id sid tid token lemma pos iajs-1818 1 1 microsoft microsoft PROPN iajs-1818 1 2 word word NOUN iajs-1818 1 3 452 452 NUM iajs-1818 1 4 - - SYM iajs-1818 1 5 462 462 NUM iajs-1818 1 6 ihsciconf ihsciconf NOUN iajs-1818 1 7 2017 2017 NUM iajs-1818 1 8 special special ADJ iajs-1818 1 9 issue issue NOUN iajs-1818 1 10 ibn ibn PROPN iajs-1818 1 11 al al PROPN iajs-1818 1 12 - - PUNCT iajs-1818 1 13 haitham haitham PROPN iajs-1818 1 14 journal journal PROPN iajs-1818 1 15 for for ADP iajs-1818 1 16 pure pure ADJ iajs-1818 1 17 and and CCONJ iajs-1818 1 18 applied apply VERB iajs-1818 1 19 science science NOUN iajs-1818 1 20 https://doi.org/ https://doi.org/ NOUN iajs-1818 1 21 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 1 22 for for ADP iajs-1818 1 23 more more ADJ iajs-1818 1 24 information information NOUN iajs-1818 1 25 about about ADP iajs-1818 1 26 the the DET iajs-1818 1 27 conference conference NOUN iajs-1818 1 28 please please INTJ iajs-1818 1 29 visit visit VERB iajs-1818 1 30 the the DET iajs-1818 1 31 websites website NOUN iajs-1818 1 32 : : PUNCT iajs-1818 1 33 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 1 34     SPACE iajs-1818 1 35 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1818 1 36       SPACE iajs-1818 1 37 mathematics|452 mathematics|452 NOUN iajs-1818 1 38         SPACE iajs-1818 1 39 coclosed coclose VERB iajs-1818 1 40 rickart rickart NOUN iajs-1818 1 41 modules module NOUN iajs-1818 1 42 ghaleb ghaleb PROPN iajs-1818 1 43 ahmed ahme VERB iajs-1818 1 44 galeb.ahmud3@gmail.com galeb.ahmud3@gmail.com X iajs-1818 2 1 dept.of dept.of PROPN iajs-1818 2 2 mathematics mathematics PROPN iajs-1818 2 3 / / SYM iajs-1818 2 4 college college NOUN iajs-1818 2 5 of of ADP iajs-1818 2 6 education education NOUN iajs-1818 2 7 for for ADP iajs-1818 2 8 pure pure ADJ iajs-1818 2 9 science science NOUN iajs-1818 2 10 ( ( PUNCT iajs-1818 2 11 ibn ibn PROPN iajs-1818 2 12 - - PUNCT iajs-1818 2 13 alhaitham)/ alhaitham)/ PROPN iajs-1818 2 14 university university NOUN iajs-1818 2 15 of of ADP iajs-1818 2 16 baghdad baghdad PROPN iajs-1818 2 17 abstract abstract ADV iajs-1818 2 18 let let VERB iajs-1818 2 19 𝑀 𝑀 PROPN iajs-1818 2 20 be be AUX iajs-1818 2 21 a a DET iajs-1818 2 22 right right ADJ iajs-1818 2 23 module module NOUN iajs-1818 2 24 over over ADP iajs-1818 2 25 an an DET iajs-1818 2 26 arbitrary arbitrary ADJ iajs-1818 2 27 ring ring NOUN iajs-1818 2 28 𝑅 𝑅 NOUN iajs-1818 2 29 with with ADP iajs-1818 2 30 identity identity NOUN iajs-1818 2 31 and and CCONJ iajs-1818 2 32 s s PART iajs-1818 2 33 end end NOUN iajs-1818 2 34 𝑀 𝑀 PROPN iajs-1818 2 35 . . PUNCT iajs-1818 3 1 in in ADP iajs-1818 3 2 this this DET iajs-1818 3 3 work work NOUN iajs-1818 3 4 , , PUNCT iajs-1818 3 5 the the DET iajs-1818 3 6 coclosed coclose VERB iajs-1818 3 7 rickart rickart NOUN iajs-1818 3 8 modules module NOUN iajs-1818 3 9 as as ADP iajs-1818 3 10 a a DET iajs-1818 3 11 generalization generalization NOUN iajs-1818 3 12 of of ADP iajs-1818 3 13 rickart rickart NOUN iajs-1818 3 14 modules module NOUN iajs-1818 3 15 is be AUX iajs-1818 3 16 given give VERB iajs-1818 3 17 . . PUNCT iajs-1818 4 1 we we PRON iajs-1818 4 2 say say VERB iajs-1818 4 3 a a DET iajs-1818 4 4 module module NOUN iajs-1818 4 5 𝑀 𝑀 PROPN iajs-1818 4 6 over over ADP iajs-1818 4 7 𝑅 𝑅 PROPN iajs-1818 4 8 coclosed coclosed ADJ iajs-1818 4 9 rickart rickart NOUN iajs-1818 4 10 if if SCONJ iajs-1818 4 11 for for ADP iajs-1818 4 12 each each DET iajs-1818 4 13 𝑓 𝑓 DET iajs-1818 4 14 ∈ ∈ PROPN iajs-1818 4 15 end end NOUN iajs-1818 4 16 𝑀 𝑀 PROPN iajs-1818 4 17 , , PUNCT iajs-1818 4 18 ker ker X iajs-1818 4 19 𝑓 𝑓 PRON iajs-1818 4 20 is be AUX iajs-1818 4 21 a a DET iajs-1818 4 22 coclosed coclosed ADJ iajs-1818 4 23 submodule submodule NOUN iajs-1818 4 24 of of ADP iajs-1818 4 25 𝑀. 