERB iajs-3089 2 112 𝑀 𝑀 PROPN iajs-3089 2 113 is be AUX iajs-3089 2 114 said say VERB iajs-3089 2 115 to to PART iajs-3089 2 116 be be AUX iajs-3089 2 117 p p ADJ iajs-3089 2 118 - - PUNCT iajs-3089 2 119 small small ADJ iajs-3089 2 120 retractable retractable NOUN iajs-3089 2 121 if if SCONJ iajs-3089 2 122 π»π‘œπ‘š(𝑀 π»π‘œπ‘š(𝑀 PROPN iajs-3089 2 123 , , PUNCT iajs-3089 2 124 𝐾 𝐾 PROPN iajs-3089 2 125 ) ) PUNCT iajs-3089 2 126 β‰  β‰  PROPN iajs-3089 2 127 0 0 NUM iajs-3089 2 128 , , PUNCT iajs-3089 2 129 for for ADP iajs-3089 2 130 every every DET iajs-3089 2 131 nonzero nonzero ADJ iajs-3089 2 132 p p ADJ iajs-3089 2 133 - - PUNCT iajs-3089 2 134 small small ADJ iajs-3089 2 135 submodule submodule NOUN iajs-3089 2 136 𝐾of 𝐾of PROPN iajs-3089 2 137 𝑀. 𝑀. PROPN iajs-3089 2 138 equivalently equivalently ADV iajs-3089 2 139 , , PUNCT iajs-3089 2 140 𝑀 𝑀 PROPN iajs-3089 2 141 is be AUX iajs-3089 2 142 p p ADJ iajs-3089 2 143 - - PUNCT iajs-3089 2 144 small small ADJ iajs-3089 2 145 retractable retractable NOUN iajs-3089 2 146 if if SCONJ iajs-3089 2 147 there there PRON iajs-3089 2 148 exists exist VERB iajs-3089 2 149 a a DET iajs-3089 2 150 homomorphism homomorphism NOUN iajs-3089 2 151 𝑓 𝑓 X iajs-3089 2 152 : : PUNCT iajs-3089 2 153 𝑀 𝑀 PROPN iajs-3089 2 154 ⟢ ⟢ NOUN iajs-3089 2 155 𝑁 𝑁 PROPN iajs-3089 2 156 whenever whenever SCONJ iajs-3089 2 157 0 0 NUM iajs-3089 2 158 β‰  β‰  PROPN iajs-3089 2 159 𝑁 𝑁 PROPN iajs-3089 2 160 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 2 161 𝑀 𝑀 PROPN iajs-3089 2 162 as as ADP iajs-3089 2 163 a a DET iajs-3089 2 164 generalization generalization NOUN iajs-3089 2 165 of of ADP iajs-3089 2 166 compressible compressible ADJ iajs-3089 2 167 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 2 168 and and CCONJ iajs-3089 2 169 retractable retractable ADJ iajs-3089 2 170 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 2 171 respectively respectively ADV iajs-3089 2 172 and and CCONJ iajs-3089 2 173 give give VERB iajs-3089 2 174 some some PRON iajs-3089 2 175 of of ADP iajs-3089 2 176 their their PRON iajs-3089 2 177 advantages advantage NOUN iajs-3089 2 178 characterizations characterization NOUN iajs-3089 2 179 and and CCONJ iajs-3089 2 180 examples example NOUN iajs-3089 2 181 . . PUNCT iajs-3089 3 1 keywords keyword NOUN iajs-3089 3 2 : : PUNCT iajs-3089 3 3 compressible compressible ADJ iajs-3089 3 4 module module NOUN iajs-3089 3 5 , , PUNCT iajs-3089 3 6 retractable retractable ADJ iajs-3089 3 7 module module NOUN iajs-3089 3 8 , , PUNCT iajs-3089 3 9 small small ADJ iajs-3089 3 10 submodule submodule NOUN iajs-3089 3 11 , , PUNCT iajs-3089 3 12 p p ADJ iajs-3089 3 13 - - PUNCT iajs-3089 3 14 small small ADJ iajs-3089 3 15 submodule submodule NOUN iajs-3089 3 16 , , PUNCT iajs-3089 3 17 psmall psmall NOUN iajs-3089 3 18 compressible compressible ADJ iajs-3089 3 19 module module NOUN iajs-3089 3 20 , , PUNCT iajs-3089 3 21 p p ADJ iajs-3089 3 22 - - PUNCT iajs-3089 3 23 small small ADJ iajs-3089 3 24 retractable retractable ADJ iajs-3089 3 25 module module NOUN iajs-3089 3 26 . . PUNCT iajs-3089 4 1 hollow hollow ADJ iajs-3089 4 2 module module NOUN iajs-3089 4 3 , , PUNCT iajs-3089 4 4 ps ps NOUN iajs-3089 4 5 - - PUNCT iajs-3089 4 6 hollow hollow NOUN iajs-3089 4 7 module module NOUN iajs-3089 4 8 . . PUNCT iajs-3089 5 1 1 1 X iajs-3089 5 2 . . X iajs-3089 5 3 introduction introduction NOUN iajs-3089 5 4 let let VERB iajs-3089 5 5 r r PRON iajs-3089 5 6 be be AUX iajs-3089 5 7 a a DET iajs-3089 5 8 commutative commutative ADJ iajs-3089 5 9 ring ring NOUN iajs-3089 5 10 with with ADP iajs-3089 5 11 1 1 NUM iajs-3089 5 12 and and CCONJ iajs-3089 5 13 m m AUX iajs-3089 5 14 be be AUX iajs-3089 5 15 left leave VERB iajs-3089 5 16 unitary unitary ADJ iajs-3089 5 17 𝑅 𝑅 PROPN iajs-3089 5 18 βˆ’ βˆ’ PROPN iajs-3089 5 19 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 5 20 authors author NOUN iajs-3089 5 21 introduced introduce VERB iajs-3089 5 22 and and CCONJ iajs-3089 5 23 studied study VERB iajs-3089 5 24 concept concept NOUN iajs-3089 5 25 small small ADJ iajs-3089 5 26 submodules submodule NOUN iajs-3089 5 27 . . PUNCT iajs-3089 6 1 a a DET iajs-3089 6 2 proper proper ADJ iajs-3089 6 3 submodule submodule NOUN iajs-3089 6 4 𝑁 𝑁 PROPN iajs-3089 6 5 of of ADP iajs-3089 6 6 an an DET iajs-3089 6 7 𝑅 𝑅 PROPN iajs-3089 6 8 βˆ’module βˆ’module PROPN iajs-3089 6 9 𝑀 𝑀 PROPN iajs-3089 6 10 is be AUX iajs-3089 6 11 termed term VERB iajs-3089 6 12 a a DET iajs-3089 6 13 small small ADJ iajs-3089 6 14 submodule submodule NOUN iajs-3089 6 15 ( ( PUNCT iajs-3089 6 16 𝑁 𝑁 PROPN iajs-3089 6 17 β‰ͺ β‰ͺ ADJ iajs-3089 6 18 𝑀 𝑀 PROPN iajs-3089 6 19 ) ) PUNCT iajs-3089 6 20 , , PUNCT iajs-3089 6 21 if if SCONJ iajs-3089 6 22 𝑁 𝑁 PROPN iajs-3089 6 23 + + CCONJ iajs-3089 6 24 𝐿 𝐿 PROPN iajs-3089 6 25 β‰  β‰  PROPN iajs-3089 6 26 𝑀 𝑀 PROPN iajs-3089 6 27 for for ADP iajs-3089 6 28 every every DET iajs-3089 6 29 submodule submodule NOUN iajs-3089 6 30 𝐿 𝐿 PROPN iajs-3089 6 31 of of ADP iajs-3089 6 32 𝑀[1 𝑀[1 PROPN iajs-3089 6 33 ] ] PUNCT iajs-3089 6 34 . . PUNCT iajs-3089 7 1 a a DET iajs-3089 7 2 proper proper ADJ iajs-3089 7 3 submodule submodule NOUN iajs-3089 7 4 𝑁 𝑁 PROPN iajs-3089 7 5 of of ADP iajs-3089 7 6 𝑀 𝑀 PROPN iajs-3089 7 7 is be AUX iajs-3089 7 8 said say VERB iajs-3089 7 9 to to PART iajs-3089 7 10 be be AUX iajs-3089 7 11 prime prime ADJ iajs-3089 7 12 if if SCONJ iajs-3089 7 13 whenever whenever SCONJ iajs-3089 7 14 π‘Ÿ π‘Ÿ X iajs-3089 7 15 ∈ ∈ PROPN iajs-3089 7 16 𝑅 𝑅 PROPN iajs-3089 7 17 , , PUNCT iajs-3089 7 18 π‘š π‘š PROPN iajs-3089 7 19 ∈ ∈ PROPN iajs-3089 7 20 𝑀 𝑀 PROPN iajs-3089 7 21 such such ADJ iajs-3089 7 22 that that SCONJ iajs-3089 7 23 π‘Ÿ. π‘Ÿ. NOUN iajs-3089 7 24 π‘š π‘š ADP iajs-3089 7 25 ∈ ∈ PRON iajs-3089 8 1 𝑁 𝑁 PROPN iajs-3089 8 2 π‘–π‘šπ‘π‘™π‘–π‘’π‘  π‘–π‘šπ‘π‘™π‘–π‘’π‘  NOUN iajs-3089 8 3 either either CCONJ iajs-3089 8 4 π‘š π‘š PROPN iajs-3089 8 5 ∈ ∈ PRON iajs-3089 8 6 𝑁 𝑁 PROPN iajs-3089 8 7 or or CCONJ iajs-3089 8 8 ∈ ∈ PROPN iajs-3089 8 9 [ [ X iajs-3089 8 10 𝑁 𝑁 NOUN iajs-3089 8 11 : : PUNCT iajs-3089 8 12 𝑀 𝑀 PROPN iajs-3089 8 13 ] ] PUNCT iajs-3089 8 14 ; ; PUNCT iajs-3089 8 15 [ [ X iajs-3089 8 16 𝑁 𝑁 PROPN iajs-3089 8 17 : : PUNCT iajs-3089 8 18 𝑀 𝑀 PROPN iajs-3089 8 19 ] ] X iajs-3089 8 20 = = PUNCT iajs-3089 8 21 { { PUNCT iajs-3089 8 22 π‘Ÿ π‘Ÿ X iajs-3089 8 23 ∈ ∈ PROPN iajs-3089 8 24 𝑅 𝑅 PROPN iajs-3089 8 25 : : PUNCT iajs-3089 8 26 π‘Ÿπ‘€ π‘Ÿπ‘€ ADJ iajs-3089 8 27 βŠ† βŠ† NUM iajs-3089 8 28 𝑁}[2 𝑁}[2 NOUN iajs-3089 8 29 ] ] PUNCT iajs-3089 8 30 . . PUNCT iajs-3089 9 1 in in ADP iajs-3089 9 2 [ [ X iajs-3089 9 3 3 3 X iajs-3089 9 4 ] ] X iajs-3089 9 5 iman iman NOUN iajs-3089 9 6 m.a.hadi m.a.hadi NOUN iajs-3089 9 7 and and CCONJ iajs-3089 9 8 tammader tammader VERB iajs-3089 9 9 a.ibrahiem a.ibrahiem PROPN iajs-3089 9 10 introduced introduce VERB iajs-3089 9 11 and and CCONJ iajs-3089 9 12 studied study VERB iajs-3089 9 13 the the DET iajs-3089 9 14 concept concept NOUN iajs-3089 9 15 of of ADP iajs-3089 9 16 p p NOUN iajs-3089 9 17 - - PUNCT iajs-3089 9 18 small small ADJ iajs-3089 9 19 submodules submodule NOUN iajs-3089 9 20 , , PUNCT iajs-3089 9 21 where where SCONJ iajs-3089 9 22 a a DET iajs-3089 9 23 submodule submodule NOUN iajs-3089 9 24 𝑁 𝑁 PROPN iajs-3089 9 25 of of ADP iajs-3089 9 26 an an DET iajs-3089 9 27 𝑅 𝑅 PROPN iajs-3089 9 28 βˆ’ βˆ’ PROPN iajs-3089 9 29 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 9 30 𝑀 𝑀 PROPN iajs-3089 9 31 is be AUX iajs-3089 9 32 called call VERB iajs-3089 9 33 p p ADJ iajs-3089 9 34 - - PUNCT iajs-3089 9 35 small small ADJ iajs-3089 9 36 submodule submodule NOUN iajs-3089 9 37 𝑁 𝑁 PROPN iajs-3089 9 38 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 9 39 𝑀 𝑀 NOUN iajs-3089 9 40 if if SCONJ iajs-3089 9 41 𝑁 𝑁 PROPN iajs-3089 9 42 + + CCONJ iajs-3089 9 43 𝑃 𝑃 PROPN iajs-3089 9 44 β‰  β‰  PROPN iajs-3089 9 45 𝑀 𝑀 PROPN iajs-3089 9 46 for for ADP iajs-3089 9 47 any any DET iajs-3089 9 48 prime prime ADJ iajs-3089 9 49 submodule submodule NOUN iajs-3089 9 50 𝑃 𝑃 PROPN iajs-3089 9 51 of of ADP iajs-3089 9 52 𝑀. 𝑀. PROPN iajs-3089 9 53 an an DET iajs-3089 9 54 𝑅 𝑅 PROPN iajs-3089 9 55 βˆ’ βˆ’ PROPN iajs-3089 9 56 π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘€ π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘€ PROPN iajs-3089 9 57 is be AUX iajs-3089 9 58 called call VERB iajs-3089 9 59 compressible compressible ADJ iajs-3089 9 60 if if SCONJ iajs-3089 9 61 𝑀 𝑀 PROPN iajs-3089 9 62 can can AUX iajs-3089 9 63 be be AUX iajs-3089 9 64 embedded embed VERB iajs-3089 9 65 in in ADP iajs-3089 9 66 every every DET iajs-3089 9 67 non non ADJ iajs-3089 9 68 - - ADJ iajs-3089 9 69 zero zero NUM iajs-3089 9 70 submodule submodule NOUN iajs-3089 9 71 . . PUNCT iajs-3089 10 1 an an DET iajs-3089 10 2 𝑅 𝑅 PROPN iajs-3089 10 3 βˆ’ βˆ’ PROPN iajs-3089 10 4 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 10 5 𝑀 𝑀 PROPN iajs-3089 10 6 is be AUX iajs-3089 10 7 said say VERB iajs-3089 10 8 to to PART iajs-3089 10 9 be be AUX iajs-3089 10 10 p p ADJ iajs-3089 10 11 - - PUNCT iajs-3089 10 12 small small ADJ iajs-3089 10 13 compressible compressible NOUN iajs-3089 10 14 if if SCONJ iajs-3089 10 15 𝑀 𝑀 PROPN iajs-3089 10 16 can can AUX iajs-3089 10 17 be be AUX iajs-3089 10 18 embedded embed VERB iajs-3089 10 19 in in ADP iajs-3089 10 20 every every PRON iajs-3089 10 21 of of ADP iajs-3089 10 22 it it PRON iajs-3089 10 23 is be AUX iajs-3089 10 24 nonzero nonzero ADJ iajs-3089 10 25 p p ADJ iajs-3089 10 26 - - PUNCT iajs-3089 10 27 small small ADJ iajs-3089 10 28 submodule submodule NOUN iajs-3089 10 29 of of ADP iajs-3089 10 30 𝑀. 𝑀. PROPN iajs-3089 10 31 equivalently equivalently ADV iajs-3089 10 32 , , PUNCT iajs-3089 10 33 𝑀 𝑀 PROPN iajs-3089 10 34 is be AUX iajs-3089 10 35 p p ADJ iajs-3089 10 36 - - PUNCT iajs-3089 10 37 small small ADJ iajs-3089 10 38 compressible compressible NOUN iajs-3089 10 39 if if SCONJ iajs-3089 10 40 there there PRON iajs-3089 10 41 exists exist VERB iajs-3089 10 42 a a DET iajs-3089 10 43 monomorphism monomorphism NOUN iajs-3089 10 44 𝑓 𝑓 X iajs-3089 10 45 : : PUNCT iajs-3089 10 46 𝑀 𝑀 PROPN iajs-3089 10 47 ⟢ ⟢ NOUN iajs-3089 10 48 𝑁 𝑁 PROPN iajs-3089 10 49 whenever whenever SCONJ iajs-3089 10 50 0 0 NUM iajs-3089 10 51 β‰  β‰  PROPN iajs-3089 10 52 𝑁 𝑁 PROPN iajs-3089 10 53 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 10 54 𝑀. 𝑀. PROPN iajs-3089 10 55 doi.org/10.30526/36.3.3089 doi.org/10.30526/36.3.3089 NOUN iajs-3089 10 56 article article NOUN iajs-3089 10 57 history history NOUN iajs-3089 10 58 : : PUNCT iajs-3089 10 59 received receive VERB iajs-3089 10 60 27 27 NUM iajs-3089 10 61 october october PROPN iajs-3089 10 62 2022 2022 NUM iajs-3089 10 63 , , PUNCT iajs-3089 10 64 accepted accept VERB iajs-3089 10 65 19 19 NUM iajs-3089 10 66 december december PROPN iajs-3089 10 67 2022 2022 NUM iajs-3089 10 68 , , PUNCT iajs-3089 10 69 published publish VERB iajs-3089 10 70 in in ADP iajs-3089 10 71 july july PROPN iajs-3089 10 72 2023 2023 NUM iajs-3089 10 73 . . PUNCT iajs-3089 11 1 ibn ibn PROPN iajs-3089 11 2 al al PROPN iajs-3089 11 3 - - PUNCT iajs-3089 11 4 haitham haitham PROPN iajs-3089 11 5 journal journal PROPN iajs-3089 11 6 for for ADP iajs-3089 11 7 pure pure ADJ iajs-3089 11 8 and and CCONJ iajs-3089 11 9 applied applied ADJ iajs-3089 11 10 sciences sciences PROPN iajs-3089 11 11 journal journal PROPN iajs-3089 11 12 homepage homepage NOUN iajs-3089 11 13 : : PUNCT iajs-3089 11 14 jih.uobaghdad.edu.iq jih.uobaghdad.edu.iq VERB iajs-3089 11 15 p p ADJ iajs-3089 11 16 - - PUNCT iajs-3089 11 17 small small ADJ iajs-3089 11 18 compressible compressible ADJ iajs-3089 11 19 modules module NOUN iajs-3089 11 20 and and CCONJ iajs-3089 11 21 p p ADJ iajs-3089 11 22 - - PUNCT iajs-3089 11 23 small small ADJ iajs-3089 11 24 retractable retractable ADJ iajs-3089 11 25 modules module NOUN iajs-3089 11 26 mohammed mohammed PROPN iajs-3089 11 27 baqer baqer PROPN iajs-3089 11 28 hashim hashim PROPN iajs-3089 11 29 al al PROPN iajs-3089 11 30 hakeem hakeem PROPN iajs-3089 11 31 * * PROPN iajs-3089 11 32 department department PROPN iajs-3089 11 33 of of ADP iajs-3089 11 34 mathematics mathematics PROPN iajs-3089 11 35 , , PUNCT iajs-3089 11 36 college college NOUN iajs-3089 11 37 of of ADP iajs-3089 11 38 science science PROPN iajs-3089 11 39 , , PUNCT iajs-3089 11 40 baghdad baghdad PROPN iajs-3089 11 41 university university PROPN iajs-3089 11 42 , , PUNCT iajs-3089 11 43 baghdad baghdad PROPN iajs-3089 11 44 , , PUNCT iajs-3089 11 45 iraq iraq PROPN iajs-3089 11 46 . . PUNCT iajs-3089 12 1 mbhmsc2015110@gmail.com mbhmsc2015110@gmail.com PROPN iajs-3089 12 2 nuhad nuhad PROPN iajs-3089 12 3 s. s. PROPN iajs-3089 12 4 al al PROPN iajs-3089 12 5 - - PUNCT iajs-3089 12 6 mothafar mothafar PROPN iajs-3089 12 7 department department NOUN iajs-3089 12 8 of of ADP iajs-3089 12 9 mathematics mathematic NOUN iajs-3089 12 10 , , PUNCT iajs-3089 12 11 college college NOUN iajs-3089 12 12 of of ADP iajs-3089 12 13 science science PROPN iajs-3089 12 14 , , PUNCT iajs-3089 12 15 baghdad baghdad PROPN iajs-3089 12 16 university university PROPN iajs-3089 12 17 , , PUNCT iajs-3089 12 18 baghdad baghdad PROPN iajs-3089 12 19 , , PUNCT iajs-3089 12 20 iraq iraq PROPN iajs-3089 12 21 . . PUNCT iajs-3089 13 1 nuhad.salim@sc.uobaghdad.edu.iq nuhad.salim@sc.uobaghdad.edu.iq PROPN iajs-3089 13 2 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3089 13 3 mailto:mbhmsc2015110@gmail.com mailto:mbhmsc2015110@gmail.com PROPN iajs-3089 13 4 mailto:mbhmsc2015110@gmail.com mailto:mbhmsc2015110@gmail.com PROPN iajs-3089 13 5 mailto:mbhmsc2015110@gmail.com mailto:mbhmsc2015110@gmail.com PROPN iajs-3089 13 6 mailto:nuhad.salim@sc.uobaghdad.edu.iq mailto:nuhad.salim@sc.uobaghdad.edu.iq PROPN iajs-3089 13 7 mailto:nuhad.salim@sc.uobaghdad.edu.iq mailto:nuhad.salim@sc.uobaghdad.edu.iq NOUN iajs-3089 13 8 ihjpas ihjpa NOUN iajs-3089 13 9 . . PUNCT iajs-3089 14 1 36 36 NUM iajs-3089 14 2 ( ( PUNCT iajs-3089 14 3 3 3 NUM iajs-3089 14 4 ) ) PUNCT iajs-3089 14 5 2023 2023 NUM iajs-3089 14 6 366 366 NUM iajs-3089 14 7 in in ADP iajs-3089 14 8 this this DET iajs-3089 14 9 paper paper NOUN iajs-3089 14 10 we we PRON iajs-3089 14 11 introduce introduce VERB iajs-3089 14 12 and and CCONJ iajs-3089 14 13 study study VERB iajs-3089 14 14 the the DET iajs-3089 14 15 concept concept NOUN iajs-3089 14 16 of of ADP iajs-3089 14 17 p p NOUN iajs-3089 14 18 - - PUNCT iajs-3089 14 19 small small ADJ iajs-3089 14 20 compressible compressible NOUN iajs-3089 14 21 as as ADP iajs-3089 14 22 a a DET iajs-3089 14 23 generalization generalization NOUN iajs-3089 14 24 of of ADP iajs-3089 14 25 compressible compressible ADJ iajs-3089 14 26 module module NOUN iajs-3089 14 27 , , PUNCT iajs-3089 14 28 and and CCONJ iajs-3089 14 29 we we PRON iajs-3089 14 30 give give VERB iajs-3089 14 31 some some DET iajs-3089 14 32 properties property NOUN iajs-3089 14 33 , , PUNCT iajs-3089 14 34 characterization characterization NOUN iajs-3089 14 35 and and CCONJ iajs-3089 14 36 examples example NOUN iajs-3089 14 37 . . PUNCT iajs-3089 15 1 in in ADP iajs-3089 15 2 addition addition NOUN iajs-3089 15 3 , , PUNCT iajs-3089 15 4 we we PRON iajs-3089 15 5 see see VERB iajs-3089 15 6 that that SCONJ iajs-3089 15 7 under under ADP iajs-3089 15 8 condition condition NOUN iajs-3089 15 9 . . PUNCT iajs-3089 16 1 p p X iajs-3089 16 2 - - PUNCT iajs-3089 