id sid tid token lemma pos american_scientific_journal-6580 1 1 232 232 NUM american_scientific_journal-6580 1 2 american american ADJ american_scientific_journal-6580 1 3 scientific scientific ADJ american_scientific_journal-6580 1 4 research research NOUN american_scientific_journal-6580 1 5 journal journal NOUN american_scientific_journal-6580 1 6 for for ADP american_scientific_journal-6580 1 7 engineering engineering NOUN american_scientific_journal-6580 1 8 , , PUNCT american_scientific_journal-6580 1 9 technology technology NOUN american_scientific_journal-6580 1 10 , , PUNCT american_scientific_journal-6580 1 11 and and CCONJ american_scientific_journal-6580 1 12 sciences science NOUN american_scientific_journal-6580 1 13 ( ( PUNCT american_scientific_journal-6580 1 14 asrjets asrjet NOUN american_scientific_journal-6580 1 15 ) ) PUNCT american_scientific_journal-6580 1 16 issn issn PROPN american_scientific_journal-6580 1 17 ( ( PUNCT american_scientific_journal-6580 1 18 print print NOUN american_scientific_journal-6580 1 19 ) ) PUNCT american_scientific_journal-6580 1 20 2313 2313 NUM american_scientific_journal-6580 1 21 - - SYM american_scientific_journal-6580 1 22 4410 4410 NUM american_scientific_journal-6580 1 23 , , PUNCT american_scientific_journal-6580 1 24 issn issn PROPN american_scientific_journal-6580 1 25 ( ( PUNCT american_scientific_journal-6580 1 26 online online ADJ american_scientific_journal-6580 1 27 ) ) PUNCT american_scientific_journal-6580 1 28 2313 2313 NUM american_scientific_journal-6580 1 29 - - SYM american_scientific_journal-6580 1 30 4402 4402 NUM american_scientific_journal-6580 1 31 © © PROPN american_scientific_journal-6580 1 32 global global ADJ american_scientific_journal-6580 1 33 society society NOUN american_scientific_journal-6580 1 34 of of ADP american_scientific_journal-6580 1 35 scientific scientific ADJ american_scientific_journal-6580 1 36 research research NOUN american_scientific_journal-6580 1 37 and and CCONJ american_scientific_journal-6580 1 38 researchers researcher NOUN american_scientific_journal-6580 1 39 http://asrjetsjournal.org/ http://asrjetsjournal.org/ VERB american_scientific_journal-6580 1 40 weak weak ADJ american_scientific_journal-6580 1 41 ss ss NOUN american_scientific_journal-6580 1 42 - - PUNCT american_scientific_journal-6580 1 43 lifting lift VERB american_scientific_journal-6580 1 44 modules module NOUN american_scientific_journal-6580 1 45 burcu burcu PROPN american_scientific_journal-6580 1 46 nişancı nişancı NOUN american_scientific_journal-6580 1 47 türkmen türkman NOUN american_scientific_journal-6580 1 48 * * PROPN american_scientific_journal-6580 1 49 amasya amasya NOUN american_scientific_journal-6580 1 50 university university NOUN american_scientific_journal-6580 1 51 , , PUNCT american_scientific_journal-6580 1 52 faculty faculty NOUN american_scientific_journal-6580 1 53 of of ADP american_scientific_journal-6580 1 54 arts art NOUN american_scientific_journal-6580 1 55 and and CCONJ american_scientific_journal-6580 1 56 sciences science NOUN american_scientific_journal-6580 1 57 , , PUNCT american_scientific_journal-6580 1 58 department department NOUN american_scientific_journal-6580 1 59 of of ADP american_scientific_journal-6580 1 60 mathematics mathematic NOUN american_scientific_journal-6580 1 61 , , PUNCT american_scientific_journal-6580 1 62 amasya amasya NOUN american_scientific_journal-6580 1 63 , , PUNCT american_scientific_journal-6580 1 64 05100 05100 NUM american_scientific_journal-6580 1 65 , , PUNCT american_scientific_journal-6580 1 66 turkey turkey NOUN american_scientific_journal-6580 1 67 email email NOUN american_scientific_journal-6580 1 68 : : PUNCT american_scientific_journal-6580 1 69 burcunisancie@hotmail.com burcunisancie@hotmail.com X american_scientific_journal-6580 2 1 abstract abstract ADV american_scientific_journal-6580 2 2 let let VERB american_scientific_journal-6580 2 3 m m PRON american_scientific_journal-6580 2 4 be be AUX american_scientific_journal-6580 2 5 a a DET american_scientific_journal-6580 2 6 module module NOUN american_scientific_journal-6580 2 7 . . PUNCT american_scientific_journal-6580 3 1 m m PROPN american_scientific_journal-6580 3 2 is be AUX american_scientific_journal-6580 3 3 called call VERB american_scientific_journal-6580 3 4 weak weak ADJ american_scientific_journal-6580 3 5 ss ss NOUN american_scientific_journal-6580 3 6 - - PUNCT american_scientific_journal-6580 3 7 lifting lifting NOUN american_scientific_journal-6580 3 8 if if SCONJ american_scientific_journal-6580 3 9 it it PRON american_scientific_journal-6580 3 10 is be AUX american_scientific_journal-6580 3 11 ss ss NOUN american_scientific_journal-6580 3 12 - - PUNCT american_scientific_journal-6580 3 13 supplemented supplement VERB american_scientific_journal-6580 3 14 and and CCONJ american_scientific_journal-6580 3 15 its its PRON american_scientific_journal-6580 3 16 ss ss ADJ american_scientific_journal-6580 3 17 - - PUNCT american_scientific_journal-6580 3 18 supplement supplement NOUN american_scientific_journal-6580 3 19 submodules submodule NOUN american_scientific_journal-6580 3 20 are be AUX american_scientific_journal-6580 3 21 direct direct ADJ american_scientific_journal-6580 3 22 summand summand NOUN american_scientific_journal-6580 3 23 . . 