𝑀. PROPN iajs-1818 4 26 basic basic ADJ iajs-1818 4 27 results result NOUN iajs-1818 4 28 over over ADP iajs-1818 4 29 this this DET iajs-1818 4 30 paper paper NOUN iajs-1818 4 31 are be AUX iajs-1818 4 32 introduced introduce VERB iajs-1818 4 33 and and CCONJ iajs-1818 4 34 connections connection NOUN iajs-1818 4 35 between between ADP iajs-1818 4 36 these these DET iajs-1818 4 37 modules module NOUN iajs-1818 4 38 and and CCONJ iajs-1818 4 39 otherwise otherwise ADV iajs-1818 4 40 notions notion NOUN iajs-1818 4 41 are be AUX iajs-1818 4 42 investigated investigate VERB iajs-1818 4 43 . . PUNCT iajs-1818 5 1 keywords keyword NOUN iajs-1818 5 2 : : PUNCT iajs-1818 5 3 coclosed coclose VERB iajs-1818 5 4 rickart rickart NOUN iajs-1818 5 5 modules module NOUN iajs-1818 5 6 , , PUNCT iajs-1818 5 7 relatively relatively ADV iajs-1818 5 8 coclosed coclosed ADJ iajs-1818 5 9 rickart rickart NOUN iajs-1818 5 10 modules module NOUN iajs-1818 5 11 , , PUNCT iajs-1818 5 12 rickart rickart NOUN iajs-1818 5 13 modules module NOUN iajs-1818 5 14 , , PUNCT iajs-1818 5 15 cosemisimple cosemisimple NOUN iajs-1818 5 16 modules module NOUN iajs-1818 5 17 , , PUNCT iajs-1818 5 18 modules module NOUN iajs-1818 5 19 with with ADP iajs-1818 5 20 ccip ccip PROPN iajs-1818 5 21 . . PUNCT iajs-1818 6 1 ihsciconf ihsciconf PROPN iajs-1818 6 2 2017 2017 NUM iajs-1818 6 3 special special ADJ iajs-1818 6 4 issue issue NOUN iajs-1818 6 5 ibn ibn PROPN iajs-1818 6 6 al al PROPN iajs-1818 6 7 - - PUNCT iajs-1818 6 8 haitham haitham PROPN iajs-1818 6 9 journal journal PROPN iajs-1818 6 10 for for ADP iajs-1818 6 11 pure pure ADJ iajs-1818 6 12 and and CCONJ iajs-1818 6 13 applied apply VERB iajs-1818 6 14 science science NOUN iajs-1818 6 15 https://doi.org/ https://doi.org/ NOUN iajs-1818 6 16 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 6 17 for for ADP iajs-1818 6 18 more more ADJ iajs-1818 6 19 information information NOUN iajs-1818 6 20 about about ADP iajs-1818 6 21 the the DET iajs-1818 6 22 conference conference NOUN iajs-1818 6 23 please please INTJ iajs-1818 6 24 visit visit VERB iajs-1818 6 25 the the DET iajs-1818 6 26 websites website NOUN iajs-1818 6 27 : : PUNCT iajs-1818 6 28 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 6 29     SPACE iajs-1818 6 30 www.ihsciconf.org www.ihsciconf.org ADV iajs-1818 6 31       SPACE iajs-1818 6 32 mathematics|453 mathematics|453 NOUN iajs-1818 6 33         SPACE iajs-1818 6 34 1 1 NUM iajs-1818 6 35 . . PUNCT iajs-1818 7 1 introduction introduction NOUN iajs-1818 7 2 𝑅 𝑅 PROPN iajs-1818 7 3 is be AUX iajs-1818 7 4 denoted denote VERB iajs-1818 7 5 a a DET iajs-1818 7 6 ring ring NOUN iajs-1818 7 7 has have VERB iajs-1818 7 8 unity unity NOUN iajs-1818 7 9 and and CCONJ iajs-1818 7 10 𝑀 𝑀 PROPN iajs-1818 7 11 is be AUX iajs-1818 7 12 studied study VERB iajs-1818 7 13 as as ADP iajs-1818 7 14 a a DET iajs-1818 7 15 left left ADJ iajs-1818 7 16 sright sright NOUN iajs-1818 7 17 𝑅-bimodule 𝑅-bimodule PROPN iajs-1818 7 18 where where SCONJ iajs-1818 7 19 s s AUX iajs-1818 7 20 end end NOUN iajs-1818 7 21 𝑀 𝑀 PROPN iajs-1818 7 22 is be AUX iajs-1818 7 23 the the DET iajs-1818 7 24 endomorphism endomorphism NOUN iajs-1818 7 25 ring ring NOUN iajs-1818 7 26 of of ADP iajs-1818 7 27 𝑀. 𝑀. PROPN iajs-1818 7 28 also also ADV iajs-1818 7 29 𝑟 𝑟 X iajs-1818 7 30 𝑆𝑓 𝑆𝑓 NOUN iajs-1818 7 31 𝑟 𝑟 X iajs-1818 7 32 𝑓 𝑓 DET iajs-1818 7 33 ker ker NOUN iajs-1818 7 34 𝑓 𝑓 PRON iajs-1818 7 35 𝑚 𝑚 PROPN iajs-1818 7 36 ∈ ∈ PROPN iajs-1818 7 37 𝑀 𝑀 NOUN iajs-1818 7 38 | | ADV iajs-1818 7 39 𝑓 𝑓 ADV iajs-1818 7 40 𝑚 𝑚 PROPN iajs-1818 7 41 0 0 NUM iajs-1818 7 42 is be AUX iajs-1818 7 43 the the DET iajs-1818 7 44 right right ADJ iajs-1818 7 45 annihilator annihilator NOUN iajs-1818 7 46 of of ADP iajs-1818 7 47 each each DET iajs-1818 7 48 element element NOUN iajs-1818 7 49 𝑓 𝑓 DET iajs-1818 7 50 ∈ ∈ NOUN iajs-1818 7 51 𝑆. 𝑆. NOUN iajs-1818 7 52 in in ADP iajs-1818 7 53 [ [ X iajs-1818 7 54 2 2 NUM iajs-1818 7 55 ] ] PUNCT iajs-1818 7 56 is be AUX iajs-1818 7 57 presented present VERB iajs-1818 7 58 a a DET iajs-1818 7 59 generalization generalization NOUN iajs-1818 7 60 of of ADP iajs-1818 7 61 rickart rickart NOUN iajs-1818 7 62 modules module NOUN iajs-1818 7 63 by by ADP iajs-1818 7 64 using use VERB iajs-1818 7 65 the the DET iajs-1818 7 66 concept concept NOUN iajs-1818 7 67 of of ADP iajs-1818 7 68 purity purity NOUN iajs-1818 7 69 . . PUNCT iajs-1818 8 1 further far ADV iajs-1818 8 2 , , PUNCT iajs-1818 8 3 dual dual ADJ iajs-1818 8 4 coclosed coclose VERB iajs-1818 8 5 rickart rickart NOUN iajs-1818 8 6 modules module NOUN iajs-1818 8 7 is be AUX iajs-1818 8 8 introduced introduce VERB iajs-1818 8 9 in in ADP iajs-1818 8 10 [ [ X iajs-1818 8 11 3 3 NUM iajs-1818 8 12 ] ] PUNCT iajs-1818 8 13 . . PUNCT iajs-1818 9 1 we we PRON iajs-1818 9 2 say say VERB iajs-1818 9 3 a a DET iajs-1818 9 4 submodule submodule PROPN iajs-1818 9 5 𝐿 𝐿 PROPN iajs-1818 9 6 of of ADP iajs-1818 9 7 a a DET