16 3 small small ADJ iajs-3089 16 4 compressible compressible NOUN iajs-3089 16 5 , , PUNCT iajs-3089 16 6 small small ADJ iajs-3089 16 7 compressible compressible NOUN iajs-3089 16 8 and and CCONJ iajs-3089 16 9 compressible compressible ADJ iajs-3089 16 10 are be AUX iajs-3089 16 11 equivalent.some equivalent.some PROPN iajs-3089 16 12 of of ADP iajs-3089 16 13 their their PRON iajs-3089 16 14 advantages advantage NOUN iajs-3089 16 15 characterizations characterization NOUN iajs-3089 16 16 and and CCONJ iajs-3089 16 17 examples example NOUN iajs-3089 16 18 are be AUX iajs-3089 16 19 given give VERB iajs-3089 16 20 . . PUNCT iajs-3089 17 1 we we PRON iajs-3089 17 2 also also ADV iajs-3089 17 3 study study VERB iajs-3089 17 4 the the DET iajs-3089 17 5 relation relation NOUN iajs-3089 17 6 between between ADP iajs-3089 17 7 p p NOUN iajs-3089 17 8 - - PUNCT iajs-3089 17 9 small small ADJ iajs-3089 17 10 compressible compressible ADJ iajs-3089 17 11 module module NOUN iajs-3089 17 12 , , PUNCT iajs-3089 17 13 p p ADJ iajs-3089 17 14 - - PUNCT iajs-3089 17 15 small small ADJ iajs-3089 17 16 retractable retractable ADJ iajs-3089 17 17 module module NOUN iajs-3089 17 18 and and CCONJ iajs-3089 17 19 some some PRON iajs-3089 17 20 of of ADP iajs-3089 17 21 classes class NOUN iajs-3089 17 22 of of ADP iajs-3089 17 23 modules module NOUN iajs-3089 17 24 . . PUNCT iajs-3089 18 1 2 2 X iajs-3089 18 2 . . NUM iajs-3089 18 3 preliminaries preliminary NOUN iajs-3089 18 4 definition definition NOUN iajs-3089 18 5 ( ( PUNCT iajs-3089 18 6 2.1 2.1 NUM iajs-3089 18 7 ): ): PUNCT iajs-3089 18 8 let let VERB iajs-3089 18 9 𝑀 𝑀 PRON iajs-3089 18 10 be be AUX iajs-3089 18 11 an an DET iajs-3089 18 12 𝑅 𝑅 PROPN iajs-3089 18 13 βˆ’module βˆ’module NOUN iajs-3089 18 14 and and CCONJ iajs-3089 18 15 𝑁 𝑁 PROPN iajs-3089 18 16 ≀ ≀ ADJ iajs-3089 18 17 𝑀 𝑀 PROPN iajs-3089 18 18 : : PUNCT iajs-3089 18 19 1 1 X iajs-3089 18 20 . . X iajs-3089 19 1 𝑁 𝑁 PROPN iajs-3089 19 2 is be AUX iajs-3089 19 3 called call VERB iajs-3089 19 4 small small ADJ iajs-3089 19 5 submodule submodule NOUN iajs-3089 19 6 of of ADP iajs-3089 19 7 𝑀 𝑀 PROPN iajs-3089 19 8 , , PUNCT iajs-3089 19 9 ( ( PUNCT iajs-3089 19 10 𝑁 𝑁 PROPN iajs-3089 19 11 β‰ͺ β‰ͺ ADJ iajs-3089 19 12 𝑀 𝑀 PROPN iajs-3089 19 13 ) ) PUNCT iajs-3089 19 14 if if SCONJ iajs-3089 19 15 𝑁 𝑁 PROPN iajs-3089 19 16 + + PROPN iajs-3089 19 17 𝐾 𝐾 PROPN iajs-3089 19 18 = = SYM iajs-3089 19 19 𝑀 𝑀 PROPN iajs-3089 19 20 implies imply VERB iajs-3089 19 21 𝐾 𝐾 PROPN iajs-3089 19 22 = = PROPN iajs-3089 19 23 𝑀 𝑀 PROPN iajs-3089 19 24 , , PUNCT iajs-3089 19 25 for for ADP iajs-3089 19 26 any any DET iajs-3089 19 27 submodule submodule NOUN iajs-3089 19 28 𝐾 𝐾 PROPN iajs-3089 19 29 of of ADP iajs-3089 19 30 𝑀[1 𝑀[1 PROPN iajs-3089 19 31 ] ] PUNCT iajs-3089 19 32 . . PUNCT iajs-3089 20 1 2 2 X iajs-3089 20 2 . . X iajs-3089 20 3 an an DET iajs-3089 20 4 𝑅 𝑅 PROPN iajs-3089 20 5 βˆ’module𝑀 βˆ’module𝑀 PROPN iajs-3089 20 6 is be AUX iajs-3089 20 7 called call VERB iajs-3089 20 8 hollow hollow ADJ iajs-3089 20 9 if if SCONJ iajs-3089 20 10 every every DET iajs-3089 20 11 proper proper ADJ iajs-3089 20 12 submodule submodule NOUN iajs-3089 20 13 is be AUX iajs-3089 20 14 small small ADJ iajs-3089 20 15 in𝑀[4 in𝑀[4 NOUN iajs-3089 20 16 ] ] PUNCT iajs-3089 20 17 . . PUNCT iajs-3089 21 1 3 3 X iajs-3089 21 2 . . X iajs-3089 21 3 a a DET iajs-3089 21 4 proper proper ADJ iajs-3089 21 5 submodule submodule NOUN iajs-3089 21 6 𝑁 𝑁 PROPN iajs-3089 21 7 of of ADP iajs-3089 21 8 𝑀 𝑀 PROPN iajs-3089 21 9 is be AUX iajs-3089 21 10 called call VERB iajs-3089 21 11 prime prime ADJ iajs-3089 21 12 if if SCONJ iajs-3089 21 13 whenever whenever SCONJ iajs-3089 21 14 π‘Ÿ π‘Ÿ X iajs-3089 21 15 ∈ ∈ PROPN iajs-3089 21 16 𝑅 𝑅 PROPN iajs-3089 21 17 , , PUNCT iajs-3089 21 18 π‘š π‘š PROPN iajs-3089 21 19 ∈ ∈ PROPN iajs-3089 21 20 𝑀 𝑀 PROPN iajs-3089 21 21 implies imply VERB iajs-3089 21 22 either either CCONJ iajs-3089 21 23 π‘š π‘š PROPN iajs-3089 21 24 ∈ ∈ PRON iajs-3089 21 25 𝑁 𝑁 PROPN iajs-3089 21 26 or or CCONJ iajs-3089 21 27 π‘Ÿ π‘Ÿ PRON iajs-3089 21 28 ∈ ∈ NOUN iajs-3089 22 1 [ [ X iajs-3089 22 2 𝑁 𝑁 NOUN iajs-3089 22 3 : : PUNCT iajs-3089 22 4 𝑀 𝑀 PROPN iajs-3089 22 5 ] ] PUNCT iajs-3089 22 6 : : PUNCT iajs-3089 23 1 [ [ X iajs-3089 23 2 𝑁 𝑁 PROPN iajs-3089 23 3 : : PUNCT iajs-3089 23 4 𝑀 𝑀 PROPN iajs-3089 23 5 ] ] X iajs-3089 23 6 = = PUNCT iajs-3089 23 7 { { PUNCT iajs-3089 23 8 π‘Ÿ π‘Ÿ X iajs-3089 23 9 ∈ ∈ PROPN iajs-3089 23 10 𝑅 𝑅 PROPN iajs-3089 23 11 : : PUNCT iajs-3089 23 12 π‘Ÿπ‘€ π‘Ÿπ‘€ ADJ iajs-3089 23 13 βŠ† βŠ† NUM iajs-3089 23 14 𝑁 𝑁 PROPN iajs-3089 23 15 } } PUNCT iajs-3089 23 16 [ [ X iajs-3089 23 17 2 2 NUM iajs-3089 23 18 ] ] SYM iajs-3089 23 19 4 4 NUM iajs-3089 23 20 . . X iajs-3089 24 1 a a DET iajs-3089 24 2 proper proper ADJ iajs-3089 24 3 submodule submodule NOUN iajs-3089 24 4 𝑁 𝑁 PROPN iajs-3089 24 5 is be AUX iajs-3089 24 6 called call VERB iajs-3089 24 7 p p ADJ iajs-3089 24 8 - - PUNCT iajs-3089 24 9 small small ADJ iajs-3089 24 10 submodule submodule NOUN iajs-3089 24 11 of of ADP iajs-3089 24 12 𝑀 𝑀 PROPN iajs-3089 24 13 , , PUNCT iajs-3089 24 14 ( ( PUNCT iajs-3089 24 15 𝑁 𝑁 PROPN iajs-3089 24 16 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 24 17 𝑀 𝑀 PROPN iajs-3089 24 18 ) ) PUNCT iajs-3089 24 19 if if SCONJ iajs-3089 24 20 𝑁 𝑁 PROPN iajs-3089 24 21 + + CCONJ iajs-3089 24 22 𝑃 𝑃 PROPN iajs-3089 24 23 β‰  β‰  PROPN iajs-3089 24 24 𝑀 𝑀 PROPN iajs-3089 24 25 , , PUNCT iajs-3089 24 26 for for ADP iajs-3089 24 27 any any DET iajs-3089 24 28 prime prime ADJ iajs-3089 24 29 submodule submodule NOUN iajs-3089 24 30 𝑃 𝑃 PROPN iajs-3089 24 31 of of ADP iajs-3089 24 32 𝑀 𝑀 PROPN iajs-3089 24 33 , , PUNCT iajs-3089 24 34 [ [ X iajs-3089 24 35 3 3 NUM iajs-3089 24 36 ] ] PUNCT iajs-3089 24 37 . . PUNCT iajs-3089 25 1 5 5 X iajs-3089 25 2 . . X iajs-3089 25 3 an an DET iajs-3089 25 4 𝑅 𝑅 PROPN iajs-3089 25 5 βˆ’module𝑀 βˆ’module𝑀 PROPN iajs-3089 25 6 is be AUX iajs-3089 25 7 called call VERB iajs-3089 25 8 ps ps NOUN iajs-3089 25 9 - - PUNCT iajs-3089 25 10 hollow hollow NOUN iajs-3089 25 11 if if SCONJ iajs-3089 25 12 every every DET iajs-3089 25 13 proper proper ADJ iajs-3089 25 14 submodule submodule NOUN iajs-3089 25 15 in in ADP iajs-3089 25 16 𝑀 𝑀 PROPN iajs-3089 25 17 is be AUX iajs-3089 25 18 p p NOUN iajs-3089 25 19 - - PUNCT iajs-3089 25 20 small[3 small[3 NOUN iajs-3089 25 21 ] ] PUNCT iajs-3089 25 22 . . PUNCT iajs-3089 26 1 6 6 NUM iajs-3089 26 2 . . X iajs-3089 26 3 an an DET iajs-3089 26 4 𝑅 𝑅 PROPN iajs-3089 26 5 βˆ’ βˆ’ PROPN iajs-3089 26 6 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 26 7 𝑀 𝑀 PROPN iajs-3089 26 8 is be AUX iajs-3089 26 9 said say VERB iajs-3089 26 10 to to PART iajs-3089 26 11 be be AUX iajs-3089 26 12 small small ADJ iajs-3089 26 13 compressible compressible ADJ iajs-3089 26 14 if if SCONJ iajs-3089 26 15 𝑀 𝑀 PROPN iajs-3089 26 16 can can AUX iajs-3089 26 17 be be AUX iajs-3089 26 18 embedded embed VERB iajs-3089 26 19 in in ADP iajs-3089 26 20 every every DET iajs-3089 26 21 nonzero nonzero ADJ iajs-3089 26 22 small small ADJ iajs-3089 26 23 submodule submodule NOUN iajs-3089 26 24 of of ADP iajs-3089 26 25 𝑀. 𝑀. PROPN iajs-3089 26 26 equivalently equivalently ADV iajs-3089 26 27 , , PUNCT iajs-3089 26 28 𝑀 𝑀 PROPN iajs-3089 26 29 is be AUX iajs-3089 26 30 small small ADJ iajs-3089 26 31 compressible compressible ADJ iajs-3089 26 32 if if SCONJ iajs-3089 26 33 there there PRON iajs-3089 26 34 exists exist VERB iajs-3089 26 35 a a DET iajs-3089 26 36 monomorphism monomorphism NOUN iajs-3089 26 37 𝑓 𝑓 X iajs-3089 26 38 : : PUNCT iajs-3089 26 39 𝑀 𝑀 PROPN iajs-3089 26 40 ⟢ ⟢ NOUN iajs-3089 26 41 𝑁 𝑁 PROPN iajs-3089 26 42 whenever whenever SCONJ iajs-3089 26 43 0 0 NUM iajs-3089 26 44 β‰  β‰  PROPN iajs-3089 26 45 𝑁 𝑁 PROPN iajs-3089 26 46 β‰ͺ β‰ͺ PUNCT iajs-3089 26 47 𝑀[5 𝑀[5 PROPN iajs-3089 26 48 ] ] PUNCT iajs-3089 26 49 . . PUNCT iajs-3089 27 1 7 7 X iajs-3089 27 2 . . X iajs-3089 27 3 an an DET iajs-3089 27 4 𝑅 𝑅 PROPN iajs-3089 27 5 βˆ’ βˆ’ PROPN iajs-3089 27 6 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 27 7 𝑀 𝑀 PROPN iajs-3089 27 8 is be AUX iajs-3089 27 9 called call VERB iajs-3089 27 10 quasi quasi ADJ iajs-3089 27 11 - - NOUN iajs-3089 27 12 dedekind dedekind ADJ iajs-3089 27 13 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 27 14 if if SCONJ iajs-3089 27 15 for for ADP iajs-3089 27 16 all all DET iajs-3089 27 17 𝑓 𝑓 DET iajs-3089 27 18 ∈ ∈ PROPN iajs-3089 27 19 𝐸𝑛𝑑𝑅(𝑀 𝐸𝑛𝑑𝑅(𝑀 NOUN iajs-3089 27 20 ) ) PUNCT iajs-3089 27 21 , , PUNCT iajs-3089 27 22 𝑓 𝑓 DET iajs-3089 27 23 β‰  β‰  PROPN iajs-3089 27 24 0 0 NUM iajs-3089 27 25 implies imply VERB iajs-3089 27 26 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 27 27 = = PROPN iajs-3089 27 28 0 0 PROPN iajs-3089 27 29 , , PUNCT iajs-3089 27 30 [ [ X iajs-3089 27 31 7 7 NUM iajs-3089 27 32 ] ] PUNCT iajs-3089 27 33 . . PUNCT iajs-3089 28 1 8 8 X iajs-3089 28 2 . . PUNCT iajs-3089 28 3 an an DET iajs-3089 28 4 𝑅 𝑅 PROPN iajs-3089 28 5 βˆ’ βˆ’ PROPN iajs-3089 28 6 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 28 7 𝑀 𝑀 PROPN iajs-3089 28 8 is be AUX iajs-3089 28 9 called call VERB iajs-3089 28 10 small small ADJ iajs-3089 28 11 quasi quasi NOUN iajs-3089 28 12 - - NOUN iajs-3089 28 13 dedekind dedekind ADJ iajs-3089 28 14 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 28 15 if if SCONJ iajs-3089 28 16 for for ADP iajs-3089 28 17 all all DET iajs-3089 28 18 𝑓 𝑓 DET iajs-3089 28 19 ∈ ∈ PROPN iajs-3089 28 20 𝐸𝑛𝑑𝑅(𝑀 𝐸𝑛𝑑𝑅(𝑀 NOUN iajs-3089 28 21 ) ) PUNCT iajs-3089 28 22 , , PUNCT iajs-3089 28 23 𝑓 𝑓 DET iajs-3089 28 24 β‰  β‰  PROPN iajs-3089 28 25 0 0 NUM iajs-3089 28 26 implies imply VERB iajs-3089 28 27 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 28 28 β‰ͺ β‰ͺ PUNCT iajs-3089 28 29 𝑀 𝑀 PROPN iajs-3089 28 30 , , PUNCT iajs-3089 28 31 [ [ X iajs-3089 28 32 7 7 NUM iajs-3089 28 33 ] ] PUNCT iajs-3089 28 34 . . PUNCT iajs-3089 29 1 remark(2.2 remark(2.2 NOUN iajs-3089 29 2 ) ) PUNCT iajs-3089 29 3 : : PUNCT iajs-3089 30 1 [ [ X iajs-3089 30 2 3](1 3](1 NUM iajs-3089 30 3 ) ) PUNCT iajs-3089 30 4 ( ( PUNCT iajs-3089 30 5 2Μ… 2Μ… NOUN iajs-3089 30 6 ) ) PUNCT iajs-3089 30 7 is be AUX iajs-3089 30 8 p p ADJ iajs-3089 30 9 - - PUNCT iajs-3089 30 10 small small ADJ iajs-3089 30 11 subπ‘šπ‘œπ‘‘π‘’π‘™π‘’ subπ‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 30 12 of of ADP iajs-3089 30 13 𝑍6 𝑍6 PROPN iajs-3089 30 14 as as ADP iajs-3089 30 15 𝑍 𝑍 PROPN iajs-3089 30 16 βˆ’ βˆ’ NOUN iajs-3089 30 17 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 30 18 ( ( PUNCT iajs-3089 30 19 2 2 NUM iajs-3089 30 20 ) ) PUNCT iajs-3089 30 21 ( ( PUNCT iajs-3089 30 22 2Μ… 2Μ… NOUN iajs-3089 30 23 ) ) PUNCT iajs-3089 30 24 and and CCONJ iajs-3089 30 25 ( ( PUNCT iajs-3089 30 26 3Μ… 3Μ… NUM iajs-3089 30 27 ) ) PUNCT iajs-3089 30 28 are be AUX iajs-3089 30 29 not not PART iajs-3089 30 30 p p ADJ iajs-3089 30 31 - - PUNCT iajs-3089 30 32 small small ADJ iajs-3089 30 33 submodule submodule NOUN iajs-3089 30 34 of of ADP iajs-3089 30 35 𝑍6 𝑍6 PROPN iajs-3089 30 36 . . PUNCT iajs-3089 31 1 ( ( PUNCT iajs-3089 31 2 3 3 X iajs-3089 31 3 ) ) PUNCT iajs-3089 31 4 if if SCONJ iajs-3089 31 5 𝑀 𝑀 PROPN iajs-3089 31 6 is be AUX iajs-3089 31 7 semi semi ADJ iajs-3089 31 8 - - ADJ iajs-3089 31 9 simple simple ADJ iajs-3089 31 10 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 31 11 , , PUNCT iajs-3089 31 12 then then ADV iajs-3089 31 13 ( ( PUNCT iajs-3089 31 14 0 0 X iajs-3089 31 15 ) ) PUNCT iajs-3089 31 16 is be AUX iajs-3089 31 17 the the DET iajs-3089 31 18 only only ADJ iajs-3089 31 19 p p ADJ iajs-3089 31 20 - - PUNCT iajs-3089 31 21 small small ADJ iajs-3089 31 22 submodule submodule NOUN iajs-3089 31 23 . . PUNCT iajs-3089 32 1 remark remark NOUN iajs-3089 32 2 ( ( PUNCT iajs-3089 32 3 2.3 2.3 NUM iajs-3089 32 4 ): ): PUNCT iajs-3089 32 5 each each DET iajs-3089 32 6 small small ADJ iajs-3089 32 7 submodule submodule NOUN iajs-3089 32 8 is be AUX iajs-3089 32 9 p p NOUN iajs-3089 32 10 - - PUNCT iajs-3089 32 11 small small ADJ iajs-3089 32 12 . . PUNCT iajs-3089 33 1 but but CCONJ iajs-3089 33 2 the the DET iajs-3089 33 3 converse converse NOUN iajs-3089 33 4 is be AUX iajs-3089 33 5 not not PART iajs-3089 33 6 true true ADJ iajs-3089 33 7 in in ADP iajs-3089 33 8 general general ADJ iajs-3089 33 9 for for ADP iajs-3089 33 10 example(2Μ… example(2Μ… PROPN iajs-3089 33 11 ) ) PUNCT iajs-3089 33 12 is be AUX iajs-3089 33 13 p p ADJ iajs-3089 33 14 - - PUNCT iajs-3089 33 15 small small ADJ iajs-3089 33 16 subπ‘šπ‘œπ‘‘π‘’π‘™π‘’ subπ‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 33 17 of of ADP iajs-3089 33 18 𝑍6 𝑍6 PROPN iajs-3089 33 19 as as ADP iajs-3089 33 20 𝑍 𝑍 NOUN iajs-3089 33 21 βˆ’ βˆ’ NOUN iajs-3089 33 22 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 33 23 but but CCONJ iajs-3089 33 24 not not PART iajs-3089 33 25 small small ADJ iajs-3089 33 26 . . PUNCT iajs-3089 34 1 lemma lemma PROPN iajs-3089 34 2 ( ( PUNCT iajs-3089 34 3 2.4 2.4 NUM iajs-3089 34 4 ): ): PUNCT iajs-3089 34 5 1 1 NUM iajs-3089 34 6 . . X iajs-3089 34 7 let let VERB iajs-3089 34 8 𝑁 𝑁 PROPN iajs-3089 34 9 be be AUX iajs-3089 34 10 a a DET iajs-3089 34 11 proper proper ADJ iajs-3089 34 12 submodule submodule NOUN iajs-3089 34 13 of of ADP iajs-3089 34 14 𝑀. 𝑀. PROPN iajs-3089 34 15 if if SCONJ iajs-3089 34 16 wβŠ‚ wβŠ‚ PRON iajs-3089 34 17 𝑁 𝑁 PROPN iajs-3089 34 18 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 34 19 𝑀 𝑀 PROPN iajs-3089 34 20 , , PUNCT iajs-3089 34 21 then then ADV iajs-3089 34 22 wβ‰ͺ𝑃 wβ‰ͺ𝑃 NOUN iajs-3089 34 23 𝑀. 𝑀. PROPN iajs-3089 34 24 in in ADP iajs-3089 34 25 particular particular ADJ iajs-3089 34 26 if if SCONJ iajs-3089 34 27 w w NOUN iajs-3089 34 28 is be AUX iajs-3089 34 29 a a DET iajs-3089 34 30 direct direct ADJ iajs-3089 34 31 summand summand NOUN iajs-3089 34 32 of of ADP iajs-3089 34 33 𝑁 𝑁 PROPN iajs-3089 34 34 and and CCONJ iajs-3089 34 35 𝑁 𝑁 ADP iajs-3089 34 36 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 34 37 𝑀 𝑀 PROPN iajs-3089 34 38 , , PUNCT iajs-3089 34 39 then then ADV iajs-3089 34 40 wβ‰ͺ𝑃 wβ‰ͺ𝑃 NOUN iajs-3089 34 41 𝑀. 𝑀. PROPN iajs-3089 34 42 2 2 NUM iajs-3089 34 43 . . PUNCT iajs-3089 35 1 let let VERB iajs-3089 35 2 𝑁1 𝑁1 NOUN iajs-3089 35 3 and and CCONJ iajs-3089 35 4 𝑁2 𝑁2 NOUN iajs-3089 35 5 be be AUX iajs-3089 35 6 proper proper ADJ iajs-3089 35 7 submodules submodule NOUN iajs-3089 35 8 of of ADP iajs-3089 35 9 𝑀. 𝑀. PROPN iajs-3089 35 10 if if SCONJ iajs-3089 35 11 𝑁1 𝑁1 NOUN iajs-3089 35 12 + + ADJ iajs-3089 35 13 𝑁2 𝑁2 NOUN iajs-3089 35 14 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 35 15 𝑀 𝑀 PROPN iajs-3089 35 16 , , PUNCT iajs-3089 35 17 then then ADV iajs-3089 35 18 𝑁1 𝑁1 PROPN iajs-3089 35 19 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 35 20 𝑀 𝑀 PROPN iajs-3089 35 21 , , PUNCT iajs-3089 35 22 𝑁2 𝑁2 NOUN iajs-3089 35 23 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 35 24 𝑀 𝑀 PROPN iajs-3089 35 25 , , PUNCT iajs-3089 35 26 the the DET iajs-3089 35 27 converse converse NOUN iajs-3089 35 28 is be AUX iajs-3089 35 29 not not PART iajs-3089 35 30 true true ADJ iajs-3089 35 31 . . PUNCT iajs-3089 36 1 3 3 X iajs-3089 36 2 . . X iajs-3089 36 3 let let VERB iajs-3089 36 4 𝐴 𝐴 PROPN iajs-3089 36 5 βŠ‚ βŠ‚ PROPN iajs-3089 36 6 𝐡 𝐡 PROPN iajs-3089 36 7 βŠ‚ βŠ‚ X iajs-3089 37 1 𝑁 𝑁 PROPN iajs-3089 37 2 βŠ‚ βŠ‚ PROPN iajs-3089 37 3 𝑀. 