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PUNCT american_scientific_journal-6580 5 1 in in ADP american_scientific_journal-6580 5 2 particular particular ADJ american_scientific_journal-6580 5 3 , , PUNCT american_scientific_journal-6580 5 4 we we PRON american_scientific_journal-6580 5 5 show show VERB american_scientific_journal-6580 5 6 that that SCONJ american_scientific_journal-6580 5 7 every every DET american_scientific_journal-6580 5 8 direct direct ADJ american_scientific_journal-6580 5 9 summand summand NOUN american_scientific_journal-6580 5 10 of of ADP american_scientific_journal-6580 5 11 a a DET american_scientific_journal-6580 5 12 weak weak ADJ american_scientific_journal-6580 5 13 ss ss NOUN american_scientific_journal-6580 5 14 - - PUNCT american_scientific_journal-6580 5 15 lifting lift VERB american_scientific_journal-6580 5 16 module module NOUN american_scientific_journal-6580 5 17 is be AUX american_scientific_journal-6580 5 18 weak weak ADJ american_scientific_journal-6580 5 19 ss ss NOUN american_scientific_journal-6580 5 20 - - PUNCT american_scientific_journal-6580 5 21 lifting lifting NOUN american_scientific_journal-6580 5 22 . . PUNCT american_scientific_journal-6580 6 1 moreover moreover ADV american_scientific_journal-6580 6 2 , , PUNCT american_scientific_journal-6580 6 3 we we PRON american_scientific_journal-6580 6 4 prove prove VERB american_scientific_journal-6580 6 5 that that SCONJ american_scientific_journal-6580 6 6 a a DET american_scientific_journal-6580 6 7 ring ring NOUN american_scientific_journal-6580 6 8 r r NOUN american_scientific_journal-6580 6 9 is be AUX american_scientific_journal-6580 6 10 semiperfect semiperfect ADJ american_scientific_journal-6580 6 11 with with ADP american_scientific_journal-6580 6 12 semisimple semisimple NOUN american_scientific_journal-6580 6 13 radical radical ADJ american_scientific_journal-6580 6 14 if if SCONJ american_scientific_journal-6580 6 15 and and CCONJ american_scientific_journal-6580 6 16 only only ADV american_scientific_journal-6580 6 17 if if SCONJ american_scientific_journal-6580 6 18 every every DET american_scientific_journal-6580 6 19 projective projective NOUN american_scientific_journal-6580 6 20 left leave VERB american_scientific_journal-6580 6 21 r r NOUN american_scientific_journal-6580 6 22 - - PUNCT american_scientific_journal-6580 6 23 module module NOUN american_scientific_journal-6580 6 24 is be AUX american_scientific_journal-6580 6 25 weak weak ADJ american_scientific_journal-6580 6 26 ss ss NOUN american_scientific_journal-6580 6 27 - - PUNCT american_scientific_journal-6580 6 28 lifting lifting NOUN american_scientific_journal-6580 6 29 . . PUNCT american_scientific_journal-6580 7 1 keywords keyword NOUN american_scientific_journal-6580 7 2 : : PUNCT american_scientific_journal-6580 7 3 weak weak ADJ american_scientific_journal-6580 7 4 ( ( PUNCT american_scientific_journal-6580 7 5 ss- ss- NUM american_scientific_journal-6580 7 6 ) ) PUNCT american_scientific_journal-6580 7 7 lifting lifting NOUN american_scientific_journal-6580 7 8 module module NOUN american_scientific_journal-6580 7 9 ; ; PUNCT american_scientific_journal-6580 7 10 ss ss ADJ american_scientific_journal-6580 7 11 - - PUNCT american_scientific_journal-6580 7 12 supplemented supplement VERB american_scientific_journal-6580 7 13 module module NOUN american_scientific_journal-6580 7 14 ; ; PUNCT american_scientific_journal-6580 7 15 semiperfect semiperfect ADJ american_scientific_journal-6580 7 16 ring ring NOUN american_scientific_journal-6580 7 17 , , PUNCT american_scientific_journal-6580 7 18 local local ADJ american_scientific_journal-6580 7 19 commutative commutative ADJ american_scientific_journal-6580 7 20 principle principle ADJ american_scientific_journal-6580 7 21 ideal ideal ADJ american_scientific_journal-6580 7 22 ring ring NOUN american_scientific_journal-6580 7 23 . . PUNCT american_scientific_journal-6580 8 1 1 1 X american_scientific_journal-6580 8 2 . . X american_scientific_journal-6580 8 3 introduction introduction NOUN american_scientific_journal-6580 8 4 a a DET american_scientific_journal-6580 8 5 submodule submodule NOUN american_scientific_journal-6580 8 6 n n PROPN american_scientific_journal-6580 8 7 of of ADP american_scientific_journal-6580 8 8 m m PROPN american_scientific_journal-6580 8 9 will will AUX american_scientific_journal-6580 8 10 show show VERB american_scientific_journal-6580 8 11 that that SCONJ american_scientific_journal-6580 8 12 n n NUM american_scientific_journal-6580 8 13 m. m. 