iajs-1818 9 8 module module NOUN iajs-1818 9 9 𝑀 𝑀 PROPN iajs-1818 9 10 is be AUX iajs-1818 9 11 coclosed coclose VERB iajs-1818 9 12 in in ADP iajs-1818 9 13 𝑀 𝑀 PROPN iajs-1818 9 14 when when SCONJ iajs-1818 9 15 ≪ ≪ PUNCT iajs-1818 9 16 then then ADV iajs-1818 9 17 𝐿 𝐿 PROPN iajs-1818 9 18 𝐾 𝐾 PROPN iajs-1818 9 19 for for ADP iajs-1818 9 20 each each DET iajs-1818 9 21 𝐾 𝐾 PROPN iajs-1818 9 22 ⊆ ⊆ NUM iajs-1818 9 23 𝐿 𝐿 PROPN iajs-1818 10 1 [ [ X iajs-1818 10 2 1 1 NUM iajs-1818 10 3 ] ] PUNCT iajs-1818 10 4 . . PUNCT iajs-1818 11 1 equivalently equivalently ADV iajs-1818 11 2 , , PUNCT iajs-1818 11 3 for for ADP iajs-1818 11 4 each each DET iajs-1818 11 5 proper proper ADJ iajs-1818 11 6 submodule submodule NOUN iajs-1818 11 7 𝐾 𝐾 PROPN iajs-1818 11 8 ⊂ ⊂ PROPN iajs-1818 11 9 𝐿 𝐿 PROPN iajs-1818 11 10 , , PUNCT iajs-1818 11 11 there there PRON iajs-1818 11 12 is be VERB iajs-1818 11 13 a a DET iajs-1818 11 14 submodule submodule NOUN iajs-1818 11 15 𝑁 𝑁 PROPN iajs-1818 11 16 of of ADP iajs-1818 11 17 𝑀 𝑀 PROPN iajs-1818 11 18 where where SCONJ iajs-1818 11 19 𝐿 𝐿 PROPN iajs-1818 11 20 𝑁 𝑁 PROPN iajs-1818 11 21 𝑀 𝑀 PROPN iajs-1818 11 22 while while SCONJ iajs-1818 11 23 𝐾 𝐾 PROPN iajs-1818 11 24 𝑁 𝑁 PROPN iajs-1818 11 25 𝑀 𝑀 PROPN iajs-1818 11 26 iff iff PROPN iajs-1818 11 27 𝐿 𝐿 PROPN iajs-1818 11 28 is be AUX iajs-1818 11 29 coclosed coclose VERB iajs-1818 11 30 of of ADP iajs-1818 11 31 𝑀. 𝑀. PROPN iajs-1818 11 32 our our PRON iajs-1818 11 33 goal goal NOUN iajs-1818 11 34 of of ADP iajs-1818 11 35 this this DET iajs-1818 11 36 research research NOUN iajs-1818 11 37 is be AUX iajs-1818 11 38 to to PART iajs-1818 11 39 present present VERB iajs-1818 11 40 and and CCONJ iajs-1818 11 41 discuss discuss VERB iajs-1818 11 42 another another DET iajs-1818 11 43 generalization generalization NOUN iajs-1818 11 44 of of ADP iajs-1818 11 45 rickart rickart NOUN iajs-1818 11 46 modules module NOUN iajs-1818 11 47 namely namely ADV iajs-1818 11 48 coclosed coclose VERB iajs-1818 11 49 rickart rickart NOUN iajs-1818 11 50 modules module NOUN iajs-1818 11 51 . . PUNCT iajs-1818 12 1 a a DET iajs-1818 12 2 module module NOUN iajs-1818 12 3 𝑀 𝑀 PROPN iajs-1818 12 4 over over ADP iajs-1818 12 5 𝑅 𝑅 PROPN iajs-1818 12 6 is be AUX iajs-1818 12 7 said say VERB iajs-1818 12 8 coclosed coclose VERB iajs-1818 12 9 rickart rickart NOUN iajs-1818 12 10 when when SCONJ iajs-1818 12 11 each each DET iajs-1818 12 12 𝑓 𝑓 PRON iajs-1818 12 13 ∈ ∈ PROPN iajs-1818 12 14 end end NOUN iajs-1818 12 15 𝑀 𝑀 PROPN iajs-1818 12 16 , , PUNCT iajs-1818 12 17 ker ker X iajs-1818 12 18 𝑓 𝑓 PRON iajs-1818 12 19 is be AUX iajs-1818 12 20 a a DET iajs-1818 12 21 coclosed coclosed NOUN iajs-1818 12 22 of of ADP iajs-1818 12 23 𝑀. 𝑀. PROPN iajs-1818 12 24 other other ADJ iajs-1818 12 25 studies study NOUN iajs-1818 12 26 in in ADP iajs-1818 12 27 [ [ X iajs-1818 12 28 5],[6],[7],[8],[11 5],[6],[7],[8],[11 NUM iajs-1818 12 29 ] ] PUNCT iajs-1818 12 30 and and CCONJ iajs-1818 12 31 [ [ X iajs-1818 12 32 12 12 NUM iajs-1818 12 33 ] ] PUNCT iajs-1818 12 34 are be AUX iajs-1818 12 35 related relate VERB iajs-1818 12 36 topics topic NOUN iajs-1818 12 37 . . PUNCT iajs-1818 13 1 the the DET iajs-1818 13 2 paper paper NOUN iajs-1818 13 3 is be AUX iajs-1818 13 4 divided divide VERB iajs-1818 13 5 into into ADP iajs-1818 13 6 four four NUM iajs-1818 13 7 sections section NOUN iajs-1818 13 8 . . PUNCT iajs-1818 14 1 section section NOUN iajs-1818 14 2 two two NUM iajs-1818 14 3 , , PUNCT iajs-1818 14 4 includes include VERB iajs-1818 14 5 introducing introduce VERB iajs-1818 14 6 the the DET iajs-1818 14 7 concept concept NOUN iajs-1818 14 8 of of ADP iajs-1818 14 9 coclosed coclose VERB iajs-1818 14 10 rickart rickart NOUN iajs-1818 14 11 modules module NOUN iajs-1818 14 12 and and CCONJ iajs-1818 14 13 providing provide VERB iajs-1818 14 14 facts fact NOUN iajs-1818 14 15 of of ADP iajs-1818 14 16 this this DET iajs-1818 14 17 argument.while argument.while NOUN iajs-1818 14 18 direct direct ADJ iajs-1818 14 19 summands summand NOUN iajs-1818 14 20 of of ADP iajs-1818 14 21 coclosed coclose VERB iajs-1818 14 22 rickart rickart NOUN iajs-1818 14 23 modules module NOUN iajs-1818 14 24 are be AUX iajs-1818 14 25 shown show VERB iajs-1818 14 26 to to PART iajs-1818 14 27 inherit inherit VERB iajs-1818 14 28 the the DET iajs-1818 14 29 property property NOUN iajs-1818 14 30     SPACE iajs-1818 14 31 ( ( PUNCT iajs-1818 14 32 proposition proposition NOUN iajs-1818 14 33 2.6 2.6 NUM iajs-1818 14 34 ) ) PUNCT iajs-1818 14 35 , , PUNCT iajs-1818 14 36 this this PRON iajs-1818 14 37 is be AUX iajs-1818 14 38 not not PART iajs-1818 14 39 so so ADV iajs-1818 14 40 for for ADP iajs-1818 14 41 direct direct ADJ iajs-1818 14 42 sums sum NOUN iajs-1818 14 43 ( ( PUNCT iajs-1818 14 44 remark remark VERB iajs-1818 14 45 2.7).we 2.7).we NUM iajs-1818 14 46 obtain obtain VERB iajs-1818 14 47 a a DET iajs-1818 14 48 condition condition NOUN iajs-1818 14 49 which which PRON iajs-1818 14 50 allows allow VERB iajs-1818 14 51 direct direct ADJ iajs-1818 14 52 sums sum NOUN iajs-1818 14 53 of of ADP iajs-1818 14 54     SPACE iajs-1818 14 55 coclosed coclose VERB iajs-1818 14 56 rickart rickart NOUN iajs-1818 14 57 modules module NOUN iajs-1818 14 58 is be AUX iajs-1818 14 59 coclosed coclose VERB iajs-1818 14 60 rickart rickart NOUN iajs-1818 14 61     SPACE iajs-1818 14 62 ( ( PUNCT iajs-1818 14 63 proposition proposition NOUN iajs-1818 14 64 2.8 2.8 NUM iajs-1818 14 65 ) ) PUNCT iajs-1818 14 66 . . PUNCT iajs-1818 15 1 section section NOUN iajs-1818 15 2 three three NUM iajs-1818 15 3 is be AUX iajs-1818 15 4 devoted devote VERB iajs-1818 15 5 to to PART iajs-1818 15 6 look look VERB iajs-1818 15 7 for for ADP iajs-1818 15 8 any any DET iajs-1818 15 9 connection connection NOUN iajs-1818 15 10 between between ADP iajs-1818 15 11 coclosed coclose VERB iajs-1818 15 12 rickart rickart NOUN iajs-1818 15 13 modules module NOUN iajs-1818 15 14 and and CCONJ iajs-1818 15 15 other other ADJ iajs-1818 15 16 modules module NOUN iajs-1818 15 17 . . PUNCT iajs-1818 15 18     SPACE iajs-1818 16 1 we we PRON iajs-1818 16 2 see see VERB iajs-1818 16 3 that that PRON iajs-1818 16 4 coclosed coclose VERB iajs-1818 16 5 rickart rickart NOUN iajs-1818 16 6 modules module NOUN iajs-1818 16 7     SPACE iajs-1818 16 8 and and CCONJ iajs-1818 16 9 rickart rickart NOUN iajs-1818 16 10 modules module NOUN iajs-1818 16 11     SPACE iajs-1818 16 12 coincide coincide NOUN iajs-1818 16 13 in in ADP iajs-1818 16 14 lifiting lifiting NOUN iajs-1818 16 15 modules module NOUN iajs-1818 16 16 ( ( PUNCT iajs-1818 16 17 proposition proposition NOUN iajs-1818 16 18 3.4 3.4 NUM iajs-1818 16 19 ) ) PUNCT iajs-1818 16 20 . . PUNCT iajs-1818 17 1 in in ADP iajs-1818 17 2 section section NOUN iajs-1818 17 3 four four NUM iajs-1818 17 4 , , PUNCT iajs-1818 17 5 we we PRON iajs-1818 17 6 present present VERB iajs-1818 17 7 and and CCONJ iajs-1818 17 8 study study VERB iajs-1818 17 9 the the DET iajs-1818 17 10 concept concept NOUN iajs-1818 17 11 of of ADP iajs-1818 17 12 relatively relatively ADV iajs-1818 17 13 coclosed coclosed ADJ iajs-1818 17 14 rickart rickart NOUN iajs-1818 17 15 modules module NOUN iajs-1818 17 16 . . PUNCT iajs-1818 18 1 by by ADP iajs-1818 18 2 using use VERB iajs-1818 18 3 the the DET iajs-1818 18 4 ccip ccip NOUN iajs-1818 18 5 , , PUNCT iajs-1818 18 6 we we PRON iajs-1818 18 7 will will AUX iajs-1818 18 8 provide provide VERB iajs-1818 18 9 a a DET iajs-1818 18 10 condition condition NOUN iajs-1818 18 11 for for SCONJ iajs-1818 18 12 modules module NOUN iajs-1818 18 13 to to PART iajs-1818 18 14 be be AUX iajs-1818 18 15 relatively relatively ADV iajs-1818 18 16 coclosed coclose VERB iajs-1818 18 17 rickart rickart NOUN iajs-1818 18 18 ( ( PUNCT iajs-1818 18 19 for for ADP iajs-1818 18 20 example example NOUN iajs-1818 18 21 , , PUNCT iajs-1818 18 22 theorem theorem VERB iajs-1818 18 23 4.7 4.7 NUM iajs-1818 18 24 , , PUNCT iajs-1818 18 25     SPACE iajs-1818 18 26 proposition proposition NOUN iajs-1818 18 27 4.11 4.11 NUM iajs-1818 18 28 ) ) PUNCT iajs-1818 18 29 where where SCONJ iajs-1818 18 30 an an DET iajs-1818 18 31 𝑅-module 𝑅-module PROPN iajs-1818 18 32 𝑀 𝑀 PROPN iajs-1818 18 33 has have VERB iajs-1818 18 34 the the DET iajs-1818 18 35 coclosed coclose VERB iajs-1818 18 36 intersection intersection NOUN iajs-1818 18 37 property property NOUN iajs-1818 18 38 ( ( PUNCT iajs-1818 18 39 in in ADP iajs-1818 18 40 brief brief ADJ iajs-1818 18 41 ccip ccip PROPN iajs-1818 18 42 ) ) PUNCT iajs-1818 18 43 if if SCONJ iajs-1818 18 44 , , PUNCT iajs-1818 18 45 the the DET iajs-1818 18 46 intersection intersection NOUN iajs-1818 18 47 of of ADP iajs-1818 18 48 any any DET iajs-1818 18 49 two two NUM iajs-1818 18 50 coclosed coclosed ADJ iajs-1818 18 51 submodules submodule NOUN iajs-1818 18 52 of of ADP iajs-1818 18 53 𝑀 𝑀 PROPN iajs-1818 18 54 is be AUX iajs-1818 18 55 coclosed coclose VERB iajs-1818 18 56 [ [ X iajs-1818 18 57 4 4 NUM iajs-1818 18 58 ] ] PUNCT iajs-1818 18 59 . . PUNCT iajs-1818 19 1 many many ADJ iajs-1818 19 2 results result NOUN iajs-1818 19 3 are be AUX iajs-1818 19 4 investigated investigate VERB iajs-1818 19 5 , , PUNCT iajs-1818 19 6 we we PRON iajs-1818 19 7 see see VERB iajs-1818 19 8 that that SCONJ iajs-1818 19 9 the the DET iajs-1818 19 10 rings ring NOUN iajs-1818 19 11 𝑅 𝑅 PROPN iajs-1818 19 12 for for ADP iajs-1818 19 13 which which PRON iajs-1818 19 14 each each DET iajs-1818 19 15 right right ADJ iajs-1818 19 16 module module NOUN iajs-1818 19 17 over over ADP iajs-1818 19 18 𝑅 𝑅 PROPN iajs-1818 19 19 relatively relatively ADJ iajs-1818 19 20 rickart rickart NOUN iajs-1818 19 21 are be AUX iajs-1818 19 22 precisely precisely ADV iajs-1818 19 23 right right ADJ iajs-1818 19 24 cosemisimple cosemisimple NOUN iajs-1818 19 25 rings ring NOUN iajs-1818 19 26 ( ( PUNCT iajs-1818 19 27 theorem theorem NOUN iajs-1818 19 28 4.13 4.13 NUM iajs-1818 19 29 ) ) PUNCT iajs-1818 19 30 . . PUNCT iajs-1818 20 1 coclosed coclose VERB iajs-1818 20 2 rickart rickart NOUN iajs-1818 20 3 modules module VERB iajs-1818 20 4 a a DET iajs-1818 20 5 comprehensive comprehensive ADJ iajs-1818 20 6 study study NOUN iajs-1818 20 7 of of ADP iajs-1818 20 8 coclosed coclose VERB iajs-1818 20 9 rickart rickart NOUN iajs-1818 20 10 modules module NOUN iajs-1818 20 11 is be AUX iajs-1818 20 12 given give VERB iajs-1818 20 13 in in ADP iajs-1818 20 14 this this DET iajs-1818 20 15 section section NOUN iajs-1818 20 16 . . PUNCT iajs-1818 21 1 we we PRON iajs-1818 21 2 provide provide VERB iajs-1818 21 3 several several ADJ iajs-1818 21 4 characterizations characterization NOUN iajs-1818 21 5 and and CCONJ iajs-1818 21 6 some some DET iajs-1818 21 7 properties property NOUN iajs-1818 21 8 of of ADP iajs-1818 21 9 this this PRON iajs-1818 21 10 of of ADP iajs-1818 21 11 modules module NOUN iajs-1818 21 12 are be AUX iajs-1818 21 13 investigated investigate VERB iajs-1818 21 14 . . PUNCT iajs-1818 22 1 we we PRON iajs-1818 22 2 begin begin VERB iajs-1818 22 3 with with ADP iajs-1818 22 4 the the DET iajs-1818 22 5 next next ADJ iajs-1818 22 6 . . PUNCT iajs-1818 23 1 definition definition NOUN iajs-1818 23 2 2.1 2.1 NUM iajs-1818 23 3 . . PUNCT iajs-1818 24 1 we we PRON iajs-1818 24 2 call call VERB iajs-1818 24 3 a a DET iajs-1818 24 4 module module NOUN iajs-1818 24 5 𝑀 𝑀 PROPN iajs-1818 24 6 over over ADP iajs-1818 24 7 𝑅 𝑅 PROPN iajs-1818 24 8 is be AUX iajs-1818 24 9 coclosed coclose VERB iajs-1818 24 10 rickart rickart NOUN iajs-1818 24 11 when when SCONJ iajs-1818 24 12 each each DET iajs-1818 24 13 𝑓 𝑓 PRON iajs-1818 24 14 ∈ ∈ PROPN iajs-1818 24 15 end end NOUN iajs-1818 24 16 𝑀 𝑀 PROPN iajs-1818 24 17 , , PUNCT iajs-1818 24 18 ker ker X iajs-1818 24 19 𝑓 𝑓 PRON iajs-1818 24 20 is be AUX iajs-1818 24 21 a a DET iajs-1818 24 22 coclosed coclosed ADJ iajs-1818 24 23 submodule submodule NOUN iajs-1818 24 24 of of ADP iajs-1818 24 25 𝑀. 𝑀. PROPN iajs-1818 24 26 remarks remark NOUN iajs-1818 24 27 and and CCONJ iajs-1818 24 28 examples example NOUN iajs-1818 24 29 clearly clearly ADV iajs-1818 24 30 each each DET iajs-1818 24 31 cosemisimple cosemisimple NOUN iajs-1818 24 32 module module NOUN iajs-1818 24 33 is be AUX iajs-1818 24 34 coclosed coclose VERB iajs-1818 24 35 rickart rickart NOUN iajs-1818 24 36 , , PUNCT iajs-1818 24 37 but but CCONJ iajs-1818 24 38 not not PART iajs-1818 24 39 conversely conversely ADV iajs-1818 24 40 , , PUNCT iajs-1818 24 41 where where SCONJ iajs-1818 24 42 an an DET iajs-1818 24 43 𝑅-module 𝑅-module PROPN iajs-1818 24 44 𝑀 𝑀 PROPN iajs-1818 24 45 is be AUX iajs-1818 24 46 called call VERB iajs-1818 24 47 cosemisimple cosemisimple NOUN iajs-1818 24 48 if if SCONJ iajs-1818 24 49 each each DET iajs-1818 24 50 submodule submodule NOUN iajs-1818 24 51 of of ADP iajs-1818 24 52 𝑀 𝑀 PROPN iajs-1818 24 53 is be AUX iajs-1818 24 54 coclosed coclose VERB iajs-1818 24 55 in in ADP iajs-1818 24 56 𝑀 𝑀 PROPN iajs-1818 25 1 [ [ X iajs-1818 25 2 1 1 NUM iajs-1818 25 3 ] ] PUNCT iajs-1818 25 4 . . PUNCT iajs-1818 26 1 for for ADP iajs-1818 26 2 example example NOUN iajs-1818 26 3 , , PUNCT iajs-1818 26 4 the the DET iajs-1818 26 5 ℤ-module