𝑀. PROPN iajs-3089 37 4 if if SCONJ iajs-3089 37 5 bβ‰ͺ𝑃 bβ‰ͺ𝑃 PROPN iajs-3089 37 6 𝑁 𝑁 PROPN iajs-3089 37 7 , , PUNCT iajs-3089 37 8 then then ADV iajs-3089 37 9 𝐴 𝐴 PROPN iajs-3089 37 10 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 37 11 𝑀. 𝑀. PROPN iajs-3089 37 12 4 4 NUM iajs-3089 37 13 . . PUNCT iajs-3089 38 1 let let VERB iajs-3089 38 2 𝑀 𝑀 PROPN iajs-3089 38 3 , , PUNCT iajs-3089 38 4 𝑀′ 𝑀′ PROPN iajs-3089 38 5 be be AUX iajs-3089 38 6 𝑅 𝑅 PROPN iajs-3089 38 7 βˆ’ βˆ’ PROPN iajs-3089 38 8 π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘  π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘  NOUN iajs-3089 38 9 and and CCONJ iajs-3089 38 10 πœ“ πœ“ NOUN iajs-3089 38 11 : : PUNCT iajs-3089 38 12 𝑀 𝑀 PROPN iajs-3089 38 13 ⟢ ⟢ PROPN iajs-3089 38 14 𝑀′ 𝑀′ PROPN iajs-3089 38 15 be be AUX iajs-3089 38 16 an an DET iajs-3089 38 17 𝑅 𝑅 PROPN iajs-3089 38 18 βˆ’ βˆ’ PROPN iajs-3089 38 19 β„Žπ‘œπ‘šπ‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘š. β„Žπ‘œπ‘šπ‘œπ‘šπ‘œπ‘Ÿπ‘β„Žπ‘–π‘ π‘š. ADJ iajs-3089 38 20 if if SCONJ iajs-3089 38 21 𝐴 𝐴 PROPN iajs-3089 38 22 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 38 23 𝑀 𝑀 PROPN iajs-3089 38 24 , , PUNCT iajs-3089 38 25 then then ADV iajs-3089 38 26 πœ“(𝐴 πœ“(𝐴 PROPN iajs-3089 38 27 ) ) PUNCT iajs-3089 39 1 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 39 2 𝑀′. 𝑀′. NUM iajs-3089 39 3 3 3 NUM iajs-3089 39 4 .p .p NOUN iajs-3089 39 5 - - PUNCT iajs-3089 39 6 small small ADJ iajs-3089 39 7 compressible compressible ADJ iajs-3089 39 8 module module NOUN iajs-3089 39 9 in in ADP iajs-3089 39 10 this this DET iajs-3089 39 11 section section NOUN iajs-3089 39 12 , , PUNCT iajs-3089 39 13 we we PRON iajs-3089 39 14 introduce introduce VERB iajs-3089 39 15 the the DET iajs-3089 39 16 concept concept NOUN iajs-3089 39 17 of of ADP iajs-3089 39 18 p p NOUN iajs-3089 39 19 - - PUNCT iajs-3089 39 20 small small ADJ iajs-3089 39 21 compressible compressible ADJ iajs-3089 39 22 module module NOUN iajs-3089 39 23 as as ADP iajs-3089 39 24 a a DET iajs-3089 39 25 generalization generalization NOUN iajs-3089 39 26 of of ADP iajs-3089 39 27 compressible compressible ADJ iajs-3089 39 28 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 39 29 , , PUNCT iajs-3089 39 30 give give VERB iajs-3089 39 31 some some PRON iajs-3089 39 32 of of ADP iajs-3089 39 33 basic basic ADJ iajs-3089 39 34 properties property NOUN iajs-3089 39 35 , , PUNCT iajs-3089 39 36 examples example NOUN iajs-3089 39 37 and and CCONJ iajs-3089 39 38 characterizations characterization NOUN iajs-3089 39 39 of of ADP iajs-3089 39 40 this this DET iajs-3089 39 41 concept concept NOUN iajs-3089 39 42 . . PUNCT iajs-3089 40 1 ihjpas ihjpas PROPN iajs-3089 40 2 . . PUNCT iajs-3089 41 1 36 36 NUM iajs-3089 41 2 ( ( PUNCT iajs-3089 41 3 3 3 NUM iajs-3089 41 4 ) ) PUNCT iajs-3089 41 5 2023 2023 NUM iajs-3089 41 6 367 367 NUM iajs-3089 41 7 definition definition NOUN iajs-3089 41 8 ( ( PUNCT iajs-3089 41 9 3.1 3.1 NUM iajs-3089 41 10 ): ): PUNCT iajs-3089 41 11 an an DET iajs-3089 41 12 𝑅 𝑅 PROPN iajs-3089 41 13 βˆ’ βˆ’ PROPN iajs-3089 41 14 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 41 15 𝑀 𝑀 PROPN iajs-3089 41 16 is be AUX iajs-3089 41 17 said say VERB iajs-3089 41 18 to to PART iajs-3089 41 19 be be AUX iajs-3089 41 20 p p ADJ iajs-3089 41 21 - - PUNCT iajs-3089 41 22 small small ADJ iajs-3089 41 23 compressible compressible NOUN iajs-3089 41 24 if if SCONJ iajs-3089 41 25 𝑀 𝑀 PROPN iajs-3089 41 26 can can AUX iajs-3089 41 27 be be AUX iajs-3089 41 28 embedded embed VERB iajs-3089 41 29 in in ADP iajs-3089 41 30 every every PRON iajs-3089 41 31 of of ADP iajs-3089 41 32 it it PRON iajs-3089 41 33 is be AUX iajs-3089 41 34 nonzero nonzero ADJ iajs-3089 41 35 p p ADJ iajs-3089 41 36 - - PUNCT iajs-3089 41 37 small small ADJ iajs-3089 41 38 submodule submodule NOUN iajs-3089 41 39 of of ADP iajs-3089 41 40 𝑀. 𝑀. PROPN iajs-3089 41 41 equivalently equivalently ADV iajs-3089 41 42 , , PUNCT iajs-3089 41 43 𝑀 𝑀 PROPN iajs-3089 41 44 is be AUX iajs-3089 41 45 p p ADJ iajs-3089 41 46 - - PUNCT iajs-3089 41 47 small small ADJ iajs-3089 41 48 compressible compressible NOUN iajs-3089 41 49 if if SCONJ iajs-3089 41 50 there there PRON iajs-3089 41 51 exists exist VERB iajs-3089 41 52 a a DET iajs-3089 41 53 monomorphism monomorphism NOUN iajs-3089 41 54 𝑓 𝑓 X iajs-3089 41 55 : : PUNCT iajs-3089 41 56 𝑀 𝑀 PROPN iajs-3089 41 57 ⟢ ⟢ NOUN iajs-3089 41 58 𝑁 𝑁 PROPN iajs-3089 41 59 whenever whenever SCONJ iajs-3089 41 60 0 0 NUM iajs-3089 41 61 β‰  β‰  PROPN iajs-3089 41 62 𝑁 𝑁 ADP iajs-3089 41 63 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 41 64 𝑀. 𝑀. NOUN iajs-3089 41 65 remarks remark NOUN iajs-3089 41 66 and and CCONJ iajs-3089 41 67 examples example NOUN iajs-3089 41 68 ( ( PUNCT iajs-3089 41 69 3.2 3.2 NUM iajs-3089 41 70 ): ): PUNCT iajs-3089 41 71 1 1 NUM iajs-3089 41 72 . . X iajs-3089 42 1 it it PRON iajs-3089 42 2 ’s ’ VERB iajs-3089 42 3 obvious obvious ADJ iajs-3089 42 4 that that SCONJ iajs-3089 42 5 every every DET iajs-3089 42 6 compressible compressible ADJ iajs-3089 42 7 module module NOUN iajs-3089 42 8 is be AUX iajs-3089 42 9 p p ADJ iajs-3089 42 10 - - PUNCT iajs-3089 42 11 small small ADJ iajs-3089 42 12 compressible compressible ADJ iajs-3089 42 13 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 42 14 , , PUNCT iajs-3089 42 15 but but CCONJ iajs-3089 42 16 the the DET iajs-3089 42 17 converse converse NOUN iajs-3089 42 18 is be AUX iajs-3089 42 19 not not PART iajs-3089 42 20 true true ADJ iajs-3089 42 21 . . PUNCT iajs-3089 43 1 for for ADP iajs-3089 43 2 example example NOUN iajs-3089 43 3 𝑍6 𝑍6 PROPN iajs-3089 43 4 as as SCONJ iajs-3089 43 5 z z NOUN iajs-3089 43 6 - - PUNCT iajs-3089 43 7 module module NOUN iajs-3089 43 8 is be AUX iajs-3089 43 9 p p ADJ iajs-3089 43 10 - - PUNCT iajs-3089 43 11 small small ADJ iajs-3089 43 12 compressible compressible NOUN iajs-3089 43 13 since since SCONJ iajs-3089 43 14 ( ( PUNCT iajs-3089 43 15 0Μ… 0Μ… NOUN iajs-3089 43 16 ) ) PUNCT iajs-3089 43 17 is be AUX iajs-3089 43 18 the the DET iajs-3089 43 19 only only ADJ iajs-3089 43 20 p p ADJ iajs-3089 43 21 - - PUNCT iajs-3089 43 22 small small ADJ iajs-3089 43 23 submodule submodule NOUN iajs-3089 43 24 , , PUNCT iajs-3089 43 25 but but CCONJ iajs-3089 43 26 not not PART iajs-3089 43 27 compressible compressible ADJ iajs-3089 43 28 . . PUNCT iajs-3089 44 1 2 2 X iajs-3089 44 2 . . X iajs-3089 44 3 𝑍 𝑍 VERB iajs-3089 44 4 π‘Žπ‘  π‘Žπ‘  NOUN iajs-3089 44 5 𝑍 𝑍 PROPN iajs-3089 44 6 βˆ’ βˆ’ PROPN iajs-3089 44 7 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 44 8 is be AUX iajs-3089 44 9 p p ADJ iajs-3089 44 10 - - PUNCT iajs-3089 44 11 small small ADJ iajs-3089 44 12 compressible compressible ADJ iajs-3089 44 13 module module NOUN iajs-3089 44 14 , , PUNCT iajs-3089 44 15 because because SCONJ iajs-3089 44 16 it it PRON iajs-3089 44 17 's be AUX iajs-3089 44 18 compressible compressible ADJ iajs-3089 44 19 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 44 20 3 3 X iajs-3089 44 21 . . X iajs-3089 45 1 𝑍𝑃 𝑍𝑃 PROPN iajs-3089 45 2 π‘Žπ‘  π‘Žπ‘  ADP iajs-3089 45 3 𝑍 𝑍 PROPN iajs-3089 45 4 βˆ’ βˆ’ PROPN iajs-3089 45 5 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 45 6 is be AUX iajs-3089 45 7 p p ADJ iajs-3089 45 8 - - PUNCT iajs-3089 45 9 small small ADJ iajs-3089 45 10 compressible compressible ADJ iajs-3089 45 11 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 45 12 ; ; PUNCT iajs-3089 45 13 𝑃 𝑃 NOUN iajs-3089 45 14 𝑖𝑠 𝑖𝑠 NOUN iajs-3089 45 15 π‘Ž π‘Ž PRON iajs-3089 45 16 π‘π‘Ÿπ‘–π‘šπ‘’ π‘π‘Ÿπ‘–π‘šπ‘’ NOUN iajs-3089 45 17 π‘›π‘’π‘šπ‘π‘’π‘Ÿ. π‘›π‘’π‘šπ‘π‘’π‘Ÿ. NOUN iajs-3089 45 18 4 4 NUM iajs-3089 45 19 . . PUNCT iajs-3089 46 1 every every DET iajs-3089 46 2 simple simple ADJ iajs-3089 46 3 𝑅 𝑅 PROPN iajs-3089 46 4 βˆ’ βˆ’ PROPN iajs-3089 46 5 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 46 6 is be AUX iajs-3089 46 7 p p ADJ iajs-3089 46 8 - - PUNCT iajs-3089 46 9 small small ADJ iajs-3089 46 10 compressible compressible ADJ iajs-3089 46 11 module module NOUN iajs-3089 46 12 but but CCONJ iajs-3089 46 13 not not PART iajs-3089 46 14 conversely conversely ADV iajs-3089 46 15 , , PUNCT iajs-3089 46 16 because because SCONJ iajs-3089 46 17 𝑍 𝑍 NOUN iajs-3089 46 18 π‘Žπ‘  π‘Žπ‘  PROPN iajs-3089 46 19 𝑍 𝑍 PROPN iajs-3089 46 20 βˆ’ βˆ’ PROPN iajs-3089 46 21 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 46 22 is be AUX iajs-3089 46 23 a a DET iajs-3089 46 24 p p ADJ iajs-3089 46 25 - - PUNCT iajs-3089 46 26 small small ADJ iajs-3089 46 27 compressible compressible ADJ iajs-3089 46 28 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 46 29 but but CCONJ iajs-3089 46 30 not not PART iajs-3089 46 31 simple simple ADJ iajs-3089 46 32 . . PUNCT iajs-3089 47 1 5 5 X iajs-3089 47 2 . . X iajs-3089 47 3 𝑍4 𝑍4 NOUN iajs-3089 47 4 π‘Žπ‘  π‘Žπ‘  PROPN iajs-3089 47 5 𝑍 𝑍 PROPN iajs-3089 47 6 βˆ’ βˆ’ PROPN iajs-3089 47 7 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 47 8 is be AUX iajs-3089 47 9 not not PART iajs-3089 47 10 p p ADJ iajs-3089 47 11 - - PUNCT iajs-3089 47 12 small small ADJ iajs-3089 47 13 compressible.(because compressible.(because NOUN iajs-3089 47 14 𝑍4 𝑍4 NOUN iajs-3089 47 15 ca can AUX iajs-3089 47 16 n’t not PART iajs-3089 47 17 be be AUX iajs-3089 47 18 embedded embed VERB iajs-3089 47 19 in in ADP iajs-3089 47 20 〈 〈 NOUN iajs-3089 47 21 2Μ… 2Μ… NOUN iajs-3089 47 22 βŒͺ βŒͺ NOUN iajs-3089 47 23 and and CCONJ iajs-3089 47 24 〈 〈 NOUN iajs-3089 47 25 2Μ… 2Μ… NOUN iajs-3089 47 26 βŒͺ βŒͺ NOUN iajs-3089 47 27 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 47 28 𝑍4 𝑍4 NOUN iajs-3089 47 29 ) ) PUNCT iajs-3089 47 30 . . PUNCT iajs-3089 48 1 6 6 X iajs-3089 48 2 . . X iajs-3089 48 3 a a DET iajs-3089 48 4 homomorphic homomorphic ADJ iajs-3089 48 5 image image NOUN iajs-3089 48 6 of of ADP iajs-3089 48 7 a a DET iajs-3089 48 8 p p ADJ iajs-3089 48 9 - - PUNCT iajs-3089 48 10 small small ADJ iajs-3089 48 11 compressible compressible NOUN iajs-3089 48 12 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 48 13 need need AUX iajs-3089 48 14 not not PART iajs-3089 48 15 be be AUX iajs-3089 48 16 psmall psmall ADV iajs-3089 48 17 compressible compressible ADJ iajs-3089 48 18 in in ADP iajs-3089 48 19 general general ADJ iajs-3089 48 20 for for ADP iajs-3089 48 21 example example NOUN iajs-3089 48 22 𝑍 𝑍 VERB iajs-3089 48 23 π‘Žπ‘  π‘Žπ‘  NOUN iajs-3089 48 24 𝑍 𝑍 PROPN iajs-3089 48 25 βˆ’ βˆ’ PROPN iajs-3089 48 26 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 48 27 is be AUX iajs-3089 48 28 a a DET iajs-3089 48 29 p p ADJ iajs-3089 48 30 - - PUNCT iajs-3089 48 31 small small ADJ iajs-3089 48 32 compressible compressible ADJ iajs-3089 48 33 module module NOUN iajs-3089 48 34 and and CCONJ iajs-3089 48 35 𝑧 𝑧 DET iajs-3089 48 36 4𝑧 4𝑧 NOUN iajs-3089 49 1 ≃ ≃ NOUN iajs-3089 49 2 𝑧4 𝑧4 PROPN iajs-3089 49 3 is be AUX iajs-3089 49 4 not not PART iajs-3089 49 5 p p ADJ iajs-3089 49 6 - - PUNCT iajs-3089 49 7 small small ADJ iajs-3089 49 8 compressible compressible ADJ iajs-3089 49 9 module module NOUN iajs-3089 49 10 view view NOUN iajs-3089 49 11 remark remark NOUN iajs-3089 49 12 ( ( PUNCT iajs-3089 49 13 5 5 NUM iajs-3089 49 14 ) ) PUNCT iajs-3089 49 15 . . PUNCT iajs-3089 50 1 proposition(3.3 proposition(3.3 NOUN iajs-3089 50 2 ): ): PUNCT iajs-3089 50 3 a a DET iajs-3089 50 4 p p ADJ iajs-3089 50 5 - - PUNCT iajs-3089 50 6 small small ADJ iajs-3089 50 7 submodule submodule NOUN iajs-3089 50 8 of of ADP iajs-3089 50 9 p p NOUN iajs-3089 50 10 - - PUNCT iajs-3089 50 11 small small ADJ iajs-3089 50 12 compressible compressible ADJ iajs-3089 50 13 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 50 14 is be AUX iajs-3089 50 15 also also ADV iajs-3089 50 16 p p ADJ iajs-3089 50 17 - - PUNCT iajs-3089 50 18 small small ADJ iajs-3089 50 19 compressible compressible ADJ iajs-3089 50 20 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 50 21 proof proof NOUN iajs-3089 50 22 : : PUNCT iajs-3089 50 23 let let VERB iajs-3089 50 24 0 0 NUM iajs-3089 50 25 β‰  β‰  PROPN iajs-3089 50 26 𝐾 𝐾 PROPN iajs-3089 50 27 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 50 28 𝑀 𝑀 PROPN iajs-3089 50 29 and and CCONJ iajs-3089 50 30 𝑀 𝑀 PROPN iajs-3089 50 31 be be VERB iajs-3089 50 32 p p ADJ iajs-3089 50 33 - - ADJ iajs-3089 50 34 small small ADJ iajs-3089 50 35 compressible compressible ADJ iajs-3089 50 36 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 50 37 and and CCONJ iajs-3089 50 38 let let VERB iajs-3089 50 39 0 0 NUM iajs-3089 50 40 β‰  β‰  PROPN iajs-3089 50 41 𝐿 𝐿 PROPN iajs-3089 50 42 ≀ ≀ NOUN iajs-3089 50 43 𝐾 𝐾 PROPN iajs-3089 50 44 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 50 45 𝑀 𝑀 PROPN iajs-3089 50 46 , , PUNCT iajs-3089 50 47 then then ADV iajs-3089 50 48 𝐿 𝐿 PROPN iajs-3089 50 49 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 50 50 𝑀 𝑀 PROPN iajs-3089 51 1 [ [ X iajs-3089 51 2 3 3 NUM iajs-3089 51 3 ] ] PUNCT iajs-3089 51 4 . . PUNCT iajs-3089 52 1 since since SCONJ iajs-3089 52 2 𝑀 𝑀 PROPN iajs-3089 52 3 is be AUX iajs-3089 52 4 p p ADJ iajs-3089 52 5 - - PUNCT iajs-3089 52 6 small small ADJ iajs-3089 52 7 compressible compressible NOUN iajs-3089 52 8 , , PUNCT iajs-3089 52 9 so so ADV iajs-3089 52 10 βˆƒ βˆƒ PROPN iajs-3089 52 11 a a DET iajs-3089 52 12 monomorphism monomorphism NOUN iajs-3089 52 13 𝑓 𝑓 X iajs-3089 52 14 : : PUNCT iajs-3089 52 15 𝑀 𝑀 PROPN iajs-3089 52 16 ⟢ ⟢ PROPN iajs-3089 52 17 𝐿 𝐿 PROPN iajs-3089 52 18 and and CCONJ iajs-3089 52 19 𝑖 𝑖 SYM iajs-3089 52 20 : : PUNCT iajs-3089 52 21 𝐾 𝐾 PROPN iajs-3089 52 22 ⟢ ⟢ NOUN iajs-3089 52 23 𝑀 𝑀 PROPN iajs-3089 52 24 is be AUX iajs-3089 52 25 the the DET iajs-3089 52 26 inclusion inclusion NOUN iajs-3089 52 27 homomorphism homomorphism NOUN iajs-3089 52 28 , , PUNCT iajs-3089 52 29 then then ADV iajs-3089 52 30 𝑓 𝑓 DET iajs-3089 52 31 ∘ ∘ NUM iajs-3089 52 32 𝑖 𝑖 NOUN iajs-3089 52 33 : : PUNCT iajs-3089 52 34 𝐾 𝐾 PROPN iajs-3089 52 35 ⟢ ⟢ NOUN iajs-3089 52 36 𝐿 𝐿 PROPN iajs-3089 52 37 is be AUX iajs-3089 52 38 a a DET iajs-3089 52 39 monomorphism monomorphism NOUN iajs-3089 52 40 . . PUNCT iajs-3089 53 1 therefore therefore ADV iajs-3089 53 2 𝐾 𝐾 PROPN iajs-3089 53 3 is be AUX iajs-3089 53 4 a a DET iajs-3089 53 5 p p ADJ iajs-3089 53 6 - - PUNCT iajs-3089 53 7 small small ADJ iajs-3089 53 8 compressible compressible ADJ iajs-3089 53 9 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 53 10 proposition(3.5 proposition(3.5 NOUN iajs-3089 53 11 ): ): PUNCT iajs-3089 53 12 if if SCONJ iajs-3089 53 13 an an DET iajs-3089 53 14 𝑅 𝑅 PROPN iajs-3089 53 15 βˆ’ βˆ’ PROPN iajs-3089 53 16 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 53 17 𝑀 𝑀 PROPN iajs-3089 53 18 has have VERB iajs-3089 53 19 no no DET iajs-3089 53 20 prime prime ADJ iajs-3089 53 21 submodule submodule NOUN iajs-3089 54 1 such such ADJ iajs-3089 54 2 that that SCONJ iajs-3089 54 3 βˆƒ βˆƒ PROPN iajs-3089 54 4 a a DET iajs-3089 54 5 monomorphism monomorphism NOUN iajs-3089 54 6 𝑓 𝑓 X iajs-3089 54 7 : : PUNCT iajs-3089 54 8 𝑀 𝑀 PROPN iajs-3089 54 9 ⟢ ⟢ AUX iajs-3089 54 10 𝑁 𝑁 PROPN iajs-3089 54 11 , , PUNCT iajs-3089 54 12 βˆ€π‘ βˆ€π‘ PROPN iajs-3089 54 13 ⊊ ⊊ VERB iajs-3089 54 14 𝑀 𝑀 PROPN iajs-3089 54 15 , , PUNCT iajs-3089 54 16 then then ADV iajs-3089 54 17 𝑀 𝑀 PROPN iajs-3089 54 18 is be AUX iajs-3089 54 19 p p ADJ iajs-3089 54 20 - - PUNCT iajs-3089 54 21 small small ADJ iajs-3089 54 22 compressible compressible NOUN iajs-3089 54 23 . . PUNCT iajs-3089 55 1 proof proof NOUN iajs-3089 55 2 : : PUNCT iajs-3089 55 3 suppose suppose VERB iajs-3089 55 4 𝑀 𝑀 PROPN iajs-3089 55 5 has have VERB iajs-3089 55 6 no no DET iajs-3089 55 7 prime prime ADJ iajs-3089 55 8 submodule submodule NOUN iajs-3089 55 9 and and CCONJ iajs-3089 55 10 let let VERB iajs-3089 55 11 𝑁 𝑁 PROPN iajs-3089 55 12 ⊊ ⊊ VERB iajs-3089 55 13 𝑀 𝑀 PROPN iajs-3089 55 14 , , PUNCT iajs-3089 55 15 then then ADV iajs-3089 55 16 𝑁 𝑁 PROPN iajs-3089 55 17 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 55 18 𝑀 𝑀 PROPN iajs-3089 55 19 ⌈3βŒ‰and ⌈3βŒ‰and NOUN iajs-3089 55 20 by by ADP iajs-3089 55 21 assumption assumption NOUN iajs-3089 55 22 𝑀 𝑀 PROPN iajs-3089 55 23 is be AUX iajs-3089 55 24 p p ADJ iajs-3089 55 25 - - PUNCT iajs-3089 55 26 small small ADJ iajs-3089 55 27 compressible compressible NOUN iajs-3089 55 28 . . PUNCT iajs-3089 56 1 proposition(3.6 proposition(3.6 ADJ iajs-3089 56 2 ): ): PUNCT iajs-3089 56 3 let let VERB iajs-3089 56 4 𝑀1 𝑀1 NOUN iajs-3089 56 5 and and CCONJ iajs-3089 56 6 𝑀2 𝑀2 PROPN iajs-3089 56 7 be be AUX iajs-3089 56 8 isomorphic isomorphic ADJ iajs-3089 56 9 𝑅 𝑅 PROPN iajs-3089 56 10 βˆ’ βˆ’ PROPN iajs-3089 56 11 π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘ . π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘ . NOUN iajs-3089 56 12 then then ADV iajs-3089 56 13 𝑀1is 𝑀1i VERB iajs-3089 56 14 p p ADJ iajs-3089 56 15 - - PUNCT iajs-3089 56 16 small small ADJ iajs-3089 56 17 compressible compressible NOUN iajs-3089 56 18 if if SCONJ iajs-3089 56 19 and and CCONJ iajs-3089 56 20 only only ADV iajs-3089 56 21 if if SCONJ iajs-3089 56 22 𝑀2is 𝑀2is VERB iajs-3089 56 23 p p ADJ iajs-3089 56 24 - - PUNCT iajs-3089 56 25 small small ADJ iajs-3089 56 26 compressible compressible NOUN iajs-3089 56 27 . . PUNCT iajs-3089 57 1 proof proof NOUN iajs-3089 57 2 : : PUNCT iajs-3089 57 3 let let VERB iajs-3089 57 4 0 0 NUM iajs-3089 57 5 β‰  β‰  PROPN iajs-3089 57 6 𝑁 𝑁 PROPN iajs-3089 57 7 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 57 8 𝑀1 𝑀1 NOUN iajs-3089 57 9 and and CCONJ iajs-3089 57 10 suppose suppose VERB iajs-3089 57 11 that that SCONJ iajs-3089 57 12 𝑀2 𝑀2 PROPN iajs-3089 57 13 is be AUX iajs-3089 57 14 p p ADJ iajs-3089 57 15 - - PUNCT iajs-3089 57 16 small small ADJ iajs-3089 57 17 compressible compressible NOUN iajs-3089 57 18 . . PUNCT iajs-3089 58 1 let let VERB iajs-3089 58 2 πœ™ πœ™ NOUN iajs-3089 58 3 : : PUNCT iajs-3089 58 4 𝑀1 𝑀1 PROPN iajs-3089 58 5 ⟢ ⟢ NOUN iajs-3089 58 6 𝑀2 𝑀2 PROPN iajs-3089 58 7 be be AUX iajs-3089 58 8 an an DET iajs-3089 58 9 isomorphism isomorphism NOUN iajs-3089 58 10 . . PUNCT iajs-3089 59 1 , , PUNCT iajs-3089 59 2 then then ADV iajs-3089 59 3 by[3 by[3 VERB iajs-3089 59 4 ] ] X iajs-3089 59 5 0 0 NUM iajs-3089 59 6 β‰  β‰  PROPN iajs-3089 59 7 πœ™(𝑁 πœ™(𝑁 NOUN iajs-3089 59 8 ) ) PUNCT iajs-3089 59 9 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 59 10 𝑀2.put 𝑀2.put PROPN iajs-3089 59 11 𝐾 𝐾 NOUN iajs-3089 59 12 = = PUNCT iajs-3089 59 13 πœ™(𝑁 πœ™(𝑁 PROPN iajs-3089 59 14 ) ) PUNCT iajs-3089 59 15 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 59 16 𝑀2 𝑀2 PROPN iajs-3089 59 17 , , PUNCT iajs-3089 59 18 so so SCONJ iajs-3089 59 19 𝛼 𝛼 ADJ iajs-3089 59 20 : : PUNCT iajs-3089 59 21 𝑀2 𝑀2 PROPN iajs-3089 59 22 ⟢ ⟢ NOUN iajs-3089 59 23 𝐾 𝐾 PROPN iajs-3089 59 24 is be AUX iajs-3089 59 25 a a DET iajs-3089 59 26 monomorphism monomorphism NOUN iajs-3089 59 27 ( ( PUNCT iajs-3089 59 28 by by ADP iajs-3089 59 29 assumption assumption NOUN iajs-3089 59 30 ) ) PUNCT iajs-3089 59 31 , , PUNCT iajs-3089 59 32 let let VERB iajs-3089 59 33 𝑔 𝑔 PROPN iajs-3089 59 34 = = VERB iajs-3089 59 35 πœ™βˆ’1 πœ™βˆ’1 PROPN iajs-3089 59 36 β”‚ β”‚ PUNCT iajs-3089 59 37 𝐾 𝐾 PROPN iajs-3089 59 38 , , PUNCT iajs-3089 59 39 then then ADV iajs-3089 59 40 𝑔 𝑔 ADJ iajs-3089 59 41 : : PUNCT iajs-3089 59 42 𝐾 𝐾 PROPN iajs-3089 59 43 ⟢ ⟢ NOUN iajs-3089 59 44 𝑀1 𝑀1 ADV iajs-3089 59 45 is be AUX iajs-3089 59 46 a a DET iajs-3089 59 47 monomorphism monomorphism NOUN iajs-3089 59 48 . . PUNCT iajs-3089 60 1 𝑔(𝐾 𝑔(𝐾 NOUN iajs-3089 60 2 ) ) PUNCT iajs-3089 61 1 = = SYM iajs-3089 61 2 πœ™βˆ’1(πœ™(𝑁 πœ™βˆ’1(πœ™(𝑁 NOUN iajs-3089 61 3 ) ) PUNCT iajs-3089 61 4 ) ) PUNCT iajs-3089 62 1 = = SYM iajs-3089 62 2 𝑁. 𝑁. PROPN iajs-3089 62 3 hence hence ADV iajs-3089 62 4 , , PUNCT iajs-3089 62 5 we we PRON iajs-3089 62 6 have have VERB iajs-3089 62 7 a a DET iajs-3089 62 8 composition composition NOUN iajs-3089 62 9 πœ“ πœ“ NOUN iajs-3089 62 10 = = PUNCT iajs-3089 62 11 𝑔 𝑔 PROPN iajs-3089 62 12 ∘ ∘ NOUN iajs-3089 62 13 𝛼 𝛼 X iajs-3089 62 14 ∘ ∘ PROPN iajs-3089 62 15 πœ™ πœ™ NOUN iajs-3089 62 16 , , PUNCT iajs-3089 62 17 hence hence ADV iajs-3089 62 18 πœ“ πœ“ NOUN iajs-3089 62 19 : : PUNCT iajs-3089 62 20 𝑀1 𝑀1 NOUN iajs-3089 62 21 ⟢ ⟢ NOUN iajs-3089 62 22 𝑁 𝑁 PROPN iajs-3089 62 23 is be AUX iajs-3089 62 24 a a DET iajs-3089 62 25 monomorphism monomorphism NOUN iajs-3089 62 26 . . PUNCT iajs-3089 63 1 therefore therefore ADV iajs-3089 63 2 𝑀1is 𝑀1i VERB iajs-3089 63 3 p p ADJ iajs-3089 63 4 - - PUNCT iajs-3089 63 5 small small ADJ iajs-3089 63 6 compressible compressible ADJ iajs-3089 63 7 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 63 8 remark(3.7 remark(3.7 NOUN iajs-3089 63 9 ): ): PUNCT iajs-3089 63 10 the the DET iajs-3089 63 11 direct direct ADJ iajs-3089 63 12 sum sum NOUN iajs-3089 63 13 of of ADP iajs-3089 63 14 p p NOUN iajs-3089 63 15 - - PUNCT iajs-3089 63 16 small small ADJ iajs-3089 63 17 compressible compressible NOUN iajs-3089 63 18 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 63 19 need need AUX iajs-3089 63 20 not not PART iajs-3089 63 21 be be AUX iajs-3089 63 22 p p ADJ iajs-3089 63 23 - - PUNCT iajs-3089 63 24 small small ADJ iajs-3089 63 25 compressible compressible NOUN iajs-3089 63 26 . . PUNCT iajs-3089 64 1 consider consider VERB iajs-3089 64 2 the the DET iajs-3089 64 3 following follow VERB iajs-3089 64 4 example example NOUN iajs-3089 64 5 let let VERB iajs-3089 64 6 𝑍4 𝑍4 PROPN iajs-3089 64 7 ≃ ≃ VERB iajs-3089 64 8 𝑍2⨁𝑍2 𝑍2⨁𝑍2 PUNCT iajs-3089 64 9 as as ADP iajs-3089 64 10 z z NOUN iajs-3089 64 11 - - PUNCT iajs-3089 64 12 module module NOUN iajs-3089 64 13 . . PUNCT iajs-3089 65 1 𝑍2 𝑍2 PROPN iajs-3089 65 2 is be AUX iajs-3089 65 3 p p ADJ iajs-3089 65 4 - - PUNCT iajs-3089 65 5 small small ADJ iajs-3089 65 6 compressible compressible ADJ iajs-3089 65 7 module module NOUN iajs-3089 65 8 , , PUNCT iajs-3089 65 9 but but CCONJ iajs-3089 65 10 𝑍4 𝑍4 NOUN iajs-3089 65 11 is be AUX iajs-3089 65 12 not not PART iajs-3089 65 13 p p ADJ iajs-3089 65 14 - - PUNCT iajs-3089 65 15 small small ADJ iajs-3089 65 16 compressible compressible ADJ iajs-3089 65 17 module module NOUN iajs-3089 65 18 see see VERB iajs-3089 65 19 remarks remark NOUN iajs-3089 65 20 and and CCONJ iajs-3089 65 21 examples example NOUN iajs-3089 65 22 ( ( PUNCT iajs-3089 65 23 2.3 2.3 NUM iajs-3089 65 24 ) ) PUNCT iajs-3089 65 25 point(5 point(5 PROPN iajs-3089 65 26 ) ) PUNCT iajs-3089 65 27 proposition(3.8):let proposition(3.8):let NOUN iajs-3089 65 28 𝑀 𝑀 PROPN iajs-3089 65 29 = = SYM iajs-3089 65 30 𝑀1⨁𝑀2 𝑀1⨁𝑀2 X iajs-3089 65 31 be be VERB iajs-3089 65 32 an an DET iajs-3089 65 33 𝑅 𝑅 NOUN iajs-3089 65 34 βˆ’module βˆ’module NOUN iajs-3089 65 35 such such ADJ iajs-3089 65 36 that that DET iajs-3089 65 37 π‘Žπ‘›π‘›π‘…π‘€1β¨π‘Žπ‘›π‘›π‘…π‘€2 π‘Žπ‘›π‘›π‘…π‘€1β¨π‘Žπ‘›π‘›π‘…π‘€2 NOUN iajs-3089 65 38 = = PUNCT iajs-3089 65 39 𝑅. 𝑅. ADV iajs-3089 65 40 if if SCONJ iajs-3089 65 41 𝑀1 𝑀1 PROPN iajs-3089 65 42 and and CCONJ iajs-3089 65 43 𝑀2 𝑀2 PROPN iajs-3089 65 44 are be AUX iajs-3089 65 45 p p ADJ iajs-3089 65 46 - - PUNCT iajs-3089 65 47 small small ADJ iajs-3089 65 48 compressible compressible ADJ iajs-3089 65 49 π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘  π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘  NOUN iajs-3089 65 50 , , PUNCT iajs-3089 65 51 then then ADV iajs-3089 65 52 𝑀 𝑀 PROPN iajs-3089 65 53 is be AUX iajs-3089 65 54 p p ADJ iajs-3089 65 55 - - PUNCT iajs-3089 65 56 small small ADJ iajs-3089 65 57 compressible compressible NOUN iajs-3089 65 58 . . PUNCT iajs-3089 66 1 proof proof NOUN iajs-3089 66 2 : : PUNCT iajs-3089 66 3 let let VERB iajs-3089 66 4 0 0 NUM iajs-3089 66 5 β‰  β‰  PROPN iajs-3089 66 6 𝑁 𝑁 PROPN iajs-3089 66 7 = = SYM iajs-3089 66 8 𝐾1⨁𝐾2 𝐾1⨁𝐾2 NUM iajs-3089 66 9 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 66 10 𝑀. 𝑀. NOUN iajs-3089 66 11 then then ADV iajs-3089 66 12 by by ADP iajs-3089 66 13 theorem theorem NOUN iajs-3089 66 14 ( ( PUNCT iajs-3089 66 15 1.12)[3 1.12)[3 VERB iajs-3089 66 16 ] ] PUNCT iajs-3089 66 17 .0 .0 NUM iajs-3089 66 18 β‰  β‰  PROPN iajs-3089 66 19 𝐾1 𝐾1 PROPN iajs-3089 66 20 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 66 21 𝑀1 𝑀1 PROPN iajs-3089 66 22 ≀ ≀ X iajs-3089 66 23 𝑀 𝑀 PROPN iajs-3089 66 24 and and CCONJ iajs-3089 66 25 0 0 NUM iajs-3089 66 26 β‰  β‰  PROPN iajs-3089 66 27 𝐾2 𝐾2 NOUN iajs-3089 66 28 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 66 29 𝑀2 𝑀2 NOUN iajs-3089 66 30 ≀ ≀ PROPN iajs-3089 66 31 𝑀. 𝑀. PROPN iajs-3089 66 32 but but CCONJ iajs-3089 66 33 𝑀1and 𝑀1and PROPN iajs-3089 66 34 𝑀2 𝑀2 PROPN iajs-3089 66 35 p p ADJ iajs-3089 66 36 - - PUNCT iajs-3089 66 37 small small ADJ iajs-3089 66 38 compressible compressible ADJ iajs-3089 66 39 π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘  π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘  NOUN iajs-3089 66 40 , , PUNCT iajs-3089 66 41 so so SCONJ iajs-3089 66 42 βˆƒ βˆƒ PROPN iajs-3089 66 43 monomorphisms monomorphism VERB iajs-3089 66 44 𝑓 𝑓 NOUN iajs-3089 66 45 : : PUNCT iajs-3089 66 46 𝑀1 𝑀1 NOUN iajs-3089 66 47 ⟢ ⟢ NOUN iajs-3089 66 48 𝐾1and 𝐾1and PROPN iajs-3089 66 49 𝑔 𝑔 PROPN iajs-3089 66 50 : : PUNCT iajs-3089 66 51 𝑀2 𝑀2 PROPN iajs-3089 66 52 ⟢ ⟢ NOUN iajs-3089 66 53 𝐾2 𝐾2 NOUN iajs-3089 66 54 . . PUNCT iajs-3089 67 1 define define VERB iajs-3089 67 2 πœ“ πœ“ NOUN iajs-3089 67 3 : : PUNCT iajs-3089 67 4 𝑀 𝑀 PROPN iajs-3089 67 5 ⟢ ⟢ AUX iajs-3089 67 6 𝑁 𝑁 PROPN iajs-3089 67 7 by by ADP iajs-3089 67 8 πœ“(π‘Ž πœ“(π‘Ž NOUN iajs-3089 67 9 , , PUNCT iajs-3089 67 10 𝑏 𝑏 NOUN iajs-3089 67 11 ) ) PUNCT iajs-3089 67 12 = = SYM iajs-3089 67 13 ( ( PUNCT iajs-3089 67 14 𝑓(π‘Ž 𝑓(π‘Ž PROPN iajs-3089 67 15 ) ) PUNCT iajs-3089 67 16 , , PUNCT iajs-3089 67 17 𝑔(𝑏 𝑔(𝑏 PROPN iajs-3089 67 18 ) ) PUNCT iajs-3089 67 19 ) ) PUNCT iajs-3089 67 20 , , PUNCT iajs-3089 67 21 it it PRON iajs-3089 67 22 can can AUX iajs-3089 67 23 be be AUX iajs-3089 67 24 easily easily ADV iajs-3089 67 25 show show VERB iajs-3089 67 26 that that SCONJ iajs-3089 67 27 πœ“is πœ“is NOUN iajs-3089 67 28 a a DET iajs-3089 67 29 monomorphism monomorphism NOUN iajs-3089 67 30 . . PUNCT iajs-3089 68 1 therefore therefore ADV iajs-3089 68 2 𝑀 𝑀 PROPN iajs-3089 68 3 is be AUX iajs-3089 68 4 p p ADJ iajs-3089 68 5 - - PUNCT iajs-3089 68 6 small small ADJ iajs-3089 68 7 compressible compressible NOUN iajs-3089 68 8 . . PUNCT iajs-3089 69 1 ihjpas ihjpas PROPN iajs-3089 69 2 . . PUNCT iajs-3089 70 1 36 36 NUM iajs-3089 70 2 ( ( PUNCT iajs-3089 70 3 3 3 NUM iajs-3089 70 4 ) ) PUNCT iajs-3089 70 5 2023 2023 NUM iajs-3089 70 6 368 368 NUM iajs-3089 70 7 proposition(3.9):let proposition(3.9):let NOUN iajs-3089 70 8 𝑀 𝑀 PROPN iajs-3089 70 9 = = SYM iajs-3089 70 10 𝑀1⨁𝑀2 𝑀1⨁𝑀2 X iajs-3089 70 11 be be VERB iajs-3089 70 12 p p ADJ iajs-3089 70 13 - - PUNCT iajs-3089 70 14 small small ADJ iajs-3089 70 15 compressible compressible NOUN iajs-3089 70 16 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ VERB iajs-3089 70 17 such such ADJ iajs-3089 70 18 that that DET iajs-3089 70 19 π‘Žπ‘›π‘›π‘…π‘€1β¨π‘Žπ‘›π‘›π‘…π‘€2 π‘Žπ‘›π‘›π‘…π‘€1β¨π‘Žπ‘›π‘›π‘…π‘€2 NOUN iajs-3089 71 1 = = PUNCT iajs-3089 71 2 . . PUNCT iajs-3089 71 3 0 0 NUM iajs-3089 72 1 β‰  β‰  PROPN iajs-3089 72 2 𝐾1 𝐾1 PROPN iajs-3089 72 3 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 72 4 𝑀1 𝑀1 PROPN iajs-3089 72 5 ≀ ≀ X iajs-3089 72 6 𝑀 𝑀 PROPN iajs-3089 72 7 and and CCONJ iajs-3089 72 8 0 0 NUM iajs-3089 72 9 β‰  β‰  PROPN iajs-3089 72 10 𝐾2 𝐾2 NOUN iajs-3089 72 11 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 72 12 𝑀2 𝑀2 PROPN iajs-3089 72 13 ≀ ≀ PROPN iajs-3089 72 14 𝑀 𝑀 PROPN iajs-3089 72 15 with with ADP iajs-3089 72 16 𝑁 𝑁 PROPN iajs-3089 72 17 = = SYM iajs-3089 72 18 𝐾1⨁𝐾2 𝐾1⨁𝐾2 NUM iajs-3089 72 19 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 72 20 𝑀 𝑀 PROPN iajs-3089 72 21 , , PUNCT iajs-3089 72 22 then then ADV iajs-3089 72 23 𝑀1 𝑀1 PROPN iajs-3089 72 24 and and CCONJ iajs-3089 72 25 𝑀2 𝑀2 PROPN iajs-3089 72 26 are be AUX iajs-3089 72 27 p p ADJ iajs-3089 72 28 - - PUNCT iajs-3089 72 29 small small ADJ iajs-3089 72 30 compressible compressible ADJ iajs-3089 72 31 π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘  π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘  NOUN iajs-3089 72 32 . . PUNCT iajs-3089 73 1 proof proof NOUN iajs-3089 73 2 : : PUNCT iajs-3089 73 3 let let VERB iajs-3089 73 4 0 0 NUM iajs-3089 73 5 β‰  β‰  PROPN iajs-3089 73 6 𝐾1 𝐾1 NOUN iajs-3089 73 7 ≀ ≀ NOUN iajs-3089 73 8 𝑁 𝑁 NOUN iajs-3089 73 9 = = SYM iajs-3089 73 10 𝐾1⨁𝐾2 𝐾1⨁𝐾2 NUM iajs-3089 73 11 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 73 12 𝑀 𝑀 PROPN iajs-3089 73 13 , , PUNCT iajs-3089 73 14 then then ADV iajs-3089 73 15 by by ADP iajs-3089 73 16 remarks remark NOUN iajs-3089 73 17 and and CCONJ iajs-3089 73 18 examples(1.2)(7)[3 examples(1.2)(7)[3 NOUN iajs-3089 73 19 ] ] PUNCT iajs-3089 73 20 𝐾1 𝐾1 PROPN iajs-3089 73 21 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 73 22 𝑀 𝑀 PROPN iajs-3089 73 23 , , PUNCT iajs-3089 73 24 but but CCONJ iajs-3089 73 25 𝑀 𝑀 PROPN iajs-3089 73 26 be be VERB iajs-3089 73 27 p p ADJ iajs-3089 73 28 - - PUNCT iajs-3089 73 29 small small ADJ iajs-3089 73 30 compressible compressible ADJ iajs-3089 73 31 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 73 32 , , PUNCT iajs-3089 73 33 so so ADV iajs-3089 73 34 βˆƒ βˆƒ PROPN iajs-3089 73 35 a a DET iajs-3089 73 36 monomorphisms monomorphism NOUN iajs-3089 73 37 𝑓 𝑓 NOUN iajs-3089 73 38 : : PUNCT iajs-3089 73 39 𝑀 𝑀 PROPN iajs-3089 73 40 ⟢ ⟢ NOUN iajs-3089 73 41 𝐾1 𝐾1 NOUN iajs-3089 74 1 and and CCONJ iajs-3089 74 2 𝐽 𝐽 NOUN iajs-3089 74 3 : : PUNCT iajs-3089 74 4 𝑀1 𝑀1 PROPN iajs-3089 74 5 ⟢ ⟢ NOUN iajs-3089 74 6 𝑀1⨁𝑀2 𝑀1⨁𝑀2 X iajs-3089 74 7 = = SYM iajs-3089 74 8 𝑀 𝑀 PROPN iajs-3089 74 9 , , PUNCT iajs-3089 74 10 hence hence ADV iajs-3089 74 11 we we PRON iajs-3089 74 12 have have VERB iajs-3089 74 13 a a DET iajs-3089 74 14 composition composition NOUN iajs-3089 74 15 . . PUNCT iajs-3089 75 1 letπœ“ letπœ“ PROPN iajs-3089 75 2 = = PUNCT iajs-3089 75 3 𝑓 𝑓 DET iajs-3089 75 4 ∘ ∘ PROPN iajs-3089 75 5 𝐽 𝐽 PROPN iajs-3089 75 6 , , PUNCT iajs-3089 75 7 thus thus ADV iajs-3089 75 8 πœ“ πœ“ NOUN iajs-3089 75 9 : : PUNCT iajs-3089 75 10 𝑀1 𝑀1 PROPN iajs-3089 75 11 ⟢ ⟢ NOUN iajs-3089 75 12 𝐾1 𝐾1 PROPN iajs-3089 75 13 is be AUX iajs-3089 75 14 a a DET iajs-3089 75 15 monomorphism monomorphism NOUN iajs-3089 75 16 . . PUNCT iajs-3089 76 1 therefore therefore ADV iajs-3089 76 2 𝑀1is 𝑀1i VERB iajs-3089 76 3 p p ADJ iajs-3089 76 4 - - PUNCT iajs-3089 76 5 small small ADJ iajs-3089 76 6 compressible compressible ADJ iajs-3089 76 7 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 76 8 the the DET iajs-3089 76 9 same same ADJ iajs-3089 76 10 way way NOUN iajs-3089 76 11 we we PRON iajs-3089 76 12 can can AUX iajs-3089 76 13 prove prove