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PUNCT american_scientific_journal-6580 9 1 a a DET american_scientific_journal-6580 9 2 submodule submodule NOUN american_scientific_journal-6580 9 3 k k PROPN american_scientific_journal-6580 9 4 of of ADP american_scientific_journal-6580 9 5 m m PROPN american_scientific_journal-6580 9 6 is be AUX american_scientific_journal-6580 9 7 called call VERB american_scientific_journal-6580 9 8 a a DET american_scientific_journal-6580 9 9 supplement supplement NOUN american_scientific_journal-6580 9 10 of of ADP american_scientific_journal-6580 9 11 n n PROPN american_scientific_journal-6580 9 12 in in ADP american_scientific_journal-6580 9 13 m m PROPN american_scientific_journal-6580 9 14 if if SCONJ american_scientific_journal-6580 9 15 m m VERB american_scientific_journal-6580 9 16 = = SYM american_scientific_journal-6580 9 17 n n PROPN american_scientific_journal-6580 9 18 + + CCONJ american_scientific_journal-6580 9 19 k k PROPN american_scientific_journal-6580 9 20 and and CCONJ american_scientific_journal-6580 9 21 n n PRON american_scientific_journal-6580 9 22 k< > X american_scientific_journal-6580 108 36 1 1 NUM american_scientific_journal-6580 108 37 such such ADJ american_scientific_journal-6580 108 38 that that SCONJ american_scientific_journal-6580 108 39 the the DET american_scientific_journal-6580 108 40 set set NOUN american_scientific_journal-6580 108 41 { { PUNCT american_scientific_journal-6580 108 42 k k NOUN american_scientific_journal-6580 108 43 k k PROPN american_scientific_journal-6580 109 1 | | ADV american_scientific_journal-6580 109 2 m m VERB american_scientific_journal-6580 109 3 l l NOUN american_scientific_journal-6580 109 4 jk jk PROPN american_scientific_journal-6580 109 5 } } PUNCT american_scientific_journal-6580 109 6 is be AUX american_scientific_journal-6580 109 7 finite finite ADJ american_scientific_journal-6580 109 8 , , PUNCT american_scientific_journal-6580 109 9 ( ( PUNCT american_scientific_journal-6580 109 10 b b X american_scientific_journal-6580 109 11 ) ) PUNCT american_scientific_journal-6580 109 12 the the DET american_scientific_journal-6580 109 13 ideals ideal NOUN american_scientific_journal-6580 109 14 { { PUNCT american_scientific_journal-6580 109 15 jk jk PROPN american_scientific_journal-6580 110 1 | | ADV american_scientific_journal-6580 110 2 k k PROPN american_scientific_journal-6580 110 3 k k PROPN american_scientific_journal-6580 110 4 } } PUNCT american_scientific_journal-6580 110 5 are be AUX american_scientific_journal-6580 110 6 linearly linearly ADV american_scientific_journal-6580 110 7 ordered order VERB american_scientific_journal-6580 110 8 by by ADP american_scientific_journal-6580 110 9 inclusion inclusion NOUN american_scientific_journal-6580 110 10 , , PUNCT american_scientific_journal-6580 110 11 and and CCONJ american_scientific_journal-6580 110 12 ( ( PUNCT american_scientific_journal-6580 110 13 c c X american_scientific_journal-6580 110 14 ) ) PUNCT american_scientific_journal-6580 111 1 if if SCONJ american_scientific_journal-6580 111 2 ji ji PROPN american_scientific_journal-6580 111 3 jh jh PROPN american_scientific_journal-6580 111 4 then then ADV american_scientific_journal-6580 111 5 mjh mjh PROPN american_scientific_journal-6580 111 6 ji ji PROPN american_scientific_journal-6580 111 7 . . PROPN american_scientific_journal-6580 111 8 proof proof PROPN american_scientific_journal-6580 111 9 . . PUNCT american_scientific_journal-6580 112 1 follows follow VERB american_scientific_journal-6580 112 2 from from ADP american_scientific_journal-6580 112 3 lemma lemma PROPN american_scientific_journal-6580 112 4 2.2 2.2 NUM american_scientific_journal-6580 112 5 and and CCONJ american_scientific_journal-6580 112 6 [ [ X american_scientific_journal-6580 112 7 3 3 NUM american_scientific_journal-6580 112 8 , , PUNCT american_scientific_journal-6580 112 9 proposition proposition NOUN american_scientific_journal-6580 112 10 3.7 3.7 NUM american_scientific_journal-6580 112 11 ] ] PUNCT american_scientific_journal-6580 112 12 . . PUNCT american_scientific_journal-6580 113 1 corollary corollary ADJ american_scientific_journal-6580 113 2 2.11 2.11 NUM american_scientific_journal-6580 113 3 . . 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NOUN american_scientific_journal-6580 114 13 the the DET american_scientific_journal-6580 114 14 following follow VERB american_scientific_journal-6580 114 15 statements statement NOUN american_scientific_journal-6580 114 16 are be AUX american_scientific_journal-6580 114 17 equivalent equivalent ADJ american_scientific_journal-6580 114 18 for for ADP american_scientific_journal-6580 114 19 a a DET american_scientific_journal-6580 114 20 projective projective ADJ american_scientific_journal-6580 114 21 r r NOUN american_scientific_journal-6580 114 22 - - PUNCT american_scientific_journal-6580 114 23 module module NOUN american_scientific_journal-6580 114 24 m m NOUN american_scientific_journal-6580 114 25 and and CCONJ american_scientific_journal-6580 114 26 rad(m rad(m NUM american_scientific_journal-6580 114 27 ) ) PUNCT american_scientific_journal-6580 114 28 soc(m soc(m PROPN american_scientific_journal-6580 114 29 ) ) PUNCT american_scientific_journal-6580 114 30 . . PUNCT american_scientific_journal-6580 115 1 ( ( PUNCT american_scientific_journal-6580 115 2 1 1 X american_scientific_journal-6580 115 3 ) ) PUNCT american_scientific_journal-6580 115 4 m m VERB american_scientific_journal-6580 115 5 is be AUX american_scientific_journal-6580 115 6 lifting lift VERB american_scientific_journal-6580 115 7 ; ; PUNCT american_scientific_journal-6580 115 8 ( ( PUNCT american_scientific_journal-6580 115 9 2 2 X american_scientific_journal-6580 115 10 ) ) PUNCT american_scientific_journal-6580 115 11 m m VERB american_scientific_journal-6580 115 12 is be AUX american_scientific_journal-6580 115 13 weak weak ADJ american_scientific_journal-6580 115 14 lifting lifting NOUN american_scientific_journal-6580 115 15 ; ; PUNCT american_scientific_journal-6580 115 16 ( ( PUNCT american_scientific_journal-6580 115 17 3 3 X american_scientific_journal-6580 115 18 ) ) PUNCT american_scientific_journal-6580 115 19 m m VERB american_scientific_journal-6580 115 20 is be AUX american_scientific_journal-6580 115 21 weak weak ADJ american_scientific_journal-6580 115 22 ss ss NOUN american_scientific_journal-6580 115 23 - - PUNCT american_scientific_journal-6580 115 24 lifting lifting NOUN american_scientific_journal-6580 115 25 ; ; PUNCT american_scientific_journal-6580 115 26 ( ( PUNCT american_scientific_journal-6580 115 27 4 4 X american_scientific_journal-6580 115 28 ) ) PUNCT american_scientific_journal-6580 115 29 m m PROPN american_scientific_journal-6580 115 30 = = NOUN american_scientific_journal-6580 115 31 rak rak ADJ american_scientific_journal-6580 115 32 and and CCONJ american_scientific_journal-6580 115 33 for for ADP american_scientific_journal-6580 115 34 every every DET american_scientific_journal-6580 115 35 pair pair NOUN american_scientific_journal-6580 115 36 ( ( PUNCT american_scientific_journal-6580 115 37 k k NOUN american_scientific_journal-6580 115 38 , , PUNCT american_scientific_journal-6580 115 39 n n CCONJ american_scientific_journal-6580 115 40 ) ) PUNCT american_scientific_journal-6580 115 41 k k NOUN american_scientific_journal-6580 116 1 x x PUNCT american_scientific_journal-6580 116 2 k k X american_scientific_journal-6580 116 3 , , PUNCT american_scientific_journal-6580 116 4 rak rak PROPN american_scientific_journal-6580 116 5 ran run VERB american_scientific_journal-6580 116 6 is be AUX american_scientific_journal-6580 116 7 weak weak ADJ american_scientific_journal-6580 116 8 ss ss NOUN american_scientific_journal-6580 116 9 - - PUNCT american_scientific_journal-6580 116 10 lifting lifting NOUN american_scientific_journal-6580 116 11 . . PUNCT american_scientific_journal-6580 117 1 □ □ PUNCT american_scientific_journal-6580 117 2 american american ADJ american_scientific_journal-6580 117 3 scientific scientific ADJ american_scientific_journal-6580 117 4 research research NOUN american_scientific_journal-6580 117 5 journal journal NOUN american_scientific_journal-6580 117 6 for for ADP american_scientific_journal-6580 117 7 engineering engineering NOUN american_scientific_journal-6580 117 8 , , PUNCT american_scientific_journal-6580 117 9 technology technology NOUN american_scientific_journal-6580 117 10 , , PUNCT american_scientific_journal-6580 117 11 and and CCONJ american_scientific_journal-6580 117 12 sciences science NOUN american_scientific_journal-6580 117 13 ( ( PUNCT american_scientific_journal-6580 117 14 asrjets asrjet NOUN american_scientific_journal-6580 117 15 ) ) PUNCT american_scientific_journal-6580 117 16 ( ( PUNCT american_scientific_journal-6580 117 17 2020 2020 NUM american_scientific_journal-6580 117 18 ) ) PUNCT american_scientific_journal-6580 117 19 volume volume NOUN american_scientific_journal-6580 117 20 74 74 NUM american_scientific_journal-6580 117 21 , , PUNCT american_scientific_journal-6580 117 22 no no DET american_scientific_journal-6580 117 23 1 1 NUM american_scientific_journal-6580 117 24 , , PUNCT american_scientific_journal-6580 117 25 pp pp ADV american_scientific_journal-6580 117 26 232 232 NUM american_scientific_journal-6580 117 27 - - SYM american_scientific_journal-6580 117 28 236 236 NUM american_scientific_journal-6580 117 29 236 236 NUM american_scientific_journal-6580 117 30 in in ADP american_scientific_journal-6580 117 31 the the DET american_scientific_journal-6580 117 32 remaining remain VERB american_scientific_journal-6580 117 33 indication indication NOUN american_scientific_journal-6580 117 34 of of ADP american_scientific_journal-6580 117 35 this this DET american_scientific_journal-6580 117 36 article article NOUN american_scientific_journal-6580 117 37 we we PRON american_scientific_journal-6580 117 38 denote denote VERB american_scientific_journal-6580 117 39 bm(k bm(k PROPN american_scientific_journal-6580 117 40 , , PUNCT american_scientific_journal-6580 117 41 k k PROPN american_scientific_journal-6580 117 42 + + PROPN american_scientific_journal-6580 117 43 1 1 X american_scientific_journal-6580 117 44 ) ) PUNCT american_scientific_journal-6580 117 45 the the DET american_scientific_journal-6580 117 46 direct direct ADJ american_scientific_journal-6580 117 47 sum sum NOUN american_scientific_journal-6580 117 48 of of ADP american_scientific_journal-6580 117 49 arbitrarily arbitrarily ADV american_scientific_journal-6580 117 50 many many ADJ american_scientific_journal-6580 117 51 copies copy NOUN american_scientific_journal-6580 117 52 of of ADP american_scientific_journal-6580 117 53 r/ r/ ADV american_scientific_journal-6580 117 54 and and CCONJ american_scientific_journal-6580 117 55 r/ r/ ADV american_scientific_journal-6580 117 56 where where SCONJ american_scientific_journal-6580 117 57 m m NOUN american_scientific_journal-6580 117 58 is be AUX american_scientific_journal-6580 117 59 a a DET american_scientific_journal-6580 117 60 maximal maximal ADJ american_scientific_journal-6580 117 61 ideal ideal NOUN american_scientific_journal-6580 117 62 of of ADP american_scientific_journal-6580 117 63 r r NOUN american_scientific_journal-6580 117 64 and and CCONJ american_scientific_journal-6580 117 65 k k PROPN american_scientific_journal-6580 117 66 is be AUX american_scientific_journal-6580 117 67 a a DET american_scientific_journal-6580 117 68 non non ADJ american_scientific_journal-6580 117 69 - - ADJ american_scientific_journal-6580 117 70 negative negative ADJ american_scientific_journal-6580 117 71 integer integer NOUN american_scientific_journal-6580 117 72 . . PUNCT american_scientific_journal-6580 118 1 theorem theorem VERB american_scientific_journal-6580 118 2 2.12 2.12 NUM american_scientific_journal-6580 118 3 . . PUNCT american_scientific_journal-6580 119 1 let let VERB american_scientific_journal-6580 119 2 r r PRON american_scientific_journal-6580 119 3 be be AUX american_scientific_journal-6580 119 4 a a DET american_scientific_journal-6580 119 5 local local ADJ american_scientific_journal-6580 119 6 commutative commutative ADJ american_scientific_journal-6580 119 7 principal principal ADJ american_scientific_journal-6580 119 8 ideal ideal NOUN american_scientific_journal-6580 119 9 ring ring NOUN american_scientific_journal-6580 119 10 with with ADP american_scientific_journal-6580 119 11 maximal maximal ADJ american_scientific_journal-6580 119 12 ideal ideal ADJ american_scientific_journal-6580 119 13 m. m. NOUN american_scientific_journal-6580 119 14 if if SCONJ american_scientific_journal-6580 119 15 m m NOUN american_scientific_journal-6580 119 16 is be AUX american_scientific_journal-6580 119 17 a a DET american_scientific_journal-6580 119 18 projective projective ADJ american_scientific_journal-6580 119 19 rmodule rmodule NOUN american_scientific_journal-6580 119 20 and and CCONJ american_scientific_journal-6580 119 21 rad(m rad(m NOUN american_scientific_journal-6580 119 22 ) ) PUNCT american_scientific_journal-6580 119 23 soc(m soc(m PROPN american_scientific_journal-6580 119 24 ) ) PUNCT american_scientific_journal-6580 119 25 , , PUNCT american_scientific_journal-6580 119 26 then then ADV american_scientific_journal-6580 119 27 the the DET american_scientific_journal-6580 119 28 following following ADJ american_scientific_journal-6580 119 29 statements statement NOUN american_scientific_journal-6580 119 30 are be AUX american_scientific_journal-6580 119 31 equivalent equivalent ADJ american_scientific_journal-6580 119 32 : : PUNCT american_scientific_journal-6580 119 33 ( ( PUNCT american_scientific_journal-6580 119 34 1 1 X american_scientific_journal-6580 119 35 ) ) PUNCT american_scientific_journal-6580 120 1 m m VERB american_scientific_journal-6580 120 2 is be AUX american_scientific_journal-6580 120 3 weak weak ADJ american_scientific_journal-6580 120 4 lifting lifting NOUN american_scientific_journal-6580 120 5 ; ; PUNCT american_scientific_journal-6580 120 6 ( ( PUNCT american_scientific_journal-6580 120 7 2 2 X american_scientific_journal-6580 120 8 ) ) PUNCT american_scientific_journal-6580 120 9 m m VERB american_scientific_journal-6580 120 10 is be AUX american_scientific_journal-6580 120 11 weak weak ADJ american_scientific_journal-6580 120 12 ss ss NOUN american_scientific_journal-6580 120 13 - - PUNCT american_scientific_journal-6580 120 14 lifting lifting NOUN american_scientific_journal-6580 120 15 ; ; PUNCT american_scientific_journal-6580 120 16 ( ( PUNCT american_scientific_journal-6580 120 17 3 3 X american_scientific_journal-6580 120 18 ) ) PUNCT american_scientific_journal-6580 120 19 m m NOUN american_scientific_journal-6580 120 20 bm(k bm(k NOUN american_scientific_journal-6580 120 21 , , PUNCT american_scientific_journal-6580 120 22 k k PROPN american_scientific_journal-6580 121 1 + + PROPN american_scientific_journal-6580 121 2 1 1 X american_scientific_journal-6580 121 3 ) ) PUNCT american_scientific_journal-6580 121 4 or or CCONJ american_scientific_journal-6580 121 5 m m VERB american_scientific_journal-6580 121 6 r r NOUN american_scientific_journal-6580 121 7 ( ( PUNCT american_scientific_journal-6580 121 8 a a NOUN american_scientific_journal-6580 121 9 ) ) PUNCT american_scientific_journal-6580 121 10 for for ADP american_scientific_journal-6580 121 11 some some DET american_scientific_journal-6580 121 12 non non ADJ american_scientific_journal-6580 121 13 - - ADJ american_scientific_journal-6580 121 14 negative negative ADJ american_scientific_journal-6580 121 15 integers integer NOUN american_scientific_journal-6580 121 16 a a DET american_scientific_journal-6580 121 17 and and CCONJ american_scientific_journal-6580 121 18 k. k. NOUN american_scientific_journal-6580 121 19 proof proof NOUN american_scientific_journal-6580 121 20 . . PUNCT american_scientific_journal-6580 122 1 clear clear ADJ american_scientific_journal-6580 122 2 by by ADP american_scientific_journal-6580 122 3 proposition proposition NOUN american_scientific_journal-6580 122 4 2.10 2.10 NUM american_scientific_journal-6580 122 5 . . PUNCT american_scientific_journal-6580 123 1 □ □ PUNCT american_scientific_journal-6580 123 2 references reference NOUN american_scientific_journal-6580 123 3 [ [ X american_scientific_journal-6580 123 4 1 1 NUM american_scientific_journal-6580 123 5 ] ] PUNCT american_scientific_journal-6580 123 6 . . PUNCT american_scientific_journal-6580 124 1 j. j. PROPN american_scientific_journal-6580 124 2 clark clark PROPN american_scientific_journal-6580 124 3 , , PUNCT american_scientific_journal-6580 124 4 c. c. PROPN american_scientific_journal-6580 124 5 lomp lomp PROPN american_scientific_journal-6580 124 6 , , PUNCT american_scientific_journal-6580 124 7 n. n. NOUN american_scientific_journal-6580 124 8 vanaja vanaja PROPN american_scientific_journal-6580 124 9 , , PUNCT american_scientific_journal-6580 124 10 and and CCONJ american_scientific_journal-6580 124 11 r. r. PROPN american_scientific_journal-6580 124 12 wisbauer wisbauer NOUN american_scientific_journal-6580 124 13 , , PUNCT american_scientific_journal-6580 124 14 lifting lift VERB american_scientific_journal-6580 124 15 modules module NOUN american_scientific_journal-6580 124 16 supplements supplement NOUN american_scientific_journal-6580 124 17 and and CCONJ american_scientific_journal-6580 124 18 projectivity projectivity NOUN american_scientific_journal-6580 124 19 in in ADP american_scientific_journal-6580 124 20 module module NOUN american_scientific_journal-6580 124 21 theory theory NOUN american_scientific_journal-6580 124 22 , , PUNCT american_scientific_journal-6580 124 23 basel basel PROPN american_scientific_journal-6580 124 24 . . PROPN american_scientific_journal-6580 124 25 boston boston PROPN american_scientific_journal-6580 124 26 . . PUNCT american_scientific_journal-6580 125 1 berlin berlin PROPN american_scientific_journal-6580 125 2 , , PUNCT american_scientific_journal-6580 125 3 birkhauser birkhaus ADJ american_scientific_journal-6580 125 4 verlag verlag NOUN american_scientific_journal-6580 125 5 , , PUNCT american_scientific_journal-6580 125 6 2000 2000 NUM american_scientific_journal-6580 125 7 . . PUNCT american_scientific_journal-6580 126 1 [ [ X american_scientific_journal-6580 126 2 2 2 NUM american_scientific_journal-6580 126 3 ] ] PUNCT american_scientific_journal-6580 126 4 . . PUNCT american_scientific_journal-6580 127 1 f. f. PROPN american_scientific_journal-6580 127 2 kasch kasch PROPN american_scientific_journal-6580 127 3 , , PUNCT american_scientific_journal-6580 127 4 modules module NOUN american_scientific_journal-6580 127 5 and and CCONJ american_scientific_journal-6580 127 6 rings ring NOUN american_scientific_journal-6580 127 7 , , PUNCT american_scientific_journal-6580 127 8 london london PROPN american_scientific_journal-6580 127 9 new new PROPN american_scientific_journal-6580 127 10 york york PROPN american_scientific_journal-6580 127 11 , , PUNCT american_scientific_journal-6580 127 12 academic academic ADJ american_scientific_journal-6580 127 13 press press NOUN american_scientific_journal-6580 127 14 , , PUNCT american_scientific_journal-6580 127 15 1982 1982 NUM american_scientific_journal-6580 127 16 . . PUNCT american_scientific_journal-6580 128 1 [ [ X american_scientific_journal-6580 128 2 3 3 NUM american_scientific_journal-6580 128 3 ] ] PUNCT american_scientific_journal-6580 128 4 . . PUNCT american_scientific_journal-6580 129 1 d. d. PROPN american_scientific_journal-6580 129 2 keskin keskin PROPN american_scientific_journal-6580 129 3 tütuncü tütuncü PROPN american_scientific_journal-6580 129 4 and and CCONJ american_scientific_journal-6580 129 5 r. r. PROPN american_scientific_journal-6580 129 6 tribak tribak PROPN american_scientific_journal-6580 129 7 , , PUNCT american_scientific_journal-6580 129 8 ” " PUNCT american_scientific_journal-6580 129 9 on on ADP american_scientific_journal-6580 129 10 lifting lift VERB american_scientific_journal-6580 129 11 modules module NOUN american_scientific_journal-6580 129 12 and and CCONJ american_scientific_journal-6580 129 13 weak weak ADJ american_scientific_journal-6580 129 14 lifting lifting NOUN american_scientific_journal-6580 129 15 modules module NOUN american_scientific_journal-6580 129 16 ” " PUNCT american_scientific_journal-6580 129 17 , , PUNCT american_scientific_journal-6580 129 18 kyungpook kyungpook PROPN american_scientific_journal-6580 129 19 math.j math.j PROPN american_scientific_journal-6580 129 20 . . PUNCT american_scientific_journal-6580 129 21 vol vol NOUN american_scientific_journal-6580 129 22 . . PROPN american_scientific_journal-6580 130 1 45 45 NUM american_scientific_journal-6580 130 2 , , PUNCT american_scientific_journal-6580 130 3 pp.445 pp.445 NOUN american_scientific_journal-6580 130 4 - - PUNCT american_scientific_journal-6580 130 5 453 453 NUM american_scientific_journal-6580 130 6 , , PUNCT american_scientific_journal-6580 130 7 ( ( PUNCT american_scientific_journal-6580 130 8 2005 2005 NUM american_scientific_journal-6580 130 9 ) ) PUNCT american_scientific_journal-6580 130 10 . . PUNCT american_scientific_journal-6580 131 1 [ [ X american_scientific_journal-6580 131 2 4 4 NUM american_scientific_journal-6580 131 3 ] ] PUNCT american_scientific_journal-6580 131 4 . . PUNCT american_scientific_journal-6580 132 1 e. e. PROPN american_scientific_journal-6580 132 2 kaynar kaynar PROPN american_scientific_journal-6580 132 3 , , PUNCT american_scientific_journal-6580 132 4 h. h. PROPN american_scientific_journal-6580 132 5 calısıcı calısıcı PROPN american_scientific_journal-6580 132 6 , , PUNCT american_scientific_journal-6580 132 7 and and CCONJ american_scientific_journal-6580 132 8 e. e. PROPN american_scientific_journal-6580 132 9 türkmen türkmen PROPN american_scientific_journal-6580 132 10 , , PUNCT