ℤ-module PROPN iajs-1818 26 6 ℤ ℤ PROPN iajs-1818 26 7 is be AUX iajs-1818 26 8 coclosed coclose VERB iajs-1818 26 9 rickart rickart NOUN iajs-1818 26 10 because because SCONJ iajs-1818 26 11 each each DET iajs-1818 26 12 𝑓 𝑓 PROPN iajs-1818 26 13 ∈ ∈ PROPN iajs-1818 26 14 end end NOUN iajs-1818 26 15 𝑀 𝑀 PROPN iajs-1818 26 16 , , PUNCT iajs-1818 26 17 ker ker NOUN iajs-1818 26 18 𝑓 𝑓 DET iajs-1818 26 19 0 0 NUM iajs-1818 26 20 is be AUX iajs-1818 26 21 a a DET iajs-1818 26 22 coclosed coclose VERB iajs-1818 26 23 submodule submodule NOUN iajs-1818 26 24 in in ADP iajs-1818 26 25 𝑀 𝑀 PROPN iajs-1818 26 26 but but CCONJ iajs-1818 26 27 it it PRON iajs-1818 26 28 is be AUX iajs-1818 26 29 not not PART iajs-1818 26 30 cosemisimple cosemisimple NOUN iajs-1818 26 31 since since SCONJ iajs-1818 26 32 the the DET iajs-1818 26 33 only only ADJ iajs-1818 26 34 proper proper ADJ iajs-1818 26 35 coclosed coclosed ADJ iajs-1818 26 36 submodule submodule NOUN iajs-1818 26 37 in in ADP iajs-1818 26 38 the the DET iajs-1818 26 39 ℤ-module ℤ-module PROPN iajs-1818 26 40 ℤ ℤ PROPN iajs-1818 26 41 is be AUX iajs-1818 26 42 the the DET iajs-1818 26 43 zero zero NUM iajs-1818 26 44 submodule submodule NOUN iajs-1818 26 45 . . PUNCT iajs-1818 27 1 ( ( PUNCT iajs-1818 27 2 1 1 X iajs-1818 27 3 ) ) PUNCT iajs-1818 27 4 it it PRON iajs-1818 27 5 is be AUX iajs-1818 27 6 obvious obvious ADJ iajs-1818 27 7 that that SCONJ iajs-1818 27 8 rickart rickart NOUN iajs-1818 27 9 module module NOUN iajs-1818 27 10 is be AUX iajs-1818 27 11 coclosed coclose VERB iajs-1818 27 12 rickart rickart NOUN iajs-1818 27 13 while while SCONJ iajs-1818 27 14 the the DET iajs-1818 27 15 reverse reverse NOUN iajs-1818 27 16 is be AUX iajs-1818 27 17 not not PART iajs-1818 27 18 hold hold NOUN iajs-1818 27 19 as as SCONJ iajs-1818 27 20 follows follow VERB iajs-1818 27 21 . . PUNCT iajs-1818 28 1 consider consider VERB iajs-1818 28 2 the the DET iajs-1818 28 3 ring ring NOUN iajs-1818 28 4 ∏ ∏ PROPN iajs-1818 28 5 f f PROPN iajs-1818 28 6 , , PUNCT iajs-1818 28 7 f f PROPN iajs-1818 28 8 f f PROPN iajs-1818 28 9 is be AUX iajs-1818 28 10 a a DET iajs-1818 28 11 field field NOUN iajs-1818 28 12 and and CCONJ iajs-1818 28 13 𝑅 𝑅 PROPN iajs-1818 28 14 is be AUX iajs-1818 28 15 not not PART iajs-1818 28 16 semisimple semisimple NOUN iajs-1818 28 17 then then ADV iajs-1818 28 18 by by ADP iajs-1818 28 19 [ [ X iajs-1818 28 20 9 9 NUM iajs-1818 28 21 , , PUNCT iajs-1818 28 22 theorem theorem VERB iajs-1818 28 23 2.25 2.25 NUM iajs-1818 28 24 ] ] PUNCT iajs-1818 28 25 , , PUNCT iajs-1818 28 26 there there PRON iajs-1818 28 27 exists exist VERB iajs-1818 28 28 a a DET iajs-1818 28 29 right right ADJ iajs-1818 28 30 module module NOUN iajs-1818 28 31 𝑀 𝑀 PROPN iajs-1818 28 32 over over ADP iajs-1818 28 33 𝑅 𝑅 PROPN iajs-1818 28 34 , , PUNCT iajs-1818 28 35 it it PRON iajs-1818 28 36 is be AUX iajs-1818 28 37 not not PART iajs-1818 28 38 rickart rickart NOUN iajs-1818 28 39 . . PUNCT iajs-1818 29 1 and and CCONJ iajs-1818 29 2 , , PUNCT iajs-1818 29 3 𝑅 𝑅 PROPN iajs-1818 29 4 is be AUX iajs-1818 29 5 commutative commutative ADJ iajs-1818 29 6 regular regular ADJ iajs-1818 29 7 ( ( PUNCT iajs-1818 29 8 in in ADP iajs-1818 29 9 sense sense NOUN iajs-1818 29 10 von von PROPN iajs-1818 29 11 neumann neumann PROPN iajs-1818 29 12 ) ) PUNCT iajs-1818 30 1 so so ADV iajs-1818 30 2 by by ADP iajs-1818 30 3 [ [ X iajs-1818 30 4 1 1 NUM iajs-1818 30 5 ] ] PUNCT iajs-1818 30 6 , , PUNCT iajs-1818 30 7 𝑅 𝑅 PROPN iajs-1818 30 8 is be AUX iajs-1818 30 9 cosemisimple cosemisimple NOUN iajs-1818 30 10 ( ( PUNCT iajs-1818 30 11 or or CCONJ iajs-1818 30 12 v v NOUN iajs-1818 30 13 - - PUNCT iajs-1818 30 14 ring ring NOUN iajs-1818 30 15 ) ) PUNCT iajs-1818 30 16 implies imply VERB iajs-1818 30 17 that that SCONJ iajs-1818 30 18 rad rad PROPN iajs-1818 30 19 0 0 NUM iajs-1818 30 20 for for ADP iajs-1818 30 21 each each DET iajs-1818 30 22 submodule submodule NOUN iajs-1818 30 23 𝑁 𝑁 PROPN iajs-1818 30 24 of of ADP iajs-1818 30 25 𝑀. 