VERB iajs-3089 76 14 𝑀2 𝑀2 PROPN iajs-3089 76 15 is be AUX iajs-3089 76 16 p p ADJ iajs-3089 76 17 - - PUNCT iajs-3089 76 18 small small ADJ iajs-3089 76 19 compressible compressible ADJ iajs-3089 76 20 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 76 21 . . PUNCT iajs-3089 77 1 remarks remark NOUN iajs-3089 77 2 and and CCONJ iajs-3089 77 3 examples example NOUN iajs-3089 77 4 ( ( PUNCT iajs-3089 77 5 3.10 3.10 NUM iajs-3089 77 6 ): ): PUNCT iajs-3089 77 7 1 1 NUM iajs-3089 77 8 . . PUNCT iajs-3089 78 1 every every DET iajs-3089 78 2 p p ADJ iajs-3089 78 3 - - PUNCT iajs-3089 78 4 small small ADJ iajs-3089 78 5 compressible compressible ADJ iajs-3089 78 6 module module NOUN iajs-3089 78 7 is be AUX iajs-3089 78 8 small small ADJ iajs-3089 78 9 compressible compressible ADJ iajs-3089 78 10 module module NOUN iajs-3089 78 11 . . PUNCT iajs-3089 79 1 proof proof NOUN iajs-3089 79 2 : : PUNCT iajs-3089 79 3 let let VERB iajs-3089 79 4 0 0 NUM iajs-3089 79 5 β‰  β‰  PROPN iajs-3089 79 6 𝑁 𝑁 PROPN iajs-3089 79 7 β‰ͺ β‰ͺ ADJ iajs-3089 79 8 𝑀 𝑀 PROPN iajs-3089 79 9 , , PUNCT iajs-3089 79 10 then then ADV iajs-3089 79 11 by by ADP iajs-3089 79 12 [ [ X iajs-3089 79 13 3 3 X iajs-3089 79 14 ] ] X iajs-3089 79 15 𝑁 𝑁 PROPN iajs-3089 79 16 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 79 17 𝑀 𝑀 PROPN iajs-3089 79 18 and and CCONJ iajs-3089 79 19 𝑀 𝑀 PROPN iajs-3089 79 20 is be AUX iajs-3089 79 21 p p ADJ iajs-3089 79 22 - - PUNCT iajs-3089 79 23 small small ADJ iajs-3089 79 24 compressible compressible ADJ iajs-3089 79 25 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 79 26 , , PUNCT iajs-3089 79 27 therefor therefor ADP iajs-3089 79 28 𝑀 𝑀 PROPN iajs-3089 79 29 is be AUX iajs-3089 79 30 small small ADJ iajs-3089 79 31 compressible compressible ADJ iajs-3089 79 32 module module NOUN iajs-3089 79 33 . . PUNCT iajs-3089 80 1 2 2 X iajs-3089 80 2 . . X iajs-3089 80 3 𝑍6as 𝑍6as AUX iajs-3089 80 4 𝑍 𝑍 NOUN iajs-3089 80 5 βˆ’ βˆ’ PROPN iajs-3089 80 6 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 80 7 is be AUX iajs-3089 80 8 small small ADJ iajs-3089 80 9 compressible compressible NOUN iajs-3089 80 10 , , PUNCT iajs-3089 80 11 since since SCONJ iajs-3089 80 12 ( ( PUNCT iajs-3089 80 13 0Μ… 0Μ… NOUN iajs-3089 80 14 ) ) PUNCT iajs-3089 80 15 the the DET iajs-3089 80 16 only only ADJ iajs-3089 80 17 p p ADJ iajs-3089 80 18 - - PUNCT iajs-3089 80 19 small small ADJ iajs-3089 80 20 submodule submodule NOUN iajs-3089 80 21 of of ADP iajs-3089 80 22 𝑍6 𝑍6 PROPN iajs-3089 80 23 . . PUNCT iajs-3089 81 1 3 3 X iajs-3089 81 2 . . X iajs-3089 81 3 𝑄 𝑄 PRON iajs-3089 81 4 π‘Žπ‘  π‘Žπ‘  AUX iajs-3089 81 5 𝑍 𝑍 PROPN iajs-3089 81 6 βˆ’ βˆ’ PROPN iajs-3089 81 7 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 81 8 is be AUX iajs-3089 81 9 not not PART iajs-3089 81 10 p p ADJ iajs-3089 81 11 - - PUNCT iajs-3089 81 12 small small ADJ iajs-3089 81 13 compressible compressible ADJ iajs-3089 81 14 module module NOUN iajs-3089 81 15 , , PUNCT iajs-3089 81 16 since since SCONJ iajs-3089 81 17 π»π‘œπ‘šπ‘…(𝑄 π»π‘œπ‘šπ‘…(𝑄 PROPN iajs-3089 81 18 , , PUNCT iajs-3089 81 19 𝑍 𝑍 NOUN iajs-3089 81 20 ) ) PUNCT iajs-3089 81 21 = = SYM iajs-3089 81 22 0 0 NUM iajs-3089 81 23 , , PUNCT iajs-3089 81 24 where where SCONJ iajs-3089 81 25 𝑍 𝑍 VERB iajs-3089 81 26 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 81 27 𝑄. 𝑄. PROPN iajs-3089 81 28 proposition(3.11 proposition(3.11 NOUN iajs-3089 81 29 ): ): PUNCT iajs-3089 81 30 let let VERB iajs-3089 81 31 𝑀 𝑀 PRON iajs-3089 81 32 be be AUX iajs-3089 81 33 an an DET iajs-3089 81 34 𝑅 𝑅 PROPN iajs-3089 81 35 βˆ’ βˆ’ PROPN iajs-3089 81 36 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 81 37 and and CCONJ iajs-3089 81 38 0 0 NUM iajs-3089 81 39 β‰  β‰  PROPN iajs-3089 81 40 π‘š π‘š PROPN iajs-3089 81 41 ∈ ∈ PROPN iajs-3089 81 42 𝑀 𝑀 PROPN iajs-3089 81 43 such such ADJ iajs-3089 81 44 that that SCONJ iajs-3089 81 45 π‘…π‘š π‘…π‘š PROPN iajs-3089 81 46 ⊊ ⊊ VERB iajs-3089 81 47 𝑀 𝑀 PROPN iajs-3089 81 48 , , PUNCT iajs-3089 81 49 then then ADV iajs-3089 81 50 𝑀 𝑀 PROPN iajs-3089 81 51 is be AUX iajs-3089 81 52 small small ADJ iajs-3089 81 53 compressible compressible ADJ iajs-3089 81 54 if if SCONJ iajs-3089 81 55 and and CCONJ iajs-3089 81 56 only only ADV iajs-3089 81 57 if if SCONJ iajs-3089 81 58 𝑀is 𝑀is PROPN iajs-3089 81 59 p p ADJ iajs-3089 81 60 - - PUNCT iajs-3089 81 61 small small ADJ iajs-3089 81 62 compressible compressible NOUN iajs-3089 81 63 . . PUNCT iajs-3089 82 1 proof proof NOUN iajs-3089 82 2 : : PUNCT iajs-3089 82 3 suppose suppose VERB iajs-3089 82 4 that that SCONJ iajs-3089 82 5 𝑀 𝑀 PROPN iajs-3089 82 6 is be AUX iajs-3089 82 7 small small ADJ iajs-3089 82 8 compressible compressible ADJ iajs-3089 82 9 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 82 10 and and CCONJ iajs-3089 82 11 let let VERB iajs-3089 82 12 𝑁 𝑁 PROPN iajs-3089 82 13 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 82 14 𝑀 𝑀 PROPN iajs-3089 82 15 , , PUNCT iajs-3089 82 16 then then ADV iajs-3089 82 17 by by ADP iajs-3089 82 18 [ [ X iajs-3089 82 19 3 3 X iajs-3089 82 20 ] ] X iajs-3089 82 21 𝑁 𝑁 PROPN iajs-3089 82 22 β‰ͺ β‰ͺ PUNCT iajs-3089 82 23 𝑀 𝑀 PROPN iajs-3089 82 24 and and CCONJ iajs-3089 82 25 since since SCONJ iajs-3089 82 26 𝑀 𝑀 PROPN iajs-3089 82 27 is be AUX iajs-3089 82 28 small small ADJ iajs-3089 82 29 compressible compressible ADJ iajs-3089 82 30 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 82 31 , , PUNCT iajs-3089 82 32 therefore therefore ADV iajs-3089 82 33 𝑀 𝑀 PROPN iajs-3089 82 34 is be AUX iajs-3089 82 35 p p ADJ iajs-3089 82 36 - - PUNCT iajs-3089 82 37 small small ADJ iajs-3089 82 38 compressible compressible NOUN iajs-3089 82 39 . . PUNCT iajs-3089 83 1 conversely conversely ADV iajs-3089 83 2 it it PRON iajs-3089 83 3 's be AUX iajs-3089 83 4 clear clear ADJ iajs-3089 83 5 by by ADP iajs-3089 83 6 remarks remark NOUN iajs-3089 83 7 and and CCONJ iajs-3089 83 8 examples example NOUN iajs-3089 83 9 ( ( PUNCT iajs-3089 83 10 3.10)point(1 3.10)point(1 NUM iajs-3089 83 11 ) ) PUNCT iajs-3089 83 12 corollary(3.12 corollary(3.12 NOUN iajs-3089 83 13 ): ): PUNCT iajs-3089 83 14 a a DET iajs-3089 83 15 small small ADJ iajs-3089 83 16 compressible compressible NOUN iajs-3089 83 17 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 83 18 𝑀 𝑀 PROPN iajs-3089 83 19 is be AUX iajs-3089 83 20 p p ADJ iajs-3089 83 21 - - PUNCT iajs-3089 83 22 small small ADJ iajs-3089 83 23 compressible compressible NOUN iajs-3089 83 24 , , PUNCT iajs-3089 83 25 if if SCONJ iajs-3089 83 26 every every DET iajs-3089 83 27 cyclic cyclic ADJ iajs-3089 83 28 submodule submodule NOUN iajs-3089 83 29 of of ADP iajs-3089 83 30 𝑀 𝑀 PROPN iajs-3089 83 31 is be AUX iajs-3089 83 32 p p ADJ iajs-3089 83 33 - - PUNCT iajs-3089 83 34 small small ADJ iajs-3089 83 35 submodule submodule NOUN iajs-3089 83 36 in in ADP iajs-3089 83 37 𝑀 𝑀 PROPN iajs-3089 83 38 . . PUNCT iajs-3089 84 1 proof proof NOUN iajs-3089 84 2 : : PUNCT iajs-3089 84 3 obviously obviously ADV iajs-3089 84 4 by by ADP iajs-3089 84 5 above above ADP iajs-3089 84 6 proposition proposition NOUN iajs-3089 84 7 . . PUNCT iajs-3089 85 1 proposition(3.13 proposition(3.13 PUNCT iajs-3089 85 2 ): ): PUNCT iajs-3089 85 3 let let VERB iajs-3089 85 4 𝑀 𝑀 PRON iajs-3089 85 5 be be AUX iajs-3089 85 6 a a DET iajs-3089 85 7 finitely finitely ADV iajs-3089 85 8 generated generate VERB iajs-3089 85 9 ( ( PUNCT iajs-3089 85 10 or or CCONJ iajs-3089 85 11 multiplication multiplication NOUN iajs-3089 85 12 ) ) PUNCT iajs-3089 85 13 𝑅 𝑅 NOUN iajs-3089 85 14 βˆ’ βˆ’ PROPN iajs-3089 85 15 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ VERB iajs-3089 85 16 . . PUNCT iajs-3089 86 1 then then ADV iajs-3089 86 2 𝑀 𝑀 PROPN iajs-3089 86 3 is be AUX iajs-3089 86 4 small small ADJ iajs-3089 86 5 compressible compressible ADJ iajs-3089 86 6 if if SCONJ iajs-3089 86 7 and and CCONJ iajs-3089 86 8 only only ADV iajs-3089 86 9 if if SCONJ iajs-3089 86 10 𝑀is 𝑀is PROPN iajs-3089 86 11 p p ADJ iajs-3089 86 12 - - PUNCT iajs-3089 86 13 small small ADJ iajs-3089 86 14 compressible compressible NOUN iajs-3089 86 15 . . PUNCT iajs-3089 87 1 proof proof NOUN iajs-3089 87 2 : : PUNCT iajs-3089 87 3 let let VERB iajs-3089 87 4 𝑁 𝑁 PROPN iajs-3089 87 5 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 87 6 𝑀. 𝑀. NOUN iajs-3089 87 7 we we PRON iajs-3089 87 8 want want VERB iajs-3089 87 9 to to PART iajs-3089 87 10 show show VERB iajs-3089 87 11 that that SCONJ iajs-3089 87 12 𝑀is 𝑀is PROPN iajs-3089 87 13 p p ADJ iajs-3089 87 14 - - PUNCT iajs-3089 87 15 small small ADJ iajs-3089 87 16 compressible compressible NOUN iajs-3089 87 17 . . PUNCT iajs-3089 88 1 since since SCONJ iajs-3089 88 2 𝑀 𝑀 PROPN iajs-3089 88 3 is be AUX iajs-3089 88 4 finitely finitely ADV iajs-3089 88 5 generated generate VERB iajs-3089 88 6 ( ( PUNCT iajs-3089 88 7 or or CCONJ iajs-3089 88 8 multiplication multiplication NOUN iajs-3089 88 9 ) ) PUNCT iajs-3089 88 10 , , PUNCT iajs-3089 88 11 then then ADV iajs-3089 88 12 by by ADP iajs-3089 88 13 proposition(1.4)[3 proposition(1.4)[3 NOUN iajs-3089 88 14 ] ] PUNCT iajs-3089 88 15 , , PUNCT iajs-3089 88 16 so so ADV iajs-3089 88 17 𝑁 𝑁 PROPN iajs-3089 88 18 β‰ͺ β‰ͺ ADJ iajs-3089 88 19 𝑀 𝑀 PROPN iajs-3089 88 20 , , PUNCT iajs-3089 88 21 but but CCONJ iajs-3089 88 22 𝑀 𝑀 PROPN iajs-3089 88 23 is be AUX iajs-3089 88 24 small small ADJ iajs-3089 88 25 compressible compressible ADJ iajs-3089 88 26 𝑅 𝑅 PROPN iajs-3089 88 27 βˆ’ βˆ’ PROPN iajs-3089 88 28 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 89 1 therefore therefore ADV iajs-3089 89 2 𝑀 𝑀 PROPN iajs-3089 89 3 is be AUX iajs-3089 89 4 p p ADJ iajs-3089 89 5 - - PUNCT iajs-3089 89 6 small small ADJ iajs-3089 89 7 compressible compressible NOUN iajs-3089 89 8 . . PUNCT iajs-3089 90 1 conversely conversely ADV iajs-3089 90 2 clear clear ADJ iajs-3089 90 3 by by ADP iajs-3089 90 4 remarks remark NOUN iajs-3089 90 5 and and CCONJ iajs-3089 90 6 examples example NOUN iajs-3089 90 7 ( ( PUNCT iajs-3089 90 8 3.10 3.10 NUM iajs-3089 90 9 ) ) PUNCT iajs-3089 90 10 point point NOUN iajs-3089 90 11 ( ( PUNCT iajs-3089 90 12 1 1 NUM iajs-3089 90 13 ) ) PUNCT iajs-3089 90 14 . . PUNCT iajs-3089 91 1 corollary(3.14 corollary(3.14 NOUN iajs-3089 91 2 ): ): PUNCT iajs-3089 91 3 let let VERB iajs-3089 91 4 𝑀 𝑀 PRON iajs-3089 91 5 be be AUX iajs-3089 91 6 a a DET iajs-3089 91 7 noetherian noetherian ADJ iajs-3089 91 8 𝑅 𝑅 PROPN iajs-3089 91 9 βˆ’ βˆ’ PROPN iajs-3089 91 10 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 91 11 then then ADV iajs-3089 91 12 𝑀 𝑀 PROPN iajs-3089 91 13 is be AUX iajs-3089 91 14 small small ADJ iajs-3089 91 15 compressible compressible ADJ iajs-3089 91 16 if if SCONJ iajs-3089 91 17 and and CCONJ iajs-3089 92 1 only only ADV iajs-3089 92 2 if if SCONJ iajs-3089 92 3 𝑀is 𝑀is PROPN iajs-3089 92 4 p p ADJ iajs-3089 92 5 - - PUNCT iajs-3089 92 6 small small ADJ iajs-3089 92 7 compressible compressible NOUN iajs-3089 92 8 . . PUNCT iajs-3089 93 1 proof proof NOUN iajs-3089 93 2 : : PUNCT iajs-3089 93 3 since since SCONJ iajs-3089 93 4 𝑀 𝑀 PROPN iajs-3089 93 5 is be AUX iajs-3089 93 6 noetherian noetherian ADJ iajs-3089 93 7 , , PUNCT iajs-3089 93 8 then then ADV iajs-3089 93 9 every every DET iajs-3089 93 10 submodule submodule NOUN iajs-3089 93 11 is be AUX iajs-3089 93 12 finitely finitely ADV iajs-3089 93 13 generated generate VERB iajs-3089 93 14 , , PUNCT iajs-3089 93 15 then then ADV iajs-3089 93 16 the the DET iajs-3089 93 17 result result NOUN iajs-3089 93 18 follows follow VERB iajs-3089 93 19 by by ADP iajs-3089 93 20 proposition(3.13 proposition(3.13 NOUN iajs-3089 93 21 ) ) PUNCT iajs-3089 93 22 . . PUNCT iajs-3089 94 1 therefore therefore ADV iajs-3089 94 2 𝑀 𝑀 PROPN iajs-3089 94 3 is be AUX iajs-3089 94 4 small small ADJ iajs-3089 94 5 compressible compressible NOUN iajs-3089 94 6 . . PUNCT iajs-3089 95 1 conversely conversely ADV iajs-3089 95 2 clear clear ADJ iajs-3089 95 3 by by ADP iajs-3089 95 4 remarks remark NOUN iajs-3089 95 5 and and CCONJ iajs-3089 95 6 examples example NOUN iajs-3089 95 7 ( ( PUNCT iajs-3089 95 8 3.10 3.10 NUM iajs-3089 95 9 ) ) PUNCT iajs-3089 95 10 point point NOUN iajs-3089 95 11 ( ( PUNCT iajs-3089 95 12 1 1 NUM iajs-3089 95 13 ) ) PUNCT iajs-3089 95 14 . . PUNCT iajs-3089 96 1 recall recall VERB iajs-3089 96 2 that that SCONJ iajs-3089 96 3 an an DET iajs-3089 96 4 𝑅 𝑅 PROPN iajs-3089 96 5 βˆ’ βˆ’ PROPN iajs-3089 96 6 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ SCONJ iajs-3089 96 7 𝑀 𝑀 PROPN iajs-3089 96 8 is be AUX iajs-3089 96 9 called call VERB iajs-3089 96 10 almost almost ADV iajs-3089 96 11 finitely finitely ADV iajs-3089 96 12 generated generate VERB iajs-3089 96 13 if if SCONJ iajs-3089 96 14 𝑀 𝑀 PROPN iajs-3089 96 15 is be AUX iajs-3089 96 16 not not PART iajs-3089 96 17 finitely finitely ADV iajs-3089 96 18 generated generate VERB iajs-3089 96 19 and and CCONJ iajs-3089 96 20 every every DET iajs-3089 96 21 proper proper ADJ iajs-3089 96 22 submodule submodule NOUN iajs-3089 96 23 of of ADP iajs-3089 96 24 of of ADP iajs-3089 96 25 𝑀 𝑀 PROPN iajs-3089 96 26 is be AUX iajs-3089 96 27 finitely finitely ADV iajs-3089 96 28 generated[6 generated[6 ADJ iajs-3089 96 29 ] ] X iajs-3089 96 30 . . PUNCT iajs-3089 97 1 proposition(3.15 proposition(3.15 NUM iajs-3089 97 2 ): ): PUNCT iajs-3089 97 3 let let VERB iajs-3089 97 4 𝑀 𝑀 PRON iajs-3089 97 5 be be AUX iajs-3089 97 6 an an DET iajs-3089 97 7 almost almost ADV iajs-3089 97 8 finitely finitely ADV iajs-3089 97 9 generated generate VERB iajs-3089 97 10 𝑅 𝑅 PROPN iajs-3089 97 11 βˆ’ βˆ’ PROPN iajs-3089 97 12 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 97 13 then then ADV iajs-3089 97 14 𝑀is 𝑀is PROPN iajs-3089 97 15 p p ADJ iajs-3089 97 16 - - PUNCT iajs-3089 97 17 small small ADJ iajs-3089 97 18 compressible compressible NOUN iajs-3089 97 19 if if SCONJ iajs-3089 97 20 and and CCONJ iajs-3089 97 21 only only ADV iajs-3089 97 22 if if SCONJ iajs-3089 97 23 𝑀is 𝑀is PROPN iajs-3089 97 24 small small ADJ iajs-3089 97 25 compressible compressible NOUN iajs-3089 97 26 . . PUNCT iajs-3089 98 1 proof proof NOUN iajs-3089 98 2 : : PUNCT iajs-3089 98 3 let let VERB iajs-3089 98 4 𝑁 𝑁 PROPN iajs-3089 98 5 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 98 6 𝑀. 