american_scientific_journal-6580 132 11 “ " PUNCT american_scientific_journal-6580 132 12 ss ss ADJ american_scientific_journal-6580 132 13 - - PUNCT american_scientific_journal-6580 132 14 supplemented supplement VERB american_scientific_journal-6580 132 15 modules module NOUN american_scientific_journal-6580 132 16 ” " PUNCT american_scientific_journal-6580 132 17 , , PUNCT american_scientific_journal-6580 132 18 commun commun PROPN american_scientific_journal-6580 132 19 . . PUNCT american_scientific_journal-6580 133 1 fac fac PROPN american_scientific_journal-6580 133 2 . . PUNCT american_scientific_journal-6580 133 3 sci sci PROPN american_scientific_journal-6580 133 4 . . PROPN american_scientific_journal-6580 133 5 univ univ PROPN american_scientific_journal-6580 133 6 . . PUNCT american_scientific_journal-6580 134 1 ank ank PROPN american_scientific_journal-6580 134 2 . . PROPN american_scientific_journal-6580 134 3 ser ser PROPN american_scientific_journal-6580 134 4 . . PUNCT american_scientific_journal-6580 135 1 a1 a1 NOUN american_scientific_journal-6580 135 2 math math NOUN american_scientific_journal-6580 135 3 . . PUNCT american_scientific_journal-6580 136 1 stat stat PROPN american_scientific_journal-6580 136 2 . . PUNCT american_scientific_journal-6580 137 1 vol vol NOUN american_scientific_journal-6580 137 2 . . PUNCT american_scientific_journal-6580 138 1 69(1 69(1 NUM american_scientific_journal-6580 138 2 ) ) PUNCT american_scientific_journal-6580 138 3 , , PUNCT american_scientific_journal-6580 138 4 pp.473 pp.473 NOUN american_scientific_journal-6580 138 5 - - ADJ american_scientific_journal-6580 138 6 485 485 NUM american_scientific_journal-6580 138 7 , , PUNCT american_scientific_journal-6580 138 8 2020 2020 NUM american_scientific_journal-6580 138 9 . . PUNCT american_scientific_journal-6580 139 1 [ [ X american_scientific_journal-6580 139 2 5 5 NUM american_scientific_journal-6580 139 3 ] ] PUNCT american_scientific_journal-6580 139 4 . . PUNCT american_scientific_journal-6580 140 1 c. c. PROPN american_scientific_journal-6580 140 2 nebiyev nebiyev PROPN american_scientific_journal-6580 140 3 and and CCONJ american_scientific_journal-6580 140 4 a. a. NOUN american_scientific_journal-6580 140 5 pancar pancar NOUN american_scientific_journal-6580 140 6 , , PUNCT american_scientific_journal-6580 140 7 “ " PUNCT american_scientific_journal-6580 140 8 on on ADP american_scientific_journal-6580 140 9 strongly strongly ADV american_scientific_journal-6580 140 10 -supplemented -supplemente VERB american_scientific_journal-6580 140 11 modules module NOUN american_scientific_journal-6580 140 12 ” " PUNCT american_scientific_journal-6580 140 13 , , PUNCT american_scientific_journal-6580 140 14 ukrainian ukrainian ADJ american_scientific_journal-6580 140 15 math.j math.j NOUN american_scientific_journal-6580 140 16 . . PUNCT american_scientific_journal-6580 140 17 vol vol NOUN american_scientific_journal-6580 140 18 . . PUNCT american_scientific_journal-6580 140 19 63(5 63(5 NOUN american_scientific_journal-6580 140 20 ) ) PUNCT american_scientific_journal-6580 140 21 , , PUNCT american_scientific_journal-6580 140 22 pp pp ADP american_scientific_journal-6580 140 23 . . PUNCT american_scientific_journal-6580 141 1 662 662 NUM american_scientific_journal-6580 141 2 - - SYM american_scientific_journal-6580 141 3 667 667 NUM american_scientific_journal-6580 141 4 , , PUNCT american_scientific_journal-6580 141 5 2011 2011 NUM american_scientific_journal-6580 141 6 . . PUNCT american_scientific_journal-6580 142 1 [ [ X american_scientific_journal-6580 142 2 6 6 NUM american_scientific_journal-6580 142 3 ] ] PUNCT american_scientific_journal-6580 142 4 . . PUNCT american_scientific_journal-6580 143 1 d.w d.w PROPN american_scientific_journal-6580 143 2 . . PROPN american_scientific_journal-6580 143 3 sharpe sharpe PROPN american_scientific_journal-6580 143 4 and and CCONJ american_scientific_journal-6580 143 5 p. p. PROPN american_scientific_journal-6580 143 6 vamos vamos PROPN american_scientific_journal-6580 143 7 , , PUNCT american_scientific_journal-6580 143 8 injective injective ADJ american_scientific_journal-6580 143 9 modules module NOUN american_scientific_journal-6580 143 10 , , PUNCT american_scientific_journal-6580 143 11 cambridge cambridge NOUN american_scientific_journal-6580 143 12 tracts tract NOUN american_scientific_journal-6580 143 13 in in ADP american_scientific_journal-6580 143 14 mathematics mathematics PROPN american_scientific_journal-6580 143 15 and and CCONJ american_scientific_journal-6580 143 16 mathematical mathematical ADJ american_scientific_journal-6580 143 17 physics physics PROPN american_scientific_journal-6580 143 18 , , PUNCT american_scientific_journal-6580 143 19 cambridge cambridge PROPN american_scientific_journal-6580 143 20 university university PROPN american_scientific_journal-6580 143 21 press press NOUN american_scientific_journal-6580 143 22 , , PUNCT american_scientific_journal-6580 143 23 1972 1972 NUM american_scientific_journal-6580 143 24 . . PUNCT american_scientific_journal-6580 144 1 [ [ X american_scientific_journal-6580 144 2 7 7 NUM american_scientific_journal-6580 144 3 ] ] PUNCT american_scientific_journal-6580 144 4 . . PUNCT american_scientific_journal-6580 145 1 r. r. PROPN