𝑀. PROPN iajs-1818 30 26 so so SCONJ iajs-1818 30 27 each each DET iajs-1818 30 28 submodule submodule NOUN iajs-1818 30 29 of of ADP iajs-1818 30 30 𝑀 𝑀 PROPN iajs-1818 30 31 is be AUX iajs-1818 30 32 coclosed coclose VERB iajs-1818 30 33 in in ADP iajs-1818 30 34 𝑀 𝑀 PROPN iajs-1818 30 35 implies imply VERB iajs-1818 30 36 it it PRON iajs-1818 30 37 is be AUX iajs-1818 30 38 a a DET iajs-1818 30 39 coclosed coclosed ADJ iajs-1818 30 40 rickart rickart NOUN iajs-1818 30 41 module module NOUN iajs-1818 30 42 . . PUNCT iajs-1818 31 1 ihsciconf ihsciconf PROPN iajs-1818 31 2 2017 2017 NUM iajs-1818 31 3 special special ADJ iajs-1818 31 4 issue issue NOUN iajs-1818 31 5 ibn ibn PROPN iajs-1818 31 6 al al PROPN iajs-1818 31 7 - - PUNCT iajs-1818 31 8 haitham haitham PROPN iajs-1818 31 9 journal journal PROPN iajs-1818 31 10 for for ADP iajs-1818 31 11 pure pure ADJ iajs-1818 31 12 and and CCONJ iajs-1818 31 13 applied apply VERB iajs-1818 31 14 science science NOUN iajs-1818 31 15 https://doi.org/ https://doi.org/ NOUN iajs-1818 31 16 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 31 17 for for ADP iajs-1818 31 18 more more ADJ iajs-1818 31 19 information information NOUN iajs-1818 31 20 about about ADP iajs-1818 31 21 the the DET iajs-1818 31 22 conference conference NOUN iajs-1818 31 23 please please INTJ iajs-1818 31 24 visit visit VERB iajs-1818 31 25 the the DET iajs-1818 31 26 websites website NOUN iajs-1818 31 27 : : PUNCT iajs-1818 31 28 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 31 29     SPACE iajs-1818 31 30 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1818 31 31       SPACE iajs-1818 31 32 mathematics|454 mathematics|454 NOUN iajs-1818 31 33         SPACE iajs-1818 31 34 ( ( PUNCT iajs-1818 31 35 2 2 NUM iajs-1818 31 36 ) ) PUNCT iajs-1818 31 37 when when SCONJ iajs-1818 31 38 𝑀 𝑀 PROPN iajs-1818 31 39 is be AUX iajs-1818 31 40 a a DET iajs-1818 31 41 coclosed coclosed ADJ iajs-1818 31 42 simple simple ADJ iajs-1818 31 43 𝑅-modul 𝑅-modul PROPN iajs-1818 31 44 yeild yeild PROPN iajs-1818 31 45 𝑀 𝑀 PROPN iajs-1818 31 46 need need AUX iajs-1818 31 47 not not PART iajs-1818 31 48 be be AUX iajs-1818 31 49 coclosed coclose VERB iajs-1818 31 50 rickart rickart NOUN iajs-1818 31 51 where where SCONJ iajs-1818 31 52 an an DET iajs-1818 31 53 𝑅-module 𝑅-module PROPN iajs-1818 31 54 𝑀 𝑀 PROPN iajs-1818 31 55 is be AUX iajs-1818 31 56 called call VERB iajs-1818 31 57 coclosed coclose VERB iajs-1818 31 58 simple simple ADJ iajs-1818 31 59 if if SCONJ iajs-1818 31 60 𝑀 𝑀 PROPN iajs-1818 31 61   VERB iajs-1818 31 62 0 0 PUNCT iajs-1818 32 1 and and CCONJ iajs-1818 32 2 it it PRON iajs-1818 32 3 has have VERB iajs-1818 32 4 no no DET iajs-1818 32 5 coclosed coclose VERB iajs-1818 32 6 submodules submodule NOUN iajs-1818 32 7 except except SCONJ iajs-1818 32 8 { { PUNCT iajs-1818 32 9 0 0 NUM iajs-1818 32 10 } } PUNCT iajs-1818 32 11 and and CCONJ iajs-1818 32 12 𝑀 𝑀 PROPN iajs-1818 32 13 4 4 NUM iajs-1818 32 14 . . PUNCT iajs-1818 33 1 study study VERB iajs-1818 33 2 ℤ ℤ PROPN iajs-1818 33 3 as as SCONJ iajs-1818 33 4 ℤ-module ℤ-module PROPN iajs-1818 33 5 is be AUX iajs-1818 33 6 coclosed coclose VERB iajs-1818 33 7 simple simple ADJ iajs-1818 33 8 , , PUNCT iajs-1818 33 9 while while SCONJ iajs-1818 33 10 it it PRON iajs-1818 33 11 is be AUX iajs-1818 33 12 not not PART iajs-1818 33 13 coclosed coclose VERB iajs-1818 33 14 rickart rickart NOUN iajs-1818 33 15 since since SCONJ iajs-1818 33 16 there there PRON iajs-1818 33 17 exists exist VERB iajs-1818 33 18 an an DET iajs-1818 33 19 endomorphism endomorphism NOUN iajs-1818 33 20 𝑓 𝑓 DET iajs-1818 33 21 ∶ ∶ NOUN iajs-1818 33 22 ℤ ℤ PROPN iajs-1818 33 23 ⟶ ⟶ NOUN iajs-1818 33 24 ℤ ℤ NOUN iajs-1818 33 25 defined define VERB iajs-1818 33 26 by by ADP iajs-1818 33 27 𝑓 𝑓 DET iajs-1818 33 28 𝑚 𝑚 PROPN iajs-1818 33 29 𝑚2 𝑚2 NOUN iajs-1818 33 30 for for ADP iajs-1818 33 31 each each DET iajs-1818 33 32 𝑚 𝑚 PROPN iajs-1818 33 33 ∈ ∈ PROPN iajs-1818 33 34 ℤ ℤ PROPN iajs-1818 33 35 , , PUNCT iajs-1818 33 36 implies imply VERB iajs-1818 33 37 ker ker PROPN iajs-1818 34 1 𝑓 𝑓 PRON iajs-1818 34 2 0 0 NUM iajs-1818 34 3 , , PUNCT iajs-1818 34 4 2 2 NUM iajs-1818 34 5 is be AUX iajs-1818 34 6 not not PART iajs-1818 34 7 a a DET iajs-1818 34 8 coclosed coclose VERB iajs-1818 34 9 submodule submodule NOUN iajs-1818 34 10 in in ADP iajs-1818 34 11 ℤ ℤ PROPN iajs-1818 34 12 . . PUNCT iajs-1818 35 1 ( ( PUNCT iajs-1818 35 2 3 3 X iajs-1818 35 3 ) ) PUNCT iajs-1818 35 4 when when SCONJ iajs-1818 35 5 𝑀 𝑀 PROPN iajs-1818 35 6 is be AUX iajs-1818 35 7 a a DET iajs-1818 35 8 quasi quasi NOUN iajs-1818 35 9 - - NOUN iajs-1818 35 10 dedekind dedekind ADJ iajs-1818 35 11 over over ADP iajs-1818 35 12 𝑅 𝑅 PROPN iajs-1818 35 13 implies imply VERB iajs-1818 