𝑀. NOUN iajs-3089 98 7 we we PRON iajs-3089 98 8 want want VERB iajs-3089 98 9 to to PART iajs-3089 98 10 show show VERB iajs-3089 98 11 that that SCONJ iajs-3089 98 12 𝑀is 𝑀is PROPN iajs-3089 98 13 p p ADJ iajs-3089 98 14 - - PUNCT iajs-3089 98 15 small small ADJ iajs-3089 98 16 compressible compressible NOUN iajs-3089 98 17 . . PUNCT iajs-3089 99 1 since since SCONJ iajs-3089 99 2 𝑀 𝑀 PROPN iajs-3089 99 3 is be AUX iajs-3089 99 4 almost almost ADV iajs-3089 99 5 finitely finitely ADV iajs-3089 99 6 generated[6 generated[6 ADJ iajs-3089 99 7 ] ] PUNCT iajs-3089 99 8 , , PUNCT iajs-3089 99 9 then then ADV iajs-3089 99 10 by by ADP iajs-3089 99 11 corollary corollary ADJ iajs-3089 99 12 ( ( PUNCT iajs-3089 99 13 1.11)[3 1.11)[3 PROPN iajs-3089 99 14 ] ] PUNCT iajs-3089 99 15 , , PUNCT iajs-3089 99 16 we we PRON iajs-3089 99 17 get get VERB iajs-3089 99 18 𝑁 𝑁 PROPN iajs-3089 99 19 β‰ͺ β‰ͺ ADJ iajs-3089 99 20 𝑀 𝑀 PROPN iajs-3089 99 21 , , PUNCT iajs-3089 99 22 but but CCONJ iajs-3089 99 23 𝑀 𝑀 PROPN iajs-3089 99 24 is be AUX iajs-3089 99 25 small small ADJ iajs-3089 99 26 compressible compressible ADJ iajs-3089 99 27 𝑅 𝑅 PROPN iajs-3089 99 28 βˆ’ βˆ’ PROPN iajs-3089 99 29 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 99 30 therefore therefore ADV iajs-3089 99 31 𝑀 𝑀 PROPN iajs-3089 99 32 is be AUX iajs-3089 99 33 p p ADJ iajs-3089 99 34 - - PUNCT iajs-3089 99 35 small small ADJ iajs-3089 99 36 compressible compressible NOUN iajs-3089 99 37 . . PUNCT iajs-3089 100 1 conversely conversely ADV iajs-3089 100 2 clear clear ADJ iajs-3089 100 3 by by ADP iajs-3089 100 4 remarks remark NOUN iajs-3089 100 5 and and CCONJ iajs-3089 100 6 examples example NOUN iajs-3089 100 7 ( ( PUNCT iajs-3089 100 8 3.10 3.10 NUM iajs-3089 100 9 ) ) PUNCT iajs-3089 100 10 point point NOUN iajs-3089 100 11 ( ( PUNCT iajs-3089 100 12 1 1 NUM iajs-3089 100 13 ) ) PUNCT iajs-3089 100 14 . . PUNCT iajs-3089 101 1 proposition(3.16 proposition(3.16 PROPN iajs-3089 101 2 ): ): PUNCT iajs-3089 101 3 let let VERB iajs-3089 101 4 𝑀 𝑀 PRON iajs-3089 101 5 be be AUX iajs-3089 101 6 a a DET iajs-3089 101 7 hollow hollow ADJ iajs-3089 101 8 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 101 9 then then ADV iajs-3089 101 10 the the DET iajs-3089 101 11 following follow VERB iajs-3089 101 12 statements statement NOUN iajs-3089 101 13 are be AUX iajs-3089 101 14 equivalent equivalent ADJ iajs-3089 101 15 : : PUNCT iajs-3089 101 16 ( ( PUNCT iajs-3089 101 17 1 1 X iajs-3089 101 18 ) ) PUNCT iajs-3089 101 19 𝑀 𝑀 PROPN iajs-3089 101 20 is be AUX iajs-3089 101 21 compressible compressible ADJ iajs-3089 101 22 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 101 23 ihjpas ihjpa NOUN iajs-3089 101 24 . . PUNCT iajs-3089 102 1 36 36 NUM iajs-3089 102 2 ( ( PUNCT iajs-3089 102 3 3 3 NUM iajs-3089 102 4 ) ) PUNCT iajs-3089 102 5 2023 2023 NUM iajs-3089 102 6 369 369 NUM iajs-3089 102 7 ( ( PUNCT iajs-3089 102 8 2 2 NUM iajs-3089 102 9 ) ) PUNCT iajs-3089 102 10 𝑀 𝑀 PROPN iajs-3089 102 11 is be AUX iajs-3089 102 12 p p ADJ iajs-3089 102 13 - - PUNCT iajs-3089 102 14 small small ADJ iajs-3089 102 15 compressible compressible ADJ iajs-3089 102 16 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 102 17 ( ( PUNCT iajs-3089 102 18 3 3 X iajs-3089 102 19 ) ) PUNCT iajs-3089 102 20 small small ADJ iajs-3089 102 21 compressible compressible ADJ iajs-3089 102 22 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 102 23 . . PUNCT iajs-3089 103 1 proof proof NOUN iajs-3089 103 2 : : PUNCT iajs-3089 103 3 ( ( PUNCT iajs-3089 103 4 1 1 X iajs-3089 103 5 ) ) PUNCT iajs-3089 103 6 ⟹ ⟹ NOUN iajs-3089 103 7 ( ( PUNCT iajs-3089 103 8 2 2 X iajs-3089 103 9 ) ) PUNCT iajs-3089 103 10 it it PRON iajs-3089 103 11 's be AUX iajs-3089 103 12 clear clear ADJ iajs-3089 103 13 by by ADP iajs-3089 103 14 remarks remark NOUN iajs-3089 103 15 and and CCONJ iajs-3089 103 16 examples example NOUN iajs-3089 103 17 ( ( PUNCT iajs-3089 103 18 3.2 3.2 NUM iajs-3089 103 19 ) ) PUNCT iajs-3089 103 20 point point NOUN iajs-3089 103 21 ( ( PUNCT iajs-3089 103 22 1 1 NUM iajs-3089 103 23 ) ) PUNCT iajs-3089 103 24 . . PUNCT iajs-3089 104 1 ( ( PUNCT iajs-3089 104 2 2 2 X iajs-3089 104 3 ) ) PUNCT iajs-3089 104 4 ⟹ ⟹ NOUN iajs-3089 104 5 ( ( PUNCT iajs-3089 104 6 3 3 X iajs-3089 104 7 ) ) PUNCT iajs-3089 104 8 it it PRON iajs-3089 104 9 's be AUX iajs-3089 104 10 clear clear ADJ iajs-3089 104 11 by by ADP iajs-3089 104 12 remarks remark NOUN iajs-3089 104 13 ( ( PUNCT iajs-3089 104 14 3.10 3.10 NUM iajs-3089 104 15 ) ) PUNCT iajs-3089 104 16 point point NOUN iajs-3089 104 17 ( ( PUNCT iajs-3089 104 18 1 1 NUM iajs-3089 104 19 ) ) PUNCT iajs-3089 104 20 ( ( PUNCT iajs-3089 104 21 3 3 X iajs-3089 104 22 ) ) PUNCT iajs-3089 104 23 ⟹ ⟹ NUM iajs-3089 105 1 ( ( PUNCT iajs-3089 105 2 1)let𝐾 1)let𝐾 NUM iajs-3089 105 3 ≀ ≀ NOUN iajs-3089 105 4 𝑀. 𝑀. PROPN iajs-3089 105 5 since since SCONJ iajs-3089 105 6 𝑀 𝑀 PROPN iajs-3089 105 7 is be AUX iajs-3089 105 8 β„Žπ‘œπ‘™π‘™π‘œπ‘€ β„Žπ‘œπ‘™π‘™π‘œπ‘€ NOUN iajs-3089 105 9 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 105 10 and and CCONJ iajs-3089 105 11 small small ADJ iajs-3089 105 12 compressible compressible ADJ iajs-3089 105 13 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 105 14 , , PUNCT iajs-3089 105 15 then then ADV iajs-3089 105 16 βˆƒ βˆƒ PROPN iajs-3089 105 17 a a DET iajs-3089 105 18 monomorphism monomorphism NOUN iajs-3089 105 19 𝑓 𝑓 X iajs-3089 105 20 : : PUNCT iajs-3089 105 21 𝑀 𝑀 PROPN iajs-3089 105 22 ⟢ ⟢ NOUN iajs-3089 105 23 𝐾. 𝐾. PROPN iajs-3089 105 24 therefor therefor ADP iajs-3089 105 25 𝑀 𝑀 PROPN iajs-3089 105 26 is be AUX iajs-3089 105 27 compressible compressible ADJ iajs-3089 105 28 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 105 29 we we PRON iajs-3089 105 30 introduce introduce VERB iajs-3089 105 31 the the DET iajs-3089 105 32 following follow VERB iajs-3089 105 33 definition definition NOUN iajs-3089 105 34 ( ( PUNCT iajs-3089 105 35 3.17 3.17 NUM iajs-3089 105 36 ): ): PUNCT iajs-3089 106 1 an an DET iajs-3089 106 2 𝑅 𝑅 PROPN iajs-3089 106 3 βˆ’ βˆ’ PROPN iajs-3089 106 4 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 106 5 𝑀 𝑀 PROPN iajs-3089 106 6 is be AUX iajs-3089 106 7 called call VERB iajs-3089 106 8 p p ADJ iajs-3089 106 9 - - PUNCT iajs-3089 106 10 small small ADJ iajs-3089 106 11 quasi quasi NOUN iajs-3089 106 12 - - NOUN iajs-3089 106 13 dedekind dedekind ADJ iajs-3089 106 14 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 106 15 if if SCONJ iajs-3089 106 16 for for ADP iajs-3089 106 17 all all DET iajs-3089 106 18 ∈ ∈ PROPN iajs-3089 106 19 ( ( PUNCT iajs-3089 106 20 𝑀 𝑀 PROPN iajs-3089 106 21 ) ) PUNCT iajs-3089 106 22 , , PUNCT iajs-3089 106 23 𝑓 𝑓 DET iajs-3089 106 24 β‰  β‰  PROPN iajs-3089 106 25 0 0 NUM iajs-3089 106 26 implies imply VERB iajs-3089 106 27 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 106 28 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 106 29 𝑀. 𝑀. NOUN iajs-3089 106 30 remark remark NOUN iajs-3089 106 31 ( ( PUNCT iajs-3089 106 32 3.18 3.18 NUM iajs-3089 106 33 ): ): PUNCT iajs-3089 106 34 it it PRON iajs-3089 106 35 's be AUX iajs-3089 106 36 clear clear ADJ iajs-3089 106 37 that that SCONJ iajs-3089 106 38 every every DET iajs-3089 106 39 quasi quasi NOUN iajs-3089 106 40 - - ADJ iajs-3089 106 41 dedekind dedekind ADJ iajs-3089 106 42 is be AUX iajs-3089 106 43 p p ADJ iajs-3089 106 44 - - PUNCT iajs-3089 106 45 small small ADJ iajs-3089 106 46 quasi quasi NOUN iajs-3089 106 47 - - NOUN iajs-3089 106 48 dedekind dedekind ADJ iajs-3089 106 49 . . PUNCT iajs-3089 107 1 proposition(3.19 proposition(3.19 PUNCT iajs-3089 107 2 ): ): PUNCT iajs-3089 107 3 if if SCONJ iajs-3089 107 4 𝑀 𝑀 PROPN iajs-3089 107 5 is be AUX iajs-3089 107 6 p p ADJ iajs-3089 107 7 - - PUNCT iajs-3089 107 8 small small ADJ iajs-3089 107 9 quasi quasi ADJ iajs-3089 107 10 - - ADJ iajs-3089 107 11 dedekind dedekind ADJ iajs-3089 107 12 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 107 13 , , PUNCT iajs-3089 107 14 then then ADV iajs-3089 107 15 𝑀 𝑀 PROPN iajs-3089 107 16 ca can AUX iajs-3089 107 17 n't not PART iajs-3089 107 18 be be AUX iajs-3089 107 19 compressible compressible ADJ iajs-3089 107 20 . . PUNCT iajs-3089 108 1 proof proof NOUN iajs-3089 108 2 : : PUNCT iajs-3089 108 3 suppose suppose VERB iajs-3089 108 4 that that SCONJ iajs-3089 108 5 𝑀 𝑀 PROPN iajs-3089 108 6 is be AUX iajs-3089 108 7 p p ADJ iajs-3089 108 8 - - PUNCT iajs-3089 108 9 small small ADJ iajs-3089 108 10 quasi quasi NOUN iajs-3089 108 11 - - ADJ iajs-3089 108 12 dedekind dedekind ADJ iajs-3089 108 13 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 108 14 and and CCONJ iajs-3089 108 15 let let VERB iajs-3089 108 16 𝑁 𝑁 PROPN iajs-3089 108 17 = = PUNCT iajs-3089 108 18 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 108 19 ≀ ≀ PROPN iajs-3089 108 20 𝑀 𝑀 PROPN iajs-3089 108 21 , , PUNCT iajs-3089 108 22 but but CCONJ iajs-3089 108 23 𝑀 𝑀 PROPN iajs-3089 108 24 is be AUX iajs-3089 108 25 psmall psmall ADV iajs-3089 108 26 quasi quasi ADJ iajs-3089 108 27 - - NOUN iajs-3089 108 28 dedekind dedekind ADJ iajs-3089 108 29 , , PUNCT iajs-3089 108 30 then then ADV iajs-3089 108 31 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 108 32 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 108 33 𝑀 𝑀 PROPN iajs-3089 108 34 , , PUNCT iajs-3089 108 35 𝑓 𝑓 DET iajs-3089 108 36 β‰  β‰  PROPN iajs-3089 108 37 0 0 NUM iajs-3089 108 38 , , PUNCT iajs-3089 108 39 thus thus ADV iajs-3089 108 40 ca can AUX iajs-3089 108 41 n't not PART iajs-3089 108 42 be be AUX iajs-3089 108 43 embedded embed VERB iajs-3089 108 44 𝑀 𝑀 PROPN iajs-3089 108 45 in in ADP iajs-3089 108 46 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 108 47 , , PUNCT iajs-3089 108 48 because because SCONJ iajs-3089 108 49 π»π‘œπ‘š(𝑀 π»π‘œπ‘š(𝑀 PROPN iajs-3089 108 50 , , PUNCT iajs-3089 108 51 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 108 52 ) ) PUNCT iajs-3089 108 53 = = SYM iajs-3089 109 1 0 0 X iajs-3089 109 2 . . PUNCT iajs-3089 109 3 therefore therefore ADV iajs-3089 109 4 𝑀 𝑀 PROPN iajs-3089 109 5 ca can AUX iajs-3089 109 6 n't not PART iajs-3089 109 7 be be AUX iajs-3089 109 8 compressible compressible ADJ iajs-3089 109 9 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 109 10 remark(3.20 remark(3.20 NOUN iajs-3089 109 11 ): ): PUNCT iajs-3089 109 12 every every DET iajs-3089 109 13 small small ADJ iajs-3089 109 14 quasi quasi NOUN iajs-3089 109 15 - - NOUN iajs-3089 109 16 dedekind dedekind ADJ iajs-3089 109 17 is be AUX iajs-3089 109 18 p p ADJ iajs-3089 109 19 - - PUNCT iajs-3089 109 20 small small ADJ iajs-3089 109 21 quasi quasi NOUN iajs-3089 109 22 - - NOUN iajs-3089 109 23 dedekind dedekind ADJ iajs-3089 109 24 . . PUNCT iajs-3089 110 1 proof proof NOUN iajs-3089 110 2 : : PUNCT iajs-3089 110 3 let let VERB iajs-3089 110 4 0 0 NUM iajs-3089 110 5 β‰  β‰  PROPN iajs-3089 110 6 𝑓 𝑓 DET iajs-3089 110 7 ∈ ∈ PROPN iajs-3089 110 8 𝐸𝑛𝑑𝑅(𝑀 𝐸𝑛𝑑𝑅(𝑀 NOUN iajs-3089 110 9 ) ) PUNCT iajs-3089 110 10 , , PUNCT iajs-3089 110 11 where where SCONJ iajs-3089 110 12 𝑀 𝑀 PROPN iajs-3089 110 13 is be AUX iajs-3089 110 14 an an DET iajs-3089 110 15 𝑅 𝑅 PROPN iajs-3089 110 16 βˆ’ βˆ’ PROPN iajs-3089 110 17 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 110 18 since since SCONJ iajs-3089 110 19 𝑀 𝑀 PROPN iajs-3089 110 20 is be AUX iajs-3089 110 21 a a DET iajs-3089 110 22 small small ADJ iajs-3089 110 23 quasi quasi NOUN iajs-3089 110 24 - - NOUN iajs-3089 110 25 dedekind dedekind ADJ iajs-3089 110 26 , , PUNCT iajs-3089 110 27 then then ADV iajs-3089 110 28 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 110 29 β‰ͺ β‰ͺ PUNCT iajs-3089 110 30 𝑀 𝑀 PROPN iajs-3089 110 31 , , PUNCT iajs-3089 110 32 hence hence ADV iajs-3089 110 33 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 110 34 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 110 35 𝑀. 𝑀. PROPN iajs-3089 110 36 thus thus ADV iajs-3089 110 37 𝑀 𝑀 PROPN iajs-3089 110 38 is be AUX iajs-3089 110 39 a a DET iajs-3089 110 40 p p ADJ iajs-3089 110 41 - - PUNCT iajs-3089 110 42 small small ADJ iajs-3089 110 43 quasi quasi ADJ iajs-3089 110 44 - - ADJ iajs-3089 110 45 dedekind dedekind ADJ iajs-3089 110 46 module module NOUN iajs-3089 110 47 . . PUNCT iajs-3089 111 1 4 4 X iajs-3089 111 2 . . X iajs-3089 111 3 p p X iajs-3089 111 4 - - PUNCT iajs-3089 111 5 small small ADJ iajs-3089 111 6 retractable retractable ADJ iajs-3089 111 7 module module NOUN iajs-3089 111 8 in in ADP iajs-3089 111 9 this this DET iajs-3089 111 10 section section NOUN iajs-3089 111 11 , , PUNCT iajs-3089 111 12 we we PRON iajs-3089 111 13 introduce introduce VERB iajs-3089 111 14 the the DET iajs-3089 111 15 concept concept NOUN iajs-3089 111 16 of of ADP iajs-3089 111 17 p p ADJ iajs-3089 111 18 - - PUNCT iajs-3089 111 19 small small ADJ iajs-3089 111 20 retractable retractable ADJ iajs-3089 111 21 module module NOUN iajs-3089 111 22 as as ADP iajs-3089 111 23 a a DET iajs-3089 111 24 generalization generalization NOUN iajs-3089 111 25 of of ADP iajs-3089 111 26 retractable retractable ADJ iajs-3089 111 27 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 111 28 , , PUNCT iajs-3089 111 29 give give VERB iajs-3089 111 30 some some PRON iajs-3089 111 31 of of ADP iajs-3089 111 32 basic basic ADJ iajs-3089 111 33 properties property NOUN iajs-3089 111 34 , , PUNCT iajs-3089 111 35 examples example NOUN iajs-3089 111 36 and and CCONJ iajs-3089 111 37 characterizations characterization NOUN iajs-3089 111 38 of of ADP iajs-3089 111 39 this this DET iajs-3089 111 40 concept concept NOUN iajs-3089 111 41 . . PUNCT iajs-3089 112 1 definition definition NOUN iajs-3089 112 2 ( ( PUNCT iajs-3089 112 3 4.1 4.1 NUM iajs-3089 112 4 ): ): PUNCT iajs-3089 112 5 an an DET iajs-3089 112 6 𝑅 𝑅 PROPN iajs-3089 112 7 βˆ’ βˆ’ PROPN iajs-3089 112 8 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 112 9 𝑀 𝑀 PROPN iajs-3089 112 10 is be AUX iajs-3089 112 11 said say VERB iajs-3089 112 12 to to PART iajs-3089 112 13 be be AUX iajs-3089 112 14 p p ADJ iajs-3089 112 15 - - PUNCT iajs-3089 112 16 small small ADJ iajs-3089 112 17 retractable retractable NOUN iajs-3089 112 18 if if SCONJ iajs-3089 112 19 𝑀 𝑀 PROPN iajs-3089 112 20 π»π‘œπ‘š(𝑀 π»π‘œπ‘š(𝑀 PROPN iajs-3089 112 21 , , PUNCT iajs-3089 112 22 𝐾 𝐾 PROPN iajs-3089 112 23 ) ) PUNCT iajs-3089 112 24 β‰  β‰  PROPN iajs-3089 112 25 0 0 NUM iajs-3089 112 26 , , PUNCT iajs-3089 112 27 for for ADP iajs-3089 112 28 every every DET iajs-3089 112 29 non non ADJ iajs-3089 112 30 - - ADJ iajs-3089 112 31 zero zero ADJ iajs-3089 112 32 p p ADJ iajs-3089 112 33 - - PUNCT iajs-3089 112 34 small small ADJ iajs-3089 112 35 submodule submodule NOUN iajs-3089 112 36 𝐾of 𝐾of PROPN iajs-3089 112 37 𝑀. 𝑀. PROPN iajs-3089 112 38 equivalently equivalently ADV iajs-3089 112 39 , , PUNCT iajs-3089 112 40 𝑀 𝑀 PROPN iajs-3089 112 41 is be AUX iajs-3089 112 42 p p ADJ iajs-3089 112 43 - - PUNCT iajs-3089 112 44 small small ADJ iajs-3089 112 45 retractable retractable NOUN iajs-3089 112 46 if if SCONJ iajs-3089 112 47 there there PRON iajs-3089 112 48 exists exist VERB iajs-3089 112 49 a a DET iajs-3089 112 50 homomorphism homomorphism NOUN iajs-3089 112 51 𝑓 𝑓 X iajs-3089 112 52 : : PUNCT iajs-3089 112 53 𝑀 𝑀 PROPN iajs-3089 112 54 ⟢ ⟢ NOUN iajs-3089 112 55 𝑁 𝑁 PROPN iajs-3089 112 56 whenever whenever SCONJ iajs-3089 112 57 0 0 NUM iajs-3089 112 58 β‰  β‰  PROPN iajs-3089 112 59 𝑁 𝑁 ADP iajs-3089 112 60 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 112 61 𝑀. 