american_scientific_journal-6580 145 2 wisbauer wisbauer PROPN american_scientific_journal-6580 145 3 , , PUNCT american_scientific_journal-6580 145 4 foundations foundation NOUN american_scientific_journal-6580 145 5 of of ADP american_scientific_journal-6580 145 6 modules module NOUN american_scientific_journal-6580 145 7 and and CCONJ american_scientific_journal-6580 145 8 rings ring NOUN american_scientific_journal-6580 145 9 theory theory NOUN american_scientific_journal-6580 145 10 , , PUNCT american_scientific_journal-6580 145 11 gordon gordon PROPN american_scientific_journal-6580 145 12 and and CCONJ american_scientific_journal-6580 145 13 breach breach NOUN american_scientific_journal-6580 145 14 , , PUNCT american_scientific_journal-6580 145 15 springer springer NOUN american_scientific_journal-6580 145 16 - - PUNCT american_scientific_journal-6580 145 17 verlag verlag PROPN american_scientific_journal-6580 145 18 , , PUNCT american_scientific_journal-6580 145 19 1991 1991 NUM american_scientific_journal-6580 145 20 . . PUNCT american_scientific_journal-6580 146 1 [ [ X american_scientific_journal-6580 146 2 8 8 NUM american_scientific_journal-6580 146 3 ] ] PUNCT american_scientific_journal-6580 146 4 . . PUNCT american_scientific_journal-6580 147 1 o. o. PROPN american_scientific_journal-6580 147 2 zarisky zarisky PROPN american_scientific_journal-6580 147 3 and and CCONJ american_scientific_journal-6580 147 4 p. p. PROPN american_scientific_journal-6580 147 5 samuel samuel PROPN american_scientific_journal-6580 147 6 , , PUNCT american_scientific_journal-6580 147 7 commutative commutative ADJ american_scientific_journal-6580 147 8 algebra algebra NOUN american_scientific_journal-6580 147 9 , , PUNCT american_scientific_journal-6580 147 10 new new ADJ american_scientific_journal-6580 147 11 - - PUNCT american_scientific_journal-6580 147 12 york york NOUN american_scientific_journal-6580 147 13 , , PUNCT american_scientific_journal-6580 147 14 heidelberg heidelberg PROPN american_scientific_journal-6580 147 15 , , PUNCT american_scientific_journal-6580 147 16 berlin berlin PROPN american_scientific_journal-6580 147 17 , , PUNCT american_scientific_journal-6580 147 18 springer springer NOUN american_scientific_journal-6580 147 19 - - PUNCT american_scientific_journal-6580 147 20 verlag verlag PROPN american_scientific_journal-6580 147 21 , , PUNCT american_scientific_journal-6580 147 22 1979 1979 NUM american_scientific_journal-6580 147 23 . . 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PUNCT american_scientific_journal-6580 151 3 zhang zhang PROPN american_scientific_journal-6580 151 4 , , PUNCT american_scientific_journal-6580 151 5 “ " PUNCT american_scientific_journal-6580 151 6 small small ADJ american_scientific_journal-6580 151 7 - - PUNCT american_scientific_journal-6580 151 8 essential essential ADJ american_scientific_journal-6580 151 9 submodules submodule NOUN american_scientific_journal-6580 151 10 and and CCONJ american_scientific_journal-6580 151 11 morita morita PROPN american_scientific_journal-6580 151 12 duality duality PROPN american_scientific_journal-6580 151 13 ” " PUNCT american_scientific_journal-6580 151 14 , , PUNCT american_scientific_journal-6580 151 15 southeast southeast ADJ american_scientific_journal-6580 151 16 asian asian ADJ american_scientific_journal-6580 151 17 bulletin bulletin NOUN american_scientific_journal-6580 151 18 of of ADP american_scientific_journal-6580 151 19 mathematics mathematic NOUN american_scientific_journal-6580 151 20 , , PUNCT american_scientific_journal-6580 151 21 vol vol NOUN american_scientific_journal-6580 151 22 . . PROPN american_scientific_journal-6580 151 23 35 35 NUM american_scientific_journal-6580 151 24 ( ( PUNCT american_scientific_journal-6580 151 25 2011 2011 NUM american_scientific_journal-6580 151 26 ) ) PUNCT american_scientific_journal-6580 151 27 , , PUNCT american_scientific_journal-6580 151 28 pp.1051 pp.1051 PROPN american_scientific_journal-6580 151 29 - - PUNCT american_scientific_journal-6580 151 30 1062 1062 NUM american_scientific_journal-6580 151 31 , , PUNCT american_scientific_journal-6580 151 32 2011 2011 NUM american_scientific_journal-6580 151 33 . . PUNCT american_scientific_journal-6580 152 1 [ [ X american_scientific_journal-6580 152 2 10 10 NUM american_scientific_journal-6580 152 3 ] ] PUNCT american_scientific_journal-6580 152 4 . . PUNCT american_scientific_journal-6580 153 1 h. h. PROPN american_scientific_journal-6580 153 2 zoschinger zoschinger PROPN american_scientific_journal-6580 153 3 , , PUNCT american_scientific_journal-6580 153 4 “ " PUNCT american_scientific_journal-6580 153 5 komplemente komplemente X american_scientific_journal-6580 153 6 als als VERB american_scientific_journal-6580 153 7 direkte direkte PROPN american_scientific_journal-6580 153 8 summanden summanden PROPN american_scientific_journal-6580 153 9 ” " PUNCT american_scientific_journal-6580 153 10 , , PUNCT american_scientific_journal-6580 153 11 arch arch NOUN american_scientific_journal-6580 153 12 . . PUNCT american_scientific_journal-6580 154 1 math math NOUN american_scientific_journal-6580 154 2 . . PUNCT american_scientific_journal-6580 155 1 vol vol NOUN american_scientific_journal-6580 155 2 . . 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