35 14 𝑀 𝑀 PROPN iajs-1818 35 15 is be AUX iajs-1818 35 16 coclosed coclose VERB iajs-1818 35 17 rickart rickart NOUN iajs-1818 35 18 , , PUNCT iajs-1818 35 19 where where SCONJ iajs-1818 35 20 𝑀 𝑀 PROPN iajs-1818 35 21 is be AUX iajs-1818 35 22 quasi quasi ADJ iajs-1818 35 23 - - ADJ iajs-1818 35 24 dedekind dedekind ADJ iajs-1818 35 25 if if SCONJ iajs-1818 35 26 every every DET iajs-1818 35 27 0 0 NUM iajs-1818 35 28 𝑓 𝑓 PRON iajs-1818 35 29 ∈ ∈ PROPN iajs-1818 35 30 end end NOUN iajs-1818 35 31 𝑀 𝑀 PROPN iajs-1818 35 32 , , PUNCT iajs-1818 35 33 ker ker NOUN iajs-1818 36 1 𝑓 𝑓 PRON iajs-1818 36 2 0 0 NUM iajs-1818 37 1 [ [ X iajs-1818 37 2 10 10 NUM iajs-1818 37 3 . . PUNCT iajs-1818 38 1 the the DET iajs-1818 38 2 reverse reverse NOUN iajs-1818 38 3 is be AUX iajs-1818 38 4 not not PART iajs-1818 38 5 hold hold ADJ iajs-1818 38 6 . . PUNCT iajs-1818 39 1 for for ADP iajs-1818 39 2 example example NOUN iajs-1818 39 3 , , PUNCT iajs-1818 39 4 ℤ ℤ PROPN iajs-1818 39 5 as as ADP iajs-1818 39 6 ℤ-module ℤ-module PROPN iajs-1818 39 7 is be AUX iajs-1818 39 8 coclosed coclose VERB iajs-1818 39 9 rickart rickart NOUN iajs-1818 39 10 while while SCONJ iajs-1818 39 11 it it PRON iajs-1818 39 12 is be AUX iajs-1818 39 13 not not PART iajs-1818 39 14 quasidedekind quasidedekind ADJ iajs-1818 39 15 . . PUNCT iajs-1818 40 1 ( ( PUNCT iajs-1818 40 2 4 4 X iajs-1818 40 3 ) ) PUNCT iajs-1818 40 4 if if SCONJ iajs-1818 40 5 𝑀 𝑀 PROPN iajs-1818 40 6 is be AUX iajs-1818 40 7 a a DET iajs-1818 40 8 coclosed coclosed ADJ iajs-1818 40 9 rickart rickart NOUN iajs-1818 40 10 coclosed coclose VERB iajs-1818 40 11 simple simple ADJ iajs-1818 40 12 module module NOUN iajs-1818 40 13 , , PUNCT iajs-1818 40 14 implies imply VERB iajs-1818 40 15 𝑀 𝑀 PROPN iajs-1818 40 16 is be AUX iajs-1818 40 17 quasi quasi ADJ iajs-1818 40 18 - - ADJ iajs-1818 40 19 dedekind dedekind ADJ iajs-1818 40 20 . . PUNCT iajs-1818 41 1 proof proof NOUN iajs-1818 41 2 . . PUNCT iajs-1818 42 1 when when SCONJ iajs-1818 42 2 𝑀 𝑀 PROPN iajs-1818 42 3 is be AUX iajs-1818 42 4 a a DET iajs-1818 42 5 coclosed coclosed ADJ iajs-1818 42 6 rickart rickart NOUN iajs-1818 42 7 over over ADP iajs-1818 42 8 𝑅 𝑅 PROPN iajs-1818 42 9 and and CCONJ iajs-1818 42 10 0 0 NUM iajs-1818 42 11 𝑓 𝑓 PRON iajs-1818 42 12 ∈ ∈ PROPN iajs-1818 42 13 end end NOUN iajs-1818 42 14 𝑀 𝑀 PROPN iajs-1818 42 15 , , PUNCT iajs-1818 42 16 implies imply VERB iajs-1818 42 17 ker ker PROPN iajs-1818 42 18 𝑓 𝑓 PRON iajs-1818 42 19 is be AUX iajs-1818 42 20 coclosed coclose VERB iajs-1818 42 21 in in ADP iajs-1818 42 22 𝑀. 𝑀. PROPN iajs-1818 42 23 but but CCONJ iajs-1818 42 24 𝑀 𝑀 PROPN iajs-1818 42 25 is be AUX iajs-1818 42 26 coclosed coclose VERB iajs-1818 42 27 simple simple ADJ iajs-1818 42 28 with with ADP iajs-1818 42 29 𝑓 𝑓 DET iajs-1818 42 30 0 0 NUM iajs-1818 42 31 , , PUNCT iajs-1818 42 32 thus thus ADV iajs-1818 42 33 ker ker VERB iajs-1818 42 34 𝑓 𝑓 DET iajs-1818 42 35 0 0 NUM iajs-1818 42 36 , , PUNCT iajs-1818 42 37 as as SCONJ iajs-1818 42 38 required require VERB iajs-1818 42 39 . . PUNCT iajs-1818 43 1 ( ( PUNCT iajs-1818 43 2 5 5 X iajs-1818 43 3 ) ) PUNCT iajs-1818 43 4 a a DET iajs-1818 43 5 homomrphic homomrphic ADJ iajs-1818 43 6 picture picture NOUN iajs-1818 43 7 of of ADP iajs-1818 43 8 a a DET iajs-1818 43 9 coclosed coclose VERB iajs-1818 43 10 rickart rickart NOUN iajs-1818 43 11 module module NOUN iajs-1818 43 12 may may AUX iajs-1818 43 13 not not PART iajs-1818 43 14 be be AUX iajs-1818 43 15 coclosed coclose VERB iajs-1818 43 16 rickart rickart NOUN iajs-1818 43 17 . . PUNCT iajs-1818 44 1 consider consider VERB iajs-1818 44 2 the the DET iajs-1818 44 3 natural natural ADJ iajs-1818 44 4 epimorphism epimorphism NOUN iajs-1818 44 5 𝑓 𝑓 DET iajs-1818 44 6 ∶ ∶ NOUN iajs-1818 44 7 ℤ ℤ NOUN iajs-1818 44 8 ⟶ ⟶ NOUN iajs-1818 44 9 ℤ ℤ PROPN iajs-1818 44 10 . . PUNCT iajs-1818 45 1 the the DET iajs-1818 45 2 ℤ ℤ PROPN iajs-1818 45 3 -module -module PROPN iajs-1818 45 4 ℤ ℤ PROPN iajs-1818 45 5 is be AUX iajs-1818 45 6 coclosed coclose VERB iajs-1818 45 7 rickart rickart NOUN iajs-1818 45 8 , , PUNCT iajs-1818 45 9 while while SCONJ iajs-1818 45 10 i i PRON iajs-1818 45 11 m m VERB iajs-1818 45 12 𝑓 𝑓 DET iajs-1818 45 13 ℤ4 ℤ4 NOUN iajs-1818 45 14 is be AUX iajs-1818 45 15 not not PART iajs-1818 45 16 a a DET iajs-1818 45 17 coclosed coclose VERB iajs-1818 45 18 rickart rickart NOUN iajs-1818 45