𝑀. NOUN iajs-3089 112 62 remarks remark NOUN iajs-3089 112 63 and and CCONJ iajs-3089 112 64 examples(4.2 examples(4.2 NOUN iajs-3089 112 65 ): ): PUNCT iajs-3089 112 66 1 1 X iajs-3089 112 67 . . X iajs-3089 113 1 it it PRON iajs-3089 113 2 ’s ’ VERB iajs-3089 113 3 obvious obvious ADJ iajs-3089 113 4 that that SCONJ iajs-3089 113 5 every every DET iajs-3089 113 6 p p ADJ iajs-3089 113 7 - - PUNCT iajs-3089 113 8 small small ADJ iajs-3089 113 9 compressible compressible ADJ iajs-3089 113 10 module module NOUN iajs-3089 113 11 is be AUX iajs-3089 113 12 p p ADJ iajs-3089 113 13 - - PUNCT iajs-3089 113 14 small small ADJ iajs-3089 113 15 retractable retractable ADJ iajs-3089 113 16 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 113 17 , , PUNCT iajs-3089 113 18 but but CCONJ iajs-3089 113 19 the the DET iajs-3089 113 20 converse converse NOUN iajs-3089 113 21 is be AUX iajs-3089 113 22 not not PART iajs-3089 113 23 true true ADJ iajs-3089 113 24 for for ADP iajs-3089 113 25 instance𝑍4 instance𝑍4 NOUN iajs-3089 113 26 is be AUX iajs-3089 113 27 p p ADJ iajs-3089 113 28 - - PUNCT iajs-3089 113 29 small small ADJ iajs-3089 113 30 retractable retractable NOUN iajs-3089 113 31 but but CCONJ iajs-3089 113 32 not not PART iajs-3089 113 33 p p ADJ iajs-3089 113 34 - - PUNCT iajs-3089 113 35 small small ADJ iajs-3089 113 36 compressible compressible ADJ iajs-3089 113 37 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 113 38 see see VERB iajs-3089 113 39 remarks remark NOUN iajs-3089 113 40 and and CCONJ iajs-3089 113 41 examples example NOUN iajs-3089 113 42 ( ( PUNCT iajs-3089 113 43 3.2 3.2 NUM iajs-3089 113 44 ) ) PUNCT iajs-3089 113 45 point(5 point(5 PROPN iajs-3089 113 46 ) ) PUNCT iajs-3089 113 47 . . PUNCT iajs-3089 114 1 2 2 X iajs-3089 114 2 . . X iajs-3089 114 3 𝑍 𝑍 VERB iajs-3089 114 4 π‘Žπ‘  π‘Žπ‘  NOUN iajs-3089 114 5 𝑍 𝑍 PROPN iajs-3089 114 6 βˆ’ βˆ’ PROPN iajs-3089 114 7 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 114 8 is be AUX iajs-3089 114 9 p p ADJ iajs-3089 114 10 - - PUNCT iajs-3089 114 11 small small ADJ iajs-3089 114 12 retractable retractable ADJ iajs-3089 114 13 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 114 14 , , PUNCT iajs-3089 114 15 because because SCONJ iajs-3089 114 16 it it PRON iajs-3089 114 17 's be AUX iajs-3089 114 18 p p NOUN iajs-3089 114 19 - - PUNCT iajs-3089 114 20 small small ADJ iajs-3089 114 21 compressible compressible ADJ iajs-3089 114 22 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 114 23 3 3 X iajs-3089 114 24 . . PUNCT iajs-3089 115 1 every every DET iajs-3089 115 2 simple simple ADJ iajs-3089 115 3 𝑅 𝑅 PROPN iajs-3089 115 4 βˆ’ βˆ’ PROPN iajs-3089 115 5 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 115 6 is be AUX iajs-3089 115 7 p p ADJ iajs-3089 115 8 - - PUNCT iajs-3089 115 9 small small ADJ iajs-3089 115 10 retractable retractable ADJ iajs-3089 115 11 module module NOUN iajs-3089 115 12 but but CCONJ iajs-3089 115 13 not not PART iajs-3089 115 14 conversely conversely ADV iajs-3089 115 15 , , PUNCT iajs-3089 115 16 because because SCONJ iajs-3089 115 17 𝑍 𝑍 NOUN iajs-3089 115 18 π‘Žπ‘  π‘Žπ‘  PROPN iajs-3089 115 19 𝑍 𝑍 PROPN iajs-3089 115 20 βˆ’ βˆ’ PROPN iajs-3089 115 21 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 115 22 is be AUX iajs-3089 115 23 a a DET iajs-3089 115 24 p p ADJ iajs-3089 115 25 - - PUNCT iajs-3089 115 26 small small ADJ iajs-3089 115 27 retractable retractable ADJ iajs-3089 115 28 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 115 29 but but CCONJ iajs-3089 115 30 not not PART iajs-3089 115 31 simple simple ADJ iajs-3089 115 32 . . PUNCT iajs-3089 116 1 4 4 X iajs-3089 116 2 . . X iajs-3089 116 3 every every DET iajs-3089 116 4 retractable retractable ADJ iajs-3089 116 5 𝑅 𝑅 PROPN iajs-3089 116 6 βˆ’ βˆ’ PROPN iajs-3089 116 7 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 116 8 is be AUX iajs-3089 116 9 p p ADJ iajs-3089 116 10 - - PUNCT iajs-3089 116 11 small small ADJ iajs-3089 116 12 retractable retractable ADJ iajs-3089 116 13 𝑅 𝑅 NOUN iajs-3089 116 14 βˆ’ βˆ’ NOUN iajs-3089 116 15 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 116 16 , , PUNCT iajs-3089 116 17 but but CCONJ iajs-3089 116 18 the the DET iajs-3089 116 19 converse converse NOUN iajs-3089 116 20 is be AUX iajs-3089 116 21 not not PART iajs-3089 116 22 true true ADJ iajs-3089 116 23 . . PUNCT iajs-3089 117 1 5 5 X iajs-3089 117 2 . . X iajs-3089 117 3 every every DET iajs-3089 117 4 semi semi ADJ iajs-3089 117 5 - - ADJ iajs-3089 117 6 simple simple ADJ iajs-3089 117 7 𝑅 𝑅 PROPN iajs-3089 117 8 βˆ’ βˆ’ PROPN iajs-3089 117 9 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 117 10 is be AUX iajs-3089 117 11 p p ADJ iajs-3089 117 12 - - PUNCT iajs-3089 117 13 small small ADJ iajs-3089 117 14 retractable retractable NOUN iajs-3089 117 15 because because SCONJ iajs-3089 117 16 it it PRON iajs-3089 117 17 is be AUX iajs-3089 117 18 retractable retractable ADJ iajs-3089 117 19 . . PUNCT iajs-3089 118 1 6 6 X iajs-3089 118 2 . . X iajs-3089 119 1 every every DET iajs-3089 119 2 compressible compressible ADJ iajs-3089 119 3 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 119 4 is be AUX iajs-3089 119 5 p p ADJ iajs-3089 119 6 - - PUNCT iajs-3089 119 7 small small ADJ iajs-3089 119 8 retractable retractable ADJ iajs-3089 119 9 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 119 10 , , PUNCT iajs-3089 119 11 but but CCONJ iajs-3089 119 12 the the DET iajs-3089 119 13 converse converse NOUN iajs-3089 119 14 is be AUX iajs-3089 119 15 not not PART iajs-3089 119 16 true true ADJ iajs-3089 119 17 for for SCONJ iajs-3089 119 18 instance instance NOUN iajs-3089 119 19 𝑍4 𝑍4 NOUN iajs-3089 119 20 is be AUX iajs-3089 119 21 p p ADJ iajs-3089 119 22 - - PUNCT iajs-3089 119 23 small small ADJ iajs-3089 119 24 retractable retractable NOUN iajs-3089 119 25 but but CCONJ iajs-3089 119 26 not not PART iajs-3089 119 27 p p ADJ iajs-3089 119 28 - - PUNCT iajs-3089 119 29 small small ADJ iajs-3089 119 30 compressible compressible ADJ iajs-3089 119 31 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 119 32 see see VERB iajs-3089 119 33 remarks remark NOUN iajs-3089 119 34 and and CCONJ iajs-3089 119 35 examples example NOUN iajs-3089 119 36 ( ( PUNCT iajs-3089 119 37 3.2)point(5 3.2)point(5 NOUN iajs-3089 119 38 ) ) PUNCT iajs-3089 119 39 . . PUNCT iajs-3089 120 1 7 7 X iajs-3089 120 2 . . X iajs-3089 120 3 a a DET iajs-3089 120 4 homomorphic homomorphic ADJ iajs-3089 120 5 image image NOUN iajs-3089 120 6 of of ADP iajs-3089 120 7 a a DET iajs-3089 120 8 p p ADJ iajs-3089 120 9 - - PUNCT iajs-3089 120 10 small small ADJ iajs-3089 120 11 retractable retractable ADJ iajs-3089 120 12 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 120 13 is be AUX iajs-3089 120 14 a a DET iajs-3089 120 15 p p ADJ iajs-3089 120 16 - - PUNCT iajs-3089 120 17 small small ADJ iajs-3089 120 18 retractable retractable ADJ iajs-3089 120 19 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 120 20 remark(4.3 remark(4.3 ADV iajs-3089 120 21 ): ): PUNCT iajs-3089 120 22 the the DET iajs-3089 120 23 direct direct ADJ iajs-3089 120 24 sum sum NOUN iajs-3089 120 25 of of ADP iajs-3089 120 26 p p NOUN iajs-3089 120 27 - - PUNCT iajs-3089 120 28 small small ADJ iajs-3089 120 29 retractable retractable ADJ iajs-3089 120 30 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 120 31 is be AUX iajs-3089 120 32 p p ADJ iajs-3089 120 33 - - PUNCT iajs-3089 120 34 small small ADJ iajs-3089 120 35 retractable retractable ADJ iajs-3089 120 36 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 120 37 proposition(4.4 proposition(4.4 NOUN iajs-3089 120 38 ): ): PUNCT iajs-3089 120 39 a a DET iajs-3089 120 40 p p ADJ iajs-3089 120 41 - - PUNCT iajs-3089 120 42 small small ADJ iajs-3089 120 43 submodule submodule NOUN iajs-3089 120 44 of of ADP iajs-3089 120 45 p p NOUN iajs-3089 120 46 - - PUNCT iajs-3089 120 47 small small ADJ iajs-3089 120 48 retractable retractable ADJ iajs-3089 120 49 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 120 50 is be AUX iajs-3089 120 51 also also ADV iajs-3089 120 52 p p ADJ iajs-3089 120 53 - - PUNCT iajs-3089 120 54 small small ADJ iajs-3089 120 55 retractable retractable ADJ iajs-3089 120 56 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 120 57 ihjpas ihjpa NOUN iajs-3089 120 58 . . PUNCT iajs-3089 121 1 36 36 NUM iajs-3089 121 2 ( ( PUNCT iajs-3089 121 3 3 3 NUM iajs-3089 121 4 ) ) PUNCT iajs-3089 121 5 2023 2023 NUM iajs-3089 121 6 370 370 NUM iajs-3089 121 7 proof proof NOUN iajs-3089 121 8 : : PUNCT iajs-3089 121 9 let let VERB iajs-3089 121 10 0 0 NUM iajs-3089 121 11 β‰  β‰  PROPN iajs-3089 121 12 𝐾 𝐾 PROPN iajs-3089 121 13 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 121 14 𝑀 𝑀 PROPN iajs-3089 121 15 and and CCONJ iajs-3089 121 16 𝑀 𝑀 PROPN iajs-3089 121 17 be be VERB iajs-3089 121 18 p p ADJ iajs-3089 121 19 - - PUNCT iajs-3089 121 20 small small ADJ iajs-3089 121 21 retractable retractable ADJ iajs-3089 121 22 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 121 23 and and CCONJ iajs-3089 121 24 let let VERB iajs-3089 121 25 0 0 NUM iajs-3089 121 26 β‰  β‰  PROPN iajs-3089 121 27 𝐿 𝐿 PROPN iajs-3089 121 28 ≀ ≀ NOUN iajs-3089 121 29 𝐾 𝐾 PROPN iajs-3089 121 30 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 121 31 𝑀 𝑀 PROPN iajs-3089 121 32 , , PUNCT iajs-3089 121 33 by by ADP iajs-3089 121 34 remarks remark NOUN iajs-3089 121 35 and and CCONJ iajs-3089 121 36 examples example NOUN iajs-3089 121 37 ( ( PUNCT iajs-3089 121 38 1.2)point(3 1.2)point(3 NUM iajs-3089 121 39 ) ) PUNCT iajs-3089 121 40 , , PUNCT iajs-3089 122 1 [ [ X iajs-3089 122 2 3 3 NUM iajs-3089 122 3 ] ] PUNCT iajs-3089 122 4 . . PUNCT iajs-3089 123 1 𝐿 𝐿 PROPN iajs-3089 123 2 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 123 3 𝑀 𝑀 PROPN iajs-3089 123 4 . . PUNCT iajs-3089 124 1 since since SCONJ iajs-3089 124 2 𝑀 𝑀 PROPN iajs-3089 124 3 is be AUX iajs-3089 124 4 p p ADJ iajs-3089 124 5 - - PUNCT iajs-3089 124 6 small small ADJ iajs-3089 124 7 retractable retractable NOUN iajs-3089 124 8 , , PUNCT iajs-3089 124 9 so so ADV iajs-3089 124 10 βˆƒ βˆƒ PROPN iajs-3089 124 11 a a DET iajs-3089 124 12 homomorphism homomorphism NOUN iajs-3089 124 13 𝑓 𝑓 X iajs-3089 124 14 : : PUNCT iajs-3089 124 15 𝑀 𝑀 PROPN iajs-3089 124 16 ⟢ ⟢ PROPN iajs-3089 124 17 𝐿 𝐿 PROPN iajs-3089 124 18 and and CCONJ iajs-3089 124 19 𝑖 𝑖 SYM iajs-3089 124 20 : : PUNCT iajs-3089 124 21 𝐾 𝐾 PROPN iajs-3089 124 22 ⟢ ⟢ NOUN iajs-3089 124 23 𝑀 𝑀 PROPN iajs-3089 124 24 is be AUX iajs-3089 124 25 the the DET iajs-3089 124 26 inclusion inclusion NOUN iajs-3089 124 27 homomorphism homomorphism NOUN iajs-3089 124 28 , , PUNCT iajs-3089 124 29 then then ADV iajs-3089 124 30 𝑓 𝑓 DET iajs-3089 124 31 ∘ ∘ NUM iajs-3089 124 32 𝑖 𝑖 NOUN iajs-3089 124 33 : : PUNCT iajs-3089 124 34 𝐾 𝐾 PROPN iajs-3089 124 35 ⟢ ⟢ NOUN iajs-3089 124 36 𝐿 𝐿 PROPN iajs-3089 124 37 be be AUX iajs-3089 124 38 a a DET iajs-3089 124 39 homomorphism homomorphism NOUN iajs-3089 124 40 . . PUNCT iajs-3089 125 1 therefore therefore ADV iajs-3089 125 2 𝐾 𝐾 PROPN iajs-3089 125 3 is be AUX iajs-3089 125 4 a a DET iajs-3089 125 5 p p ADJ iajs-3089 125 6 - - PUNCT iajs-3089 125 7 small small ADJ iajs-3089 125 8 retractable retractable ADJ iajs-3089 125 9 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 125 10 proposition(4.5 proposition(4.5 NOUN iajs-3089 125 11 ): ): PUNCT iajs-3089 125 12 let let VERB iajs-3089 125 13 𝑀1 𝑀1 NOUN iajs-3089 125 14 and and CCONJ iajs-3089 125 15 𝑀2 𝑀2 PROPN iajs-3089 125 16 be be AUX iajs-3089 125 17 isomorphic isomorphic ADJ iajs-3089 125 18 𝑅 𝑅 PROPN iajs-3089 125 19 βˆ’ βˆ’ PROPN iajs-3089 125 20 π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘ . π‘šπ‘œπ‘‘π‘’π‘™π‘’π‘ . NOUN iajs-3089 125 21 then then ADV iajs-3089 125 22 𝑀1is 𝑀1i VERB iajs-3089 125 23 p p ADJ iajs-3089 125 24 - - PUNCT iajs-3089 125 25 small small ADJ iajs-3089 125 26 retractable retractable ADJ iajs-3089 125 27 if if SCONJ iajs-3089 125 28 and and CCONJ iajs-3089 125 29 only only ADV iajs-3089 125 30 if if SCONJ iajs-3089 125 31 𝑀2is 𝑀2is VERB iajs-3089 125 32 p p ADJ iajs-3089 125 33 - - PUNCT iajs-3089 125 34 small small ADJ iajs-3089 125 35 retractable retractable ADJ iajs-3089 125 36 proof proof NOUN iajs-3089 125 37 : : PUNCT iajs-3089 125 38 let let VERB iajs-3089 125 39 0 0 NUM iajs-3089 125 40 β‰  β‰  PROPN iajs-3089 125 41 𝑁 𝑁 PROPN iajs-3089 125 42 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 125 43 𝑀1 𝑀1 NOUN iajs-3089 125 44 and and CCONJ iajs-3089 125 45 suppose suppose VERB iajs-3089 125 46 that that SCONJ iajs-3089 125 47 𝑀2 𝑀2 PROPN iajs-3089 125 48 is be AUX iajs-3089 125 49 p p ADJ iajs-3089 125 50 - - PUNCT iajs-3089 125 51 small small ADJ iajs-3089 125 52 retractable retractable NOUN iajs-3089 125 53 . . PUNCT iajs-3089 126 1 let let VERB iajs-3089 126 2 𝑓 𝑓 PRON iajs-3089 126 3 : : PUNCT iajs-3089 126 4 𝑀1 𝑀1 PROPN iajs-3089 126 5 ⟢ ⟢ NOUN iajs-3089 126 6 𝑀2 𝑀2 PROPN iajs-3089 126 7 be be AUX iajs-3089 126 8 an an DET iajs-3089 126 9 isomorphism isomorphism NOUN iajs-3089 126 10 . . PUNCT iajs-3089 127 1 then then ADV iajs-3089 127 2 by[3 by[3 VERB iajs-3089 127 3 ] ] X iajs-3089 127 4 0 0 NUM iajs-3089 127 5 β‰  β‰  PROPN iajs-3089 127 6 𝑓(𝑁 𝑓(𝑁 X iajs-3089 127 7 ) ) PUNCT iajs-3089 127 8 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 127 9 𝑀2 𝑀2 PROPN iajs-3089 127 10 . . PUNCT iajs-3089 128 1 put put VERB iajs-3089 128 2 𝐾 𝐾 NOUN iajs-3089 128 3 = = SYM iajs-3089 128 4 𝑓(𝑁 𝑓(𝑁 X iajs-3089 128 5 ) ) PUNCT iajs-3089 128 6 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 128 7 𝑀2 𝑀2 PROPN iajs-3089 128 8 , , PUNCT iajs-3089 128 9 we we PRON iajs-3089 128 10 get get VERB iajs-3089 128 11 β„Ž β„Ž NOUN iajs-3089 128 12 : : PUNCT iajs-3089 128 13 𝑀2 𝑀2 PROPN iajs-3089 128 14 ⟢ ⟢ NOUN iajs-3089 128 15 𝐾 𝐾 PROPN iajs-3089 128 16 is be AUX iajs-3089 128 17 a a DET iajs-3089 128 18 homomorphism homomorphism NOUN iajs-3089 128 19 ( ( PUNCT iajs-3089 128 20 by by ADP iajs-3089 128 21 assumption assumption NOUN iajs-3089 128 22 ) ) PUNCT iajs-3089 128 23 , , PUNCT iajs-3089 128 24 let let VERB iajs-3089 128 25 𝑔 𝑔 PROPN iajs-3089 128 26 = = VERB iajs-3089 128 27 π‘“βˆ’1 π‘“βˆ’1 NUM iajs-3089 128 28 β”‚ β”‚ ADJ iajs-3089 128 29 𝐾 𝐾 PROPN iajs-3089 128 30 , , PUNCT iajs-3089 128 31 then then ADV iajs-3089 128 32 𝑔 𝑔 ADJ iajs-3089 128 33 : : PUNCT iajs-3089 129 1 𝐾 𝐾 PROPN iajs-3089 129 2 ⟢ ⟢ NOUN iajs-3089 129 3 𝑀1 𝑀1 ADV iajs-3089 129 4 is be AUX iajs-3089 129 5 a a DET iajs-3089 129 6 monomorphism monomorphism NOUN iajs-3089 129 7 . . PUNCT iajs-3089 130 1 𝑔(𝐾 𝑔(𝐾 NOUN iajs-3089 130 2 ) ) PUNCT iajs-3089 130 3 = = SYM iajs-3089 130 4 π‘“βˆ’1(𝑓(𝑁 π‘“βˆ’1(𝑓(𝑁 NOUN iajs-3089 130 5 ) ) PUNCT iajs-3089 130 6 ) ) PUNCT iajs-3089 131 1 = = SYM iajs-3089 131 2 𝑁. 𝑁. PROPN iajs-3089 131 3 hence hence ADV iajs-3089 131 4 we we PRON iajs-3089 131 5 have have VERB iajs-3089 131 6 a a DET iajs-3089 131 7 composition composition NOUN iajs-3089 131 8 Ξ· Ξ· NOUN iajs-3089 131 9 = = PROPN iajs-3089 131 10 𝑔 𝑔 PROPN iajs-3089 131 11 ∘ ∘ NOUN iajs-3089 131 12 β„Ž β„Ž PROPN iajs-3089 131 13 ∘ ∘ PROPN iajs-3089 131 14 𝑓. 𝑓. PROPN iajs-3089 131 15 hence hence ADV iajs-3089 131 16 Ξ· Ξ· PROPN iajs-3089 131 17 : : PUNCT iajs-3089 131 18 𝑀1 𝑀1 PROPN iajs-3089 131 19 ⟢ ⟢ NOUN iajs-3089 131 20 𝑁 𝑁 PROPN iajs-3089 131 21 is be AUX iajs-3089 131 22 a a DET iajs-3089 131 23 monomorphism monomorphism NOUN iajs-3089 131 24 . . PUNCT iajs-3089 132 1 therefore therefore ADV iajs-3089 132 2 𝑀1is 𝑀1i VERB iajs-3089 132 3 p p ADJ iajs-3089 132 4 - - PUNCT iajs-3089 132 5 small small ADJ iajs-3089 132 6 retractable retractable ADJ iajs-3089 132 7 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 132 8 proposition(4.7 proposition(4.7 NOUN iajs-3089 132 9 ): ): PUNCT iajs-3089 132 10 let let VERB iajs-3089 132 11 𝑀 𝑀 PRON iajs-3089 132 12 be be AUX iajs-3089 132 13 𝑃𝑆 𝑃𝑆 NOUN iajs-3089 132 14 βˆ’ βˆ’ NOUN iajs-3089 132 15 β„Žπ‘œπ‘™π‘™π‘œπ‘€ β„Žπ‘œπ‘™π‘™π‘œπ‘€ NOUN iajs-3089 132 16 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ VERB iajs-3089 132 17 , , PUNCT iajs-3089 132 18 then then ADV iajs-3089 132 19 the the DET iajs-3089 132 20 following follow VERB iajs-3089 132 21 are be AUX iajs-3089 132 22 equivalent equivalent ADJ iajs-3089 132 23 ( ( PUNCT iajs-3089 132 24 1 1 NUM iajs-3089 132 25 ) ) PUNCT iajs-3089 132 26 𝑀 𝑀 PROPN iajs-3089 132 27 is be AUX iajs-3089 132 28 retractable retractable ADJ iajs-3089 132 29 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 132 30 ( ( PUNCT iajs-3089 132 31 2 2 X iajs-3089 132 32 ) ) PUNCT iajs-3089 132 33 𝑀 𝑀 PROPN iajs-3089 132 34 is be AUX iajs-3089 132 35 p p ADJ iajs-3089 132 36 - - PUNCT iajs-3089 132 37 small small ADJ iajs-3089 132 38 retractable retractable ADJ iajs-3089 132 39 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 132 40 proposition(4.8 proposition(4.8 NOUN iajs-3089 132 41 ): ): PUNCT iajs-3089 132 42 if if SCONJ iajs-3089 132 43 𝑀 𝑀 PROPN iajs-3089 132 44 is be AUX iajs-3089 132 45 p p ADJ iajs-3089 132 46 - - PUNCT iajs-3089 132 47 small small ADJ iajs-3089 132 48 quasi quasi ADJ iajs-3089 132 49 - - ADJ iajs-3089 132 50 dedekind dedekind ADJ iajs-3089 132 51 𝑅 𝑅 PROPN iajs-3089 132 52 βˆ’ βˆ’ PROPN iajs-3089 132 53 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 132 54 , , PUNCT iajs-3089 132 55 then then ADV iajs-3089 132 56 𝑀 𝑀 PROPN iajs-3089 132 57 ca can AUX iajs-3089 132 58 n't not PART iajs-3089 132 59 be be AUX iajs-3089 132 60 p p ADJ iajs-3089 132 61 - - PUNCT iajs-3089 132 62 small small ADJ iajs-3089 132 63 retractable retractable NOUN iajs-3089 132 64 . . PUNCT iajs-3089 133 1 proof proof NOUN iajs-3089 133 2 : : PUNCT iajs-3089 133 3 suppose suppose VERB iajs-3089 133 4 that that SCONJ iajs-3089 133 5 𝑀 𝑀 PROPN iajs-3089 133 6 is be AUX iajs-3089 133 7 p p ADJ iajs-3089 133 8 - - PUNCT iajs-3089 133 9 small small ADJ iajs-3089 133 10 quasi quasi NOUN iajs-3089 133 11 - - ADJ iajs-3089 133 12 dedekind dedekind ADJ iajs-3089 133 13 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 133 14 and and CCONJ iajs-3089 133 15 let let VERB iajs-3089 133 16 𝑁 𝑁 PROPN iajs-3089 133 17 = = PUNCT iajs-3089 133 18 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 133 19 ≀ ≀ PROPN iajs-3089 133 20 𝑀 𝑀 PROPN iajs-3089 133 21 , , PUNCT iajs-3089 133 22 but but CCONJ iajs-3089 133 23 𝑀 𝑀 PROPN iajs-3089 133 24 is be AUX iajs-3089 133 25 psmall psmall ADV iajs-3089 133 26 quasi quasi ADJ iajs-3089 133 27 - - NOUN iajs-3089 133 28 dedekind dedekind ADJ iajs-3089 133 29 , , PUNCT iajs-3089 133 30 then then ADV iajs-3089 133 31 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 133 32 β‰ͺ𝑃 β‰ͺ𝑃 PROPN iajs-3089 133 33 𝑀 𝑀 PROPN iajs-3089 133 34 , , PUNCT iajs-3089 133 35 𝑓 𝑓 DET iajs-3089 133 36 β‰  β‰  PROPN iajs-3089 133 37 0 0 NUM iajs-3089 133 38 , , PUNCT iajs-3089 133 39 thus thus ADV iajs-3089 133 40 π»π‘œπ‘š(𝑀 π»π‘œπ‘š(𝑀 PROPN iajs-3089 133 41 , , PUNCT iajs-3089 133 42 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 133 43 ) ) PUNCT iajs-3089 133 44 = = SYM iajs-3089 134 1 0 0 X iajs-3089 134 2 . . PUNCT iajs-3089 134 3 therefore therefore ADV iajs-3089 134 4 𝑀 𝑀 PROPN iajs-3089 134 5 ca can AUX iajs-3089 134 6 n't not PART iajs-3089 134 7 be be AUX iajs-3089 134 8 p p ADJ iajs-3089 134 9 - - PUNCT iajs-3089 134 10 small small ADJ iajs-3089 134 11 retractable retractable ADJ iajs-3089 134 12 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 134 13 recall recall VERB iajs-3089 134 14 that that SCONJ iajs-3089 134 15 an an DET iajs-3089 134 16 𝑅 𝑅 PROPN iajs-3089 134 17 βˆ’ βˆ’ PROPN iajs-3089 134 18 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 134 19 𝑀 𝑀 PROPN iajs-3089 134 20 is be AUX iajs-3089 134 21 called call VERB iajs-3089 134 22 monoform monoform NOUN iajs-3089 134 23 if if SCONJ iajs-3089 134 24 for for ADP iajs-3089 134 25 each each DET iajs-3089 134 26 non non ADJ iajs-3089 134 27 - - ADJ iajs-3089 134 28 zero zero NUM iajs-3089 134 29 submodule submodule NOUN iajs-3089 134 30 𝑁 𝑁 PROPN iajs-3089 134 31 of of ADP iajs-3089 134 32 𝑀 𝑀 PROPN iajs-3089 134 33 and and CCONJ iajs-3089 134 34 for for ADP iajs-3089 134 35 each each DET iajs-3089 134 36 𝑓 𝑓 PRON iajs-3089 134 37 ∈ ∈ PROPN iajs-3089 134 38 π»π‘œπ‘šπ‘…(𝑁 π»π‘œπ‘šπ‘…(𝑁 NOUN iajs-3089 134 39 , , PUNCT iajs-3089 134 40 𝑀 𝑀 PROPN iajs-3089 134 41 ) ) PUNCT iajs-3089 134 42 , , PUNCT iajs-3089 134 43 𝑓 𝑓 DET iajs-3089 134 44 β‰  β‰  PROPN iajs-3089 134 45 0 0 NUM iajs-3089 134 46 implies imply VERB iajs-3089 134 47 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 134 48 = = PROPN iajs-3089 134 49 0 0 PROPN iajs-3089 134 50 , , PUNCT iajs-3089 134 51 [ [ X iajs-3089 134 52 5 5 NUM iajs-3089 134 53 ] ] PUNCT iajs-3089 134 54 . . PUNCT iajs-3089 135 1 definition(4.9 definition(4.9 NOUN iajs-3089 135 2 ): ): PUNCT iajs-3089 135 3 an an DET iajs-3089 135 4 𝑅 𝑅 PROPN iajs-3089 135 5 βˆ’ βˆ’ PROPN iajs-3089 135 6 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ ADP iajs-3089 135 7 𝑀 𝑀 PROPN iajs-3089 135 8 is be AUX iajs-3089 135 9 called call VERB iajs-3089 135 10 p p ADJ iajs-3089 135 11 - - PUNCT iajs-3089 135 12 small small ADJ iajs-3089 135 13 monoform monoform NOUN iajs-3089 135 14 if if SCONJ iajs-3089 135 15 for for ADP iajs-3089 135 16 each each DET iajs-3089 135 17 non non ADJ iajs-3089 135 18 - - ADJ iajs-3089 135 19 zero zero NUM iajs-3089 135 20 submodule submodule NOUN iajs-3089 135 21 𝑁 𝑁 PROPN iajs-3089 135 22 of of ADP iajs-3089 135 23 𝑀 𝑀 PROPN iajs-3089 135 24 and and CCONJ iajs-3089 135 25 for for ADP iajs-3089 135 26 each each DET iajs-3089 135 27 𝑓 𝑓 PRON iajs-3089 135 28 ∈ ∈ PROPN iajs-3089 135 29 π»π‘œπ‘šπ‘…(𝑁 π»π‘œπ‘šπ‘…(𝑁 NOUN iajs-3089 135 30 , , PUNCT iajs-3089 135 31 𝑀 𝑀 PROPN iajs-3089 135 32 ) ) PUNCT iajs-3089 135 33 , , PUNCT iajs-3089 135 34 𝑓 𝑓 DET iajs-3089 135 35 β‰  β‰  PROPN iajs-3089 135 36 0 0 NUM iajs-3089 135 37 implies imply VERB iajs-3089 135 38 πΎπ‘’π‘Ÿπ‘“ πΎπ‘’π‘Ÿπ‘“ PROPN iajs-3089 135 39 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 135 40 𝑁. 𝑁. PROPN iajs-3089 135 41 remark(4.10 remark(4.10 NOUN iajs-3089 135 42 ): ): PUNCT iajs-3089 135 43 every every DET iajs-3089 135 44 p p ADJ iajs-3089 135 45 - - PUNCT iajs-3089 135 46 small small ADJ iajs-3089 135 47 compressible compressible ADJ iajs-3089 135 48 𝑅 𝑅 PROPN iajs-3089 135 49 βˆ’ βˆ’ PROPN iajs-3089 135 50 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 135 51 is be AUX iajs-3089 135 52 p p ADJ iajs-3089 135 53 - - PUNCT iajs-3089 135 54 small small ADJ iajs-3089 135 55 monoform monoform NOUN iajs-3089 135 56 , , PUNCT iajs-3089 135 57 but but CCONJ iajs-3089 135 58 not not PART iajs-3089 135 59 conversely conversely ADV iajs-3089 135 60 . . PUNCT iajs-3089 136 1 for for ADP iajs-3089 136 2 example example NOUN iajs-3089 136 3 , , PUNCT iajs-3089 136 4 𝑍8 𝑍8 NOUN iajs-3089 136 5 as as ADP iajs-3089 136 6 𝑍 𝑍 PROPN iajs-3089 136 7 βˆ’ βˆ’ PROPN iajs-3089 136 8 π‘šπ‘œπ‘‘π‘’π‘™π‘’ π‘šπ‘œπ‘‘π‘’π‘™π‘’ NOUN iajs-3089 136 9 is be AUX iajs-3089 136 10 p p ADJ iajs-3089 136 11 - - PUNCT iajs-3089 136 12 small small ADJ iajs-3089 136 13 monoform monoform NOUN iajs-3089 136 14 but but CCONJ iajs-3089 136 15 not not PART iajs-3089 136 16 p p ADJ iajs-3089 136 17 - - PUNCT iajs-3089 136 18 small small ADJ iajs-3089 136 19 compressible compressible NOUN iajs-3089 136 20 . . PUNCT iajs-3089 137 1 proposition(4.11 proposition(4.11 NOUN iajs-3089 137 2 ): ): PUNCT iajs-3089 137 3 let let VERB iajs-3089 137 4 𝑀 𝑀 PRON iajs-3089 137 5 be be AUX iajs-3089 137 6 a a DET iajs-3089 137 7 quasi quasi ADJ iajs-3089 137 8 - - ADJ iajs-3089 137 9 dedekind dedekind ADJ iajs-3089 137 10 𝑅 𝑅 PROPN iajs-3089 137 11 βˆ’ βˆ’ PROPN iajs-3089 137 12 π‘šπ‘œπ‘‘π‘’π‘™π‘’. π‘šπ‘œπ‘‘π‘’π‘™π‘’. NOUN iajs-3089 137 13 then then ADV iajs-3089 137 14 𝑀 𝑀 PROPN iajs-3089 137 15 is be AUX iajs-3089 137 16 p p ADJ iajs-3089 137 17 - - PUNCT iajs-3089 137 18 small small ADJ iajs-3089 137 19 monoform monoform NOUN iajs-3089 138 1 if if SCONJ iajs-3089 138 2 and and CCONJ iajs-3089 138 3 only only ADV iajs-3089 138 4 if if SCONJ iajs-3089 138 5 𝑀 𝑀 PROPN iajs-3089 138 6 is be AUX iajs-3089 138 7 p p ADJ iajs-3089 138 8 - - PUNCT iajs-3089 138 9 small small ADJ iajs-3089 138 10 compressible compressible NOUN iajs-3089 138 11 . . PUNCT iajs-3089 139 1 proof proof NOUN iajs-3089 139 2 : : PUNCT iajs-3089 139 3 suppose suppose VERB iajs-3089 139 4 that that SCONJ iajs-3089 139 5 𝑀 𝑀 PROPN iajs-3089 139 6 is be AUX iajs-3089 139 7 p p ADJ iajs-3089 139 8 - - PUNCT iajs-3089 139 9 small small ADJ iajs-3089 139 10 monoform monoform NOUN iajs-3089 139 11 . . PUNCT iajs-3089 140 1 let let VERB iajs-3089 140 2 0 0 NUM iajs-3089 140 3 β‰  β‰  PROPN iajs-3089 140 4 𝑁 𝑁 PROPN iajs-3089 140 5 β‰ͺ𝑃 β‰ͺ𝑃 ADJ iajs-3089 140 6 𝑀 𝑀 PROPN iajs-3089 140 7 , , PUNCT iajs-3089 140 8 then then ADV iajs-3089 140 9 0 0 NUM iajs-3089 140 10 β‰  β‰  PROPN iajs-3089 140 11 𝑓 𝑓 DET iajs-3089 140 12 ∈ ∈ PROPN iajs-3089 140 13 π»π‘œπ‘šπ‘…(𝑁 π»π‘œπ‘šπ‘…(𝑁 NOUN iajs-3089 140 14 , , PUNCT iajs-3089 140 15 𝑀 𝑀 PROPN iajs-3089 140 16 ) ) PUNCT iajs-3089 140 17 . . PUNCT iajs-3089 141 1 since since SCONJ iajs-3089 141 2 𝑀 𝑀 PROPN iajs-3089 141 3 is be AUX iajs-3089 141 4 quasi quasi ADJ iajs-3089 141 5 - - ADJ iajs-3089 141 6 dedekind dedekind ADJ iajs-3089 141 7 , , PUNCT iajs-3089 141 8 then then ADV iajs-3089 141 9 𝑓 𝑓 PRON iajs-3089 141 10 ∘ ∘ PROPN iajs-3089 141 11 𝑔 𝑔 NOUN iajs-3089 141 12 : : PUNCT iajs-3089 141 13 𝑀 𝑀 PROPN iajs-3089 141 14 ⟢ ⟢ AUX iajs-3089 141 15 𝑁 𝑁 PROPN iajs-3089 141 16 ⟢ ⟢ NOUN iajs-3089 141 17 𝑀 𝑀 PROPN iajs-3089 141 18 is be AUX iajs-3089 141 19 a a DET iajs-3089 141 20 monomorphism monomorphism NOUN iajs-3089 141 21 , , PUNCT iajs-3089 141 22 hence hence ADV iajs-3089 141 23 𝑔 𝑔 ADJ iajs-3089 141 24 : : PUNCT iajs-3089 141 25 𝑀 𝑀 PROPN iajs-3089 141 26 ⟢ ⟢ AUX iajs-3089 141 27 𝑁 𝑁 PROPN iajs-3089 141 28 is be AUX iajs-3089 141 29 a a DET iajs-3089 141 30 monomorphism monomorphism NOUN iajs-3089 141 31 . . PUNCT iajs-3089 142 1 thus thus ADV iajs-3089 142 2 𝑀 𝑀 PROPN iajs-3089 142 3 is be AUX iajs-3089 142 4 p p ADJ iajs-3089 142 5 - - PUNCT iajs-3089 142 6 small small ADJ iajs-3089 142 7 compressible compressible NOUN iajs-3089 142 8 . . PUNCT iajs-3089 143 1 conversely conversely ADV iajs-3089 143 2 it it PRON iajs-3089 143 3 is be AUX iajs-3089 143 4 clear clear ADJ iajs-3089 143 5 by by ADP iajs-3089 143 6 remark remark NOUN iajs-3089 143 7 ( ( PUNCT iajs-3089 143 8 4.10 4.10 NUM iajs-3089 143 9 ) ) PUNCT iajs-3089 143 10 . . PUNCT iajs-3089 144 1 5 5 X iajs-3089 144 2 . . X iajs-3089 144 3 conclusion conclusion NOUN iajs-3089 144 4 in in ADP iajs-3089 144 5 this this DET iajs-3089 144 6 work work NOUN iajs-3089 144 7 , , PUNCT iajs-3089 144 8 the the DET iajs-3089 144 9 class class NOUN iajs-3089 144 10 of of ADP iajs-3089 144 11 compressible compressible ADJ iajs-3089 144 12 and and CCONJ iajs-3089 144 13 retractable retractable ADJ iajs-3089 144 14 modules module NOUN iajs-3089 144 15 have have AUX iajs-3089 144 16 been be AUX iajs-3089 144 17 generalized generalize VERB iajs-3089 144 18 to to ADP iajs-3089 144 19 a a DET iajs-3089 144 20 new new ADJ iajs-3089 144 21 concepts concept NOUN iajs-3089 144 22 called call VERB iajs-3089 144 23 p p NOUN iajs-3089 144 24 - - PUNCT iajs-3089 144 25 small small ADJ iajs-3089 144 26 compressible compressible NOUN iajs-3089 144 27 and and CCONJ iajs-3089 144 28 p p ADJ iajs-3089 144 29 - - PUNCT iajs-3089 144 30 small small ADJ iajs-3089 144 31 retractable retractable ADJ iajs-3089 144 32 modules module NOUN iajs-3089 144 33 . . PUNCT iajs-3089 145 1 several several ADJ iajs-3089 145 2 characteristics characteristic NOUN iajs-3089 145 3 of of ADP iajs-3089 145 4 this this DET iajs-3089 145 5 type type NOUN iajs-3089 145 6 of of ADP iajs-3089 145 7 modules module NOUN iajs-3089 145 8 have have AUX iajs-3089 145 9 been be AUX iajs-3089 145 10 studied study VERB iajs-3089 145 11 . . PUNCT iajs-3089 146 1 sufficient sufficient ADJ iajs-3089 146 2 conditions condition NOUN iajs-3089 146 3 under under ADP iajs-3089 146 4 which which PRON iajs-3089 146 5 these these DET iajs-3089 146 6 modules module NOUN iajs-3089 146 7 with with ADP iajs-3089 146 8 compressible compressible ADJ iajs-3089 146 9 and and CCONJ iajs-3089 146 10 retractable retractable ADJ iajs-3089 146 11 are be AUX iajs-3089 146 12 discuss discus VERB iajs-3089 146 13 also also ADV iajs-3089 146 14 we we PRON iajs-3089 146 15 see see VERB iajs-3089 146 16 relations relation NOUN iajs-3089 146 17 between between ADP iajs-3089 146 18 p p NOUN iajs-3089 146 19 - - PUNCT iajs-3089 146 20 small small ADJ iajs-3089 146 21 compressible compressible ADJ iajs-3089 146 22 modules module NOUN iajs-3089 146 23 and and CCONJ iajs-3089 146 24 other other ADJ iajs-3089 146 25 related related ADJ iajs-3089 146 26 modules module NOUN iajs-3089 146 27 as as SCONJ iajs-3089 146 28 psmall psmall NOUN iajs-3089 146 29 retractable retractable ADJ iajs-3089 146 30 module module NOUN iajs-3089 146 31 p p NOUN iajs-3089 146 32 - - PUNCT iajs-3089 146 33 small small ADJ iajs-3089 146 34 quasi quasi NOUN iajs-3089 146 35 - - NOUN iajs-3089 146 36 dedekind dedekind ADJ iajs-3089 146 37 , , PUNCT iajs-3089 146 38 p p NOUN iajs-3089 146 39 - - PUNCT iajs-3089 146 40 small small ADJ iajs-3089 146 41 monoform monoform NOUN iajs-3089 146 42 . . PUNCT iajs-3089 147 1 ihjpas ihjpas PROPN iajs-3089 147 2 . . PUNCT iajs-3089 148 1 36 36 NUM iajs-3089 148 2 ( ( PUNCT iajs-3089 148 3 3 3 NUM iajs-3089 148 4 ) ) PUNCT iajs-3089 148 5 2023 2023 NUM iajs-3089 148 6 371 371 NUM iajs-3089 148 7 references reference NOUN iajs-3089 148 8 1 1 NUM iajs-3089 148 9 . . PUNCT iajs-3089 149 1 wisam wisam PROPN iajs-3089 149 2 , , PUNCT iajs-3089 149 3 a. a. PROPN iajs-3089 149 4 ali ali PROPN iajs-3089 149 5 . . PROPN iajs-3089 149 6 ; ; PUNCT iajs-3089 150 1 nuhad nuhad PROPN iajs-3089 150 2 s. s. PROPN iajs-3089 150 3 al al PROPN iajs-3089 150 4 . . PUNCT iajs-3089 151 1 mothafar mothafar ADV iajs-3089 151 2 .on .on PUNCT iajs-3089 151 3 quasi quasi ADJ iajs-3089 151 4 - - ADJ iajs-3089 151 5 small small ADJ iajs-3089 151 6 prime prime ADJ iajs-3089 151 7 submodules submodule NOUN iajs-3089 151 8 . . PUNCT iajs-3089 152 1 iraqi iraqi ADJ iajs-3089 152 2 journal journal PROPN iajs-3089 152 3 of of ADP iajs-3089 152 4 science science NOUN iajs-3089 152 5 , , PUNCT iajs-3089 152 6 2022,63 2022,63 PROPN iajs-3089 152 7 , , PUNCT iajs-3089 152 8 4,1692 4,1692 NUM iajs-3089 152 9 - - SYM iajs-3089 152 10 1699 1699 NUM iajs-3089 152 11 . . PUNCT iajs-3089 153 1 2 2 NUM iajs-3089 153 2 . . X iajs-3089 153 3 nuhad nuhad PROPN iajs-3089 153 4 , , PUNCT iajs-3089 153 5 s. s. PROPN iajs-3089 153 6 al al PROPN iajs-3089 153 7 - - PUNCT iajs-3089 153 8 mothafar mothafar PROPN iajs-3089 153 9 , , PUNCT iajs-3089 153 10 ; ; PUNCT iajs-3089 154 1 mohammed mohammed PROPN iajs-3089 154 2 b. b. PROPN iajs-3089 154 3 h. h. PROPN iajs-3089 154 4 alhakeem alhakeem PROPN iajs-3089 154 5 . . PUNCT iajs-3089 155 1 nearly nearly ADV iajs-3089 155 2 semiprime semiprime NOUN iajs-3089 155 3 submodules submodule NOUN iajs-3089 155 4 . . PUNCT iajs-3089 156 1 iraqi iraqi ADJ iajs-3089 156 2 journal journal PROPN iajs-3089 156 3 of of ADP iajs-3089 156 4 science science NOUN iajs-3089 156 5 , , PUNCT iajs-3089 156 6 2015 2015 NUM iajs-3089 156 7 , , PUNCT iajs-3089 156 8 56 56 NUM iajs-3089 156 9 , , PUNCT iajs-3089 156 10 4 4 NUM iajs-3089 156 11 , , PUNCT iajs-3089 156 12 3210 3210 NUM iajs-3089 156 13 - - SYM iajs-3089 156 14 3214 3214 NUM iajs-3089 156 15 . . PUNCT iajs-3089 157 1 3 3 X iajs-3089 157 2 . . X iajs-3089 157 3 inam inam PROPN iajs-3089 157 4 , , PUNCT iajs-3089 157 5 m.a m.a PROPN iajs-3089 157 6 . . PROPN iajs-3089 157 7 hadi hadi PROPN iajs-3089 157 8 . . PUNCT iajs-3089 157 9 ; ; PUNCT iajs-3089 157 10 tamadher tamadher NOUN iajs-3089 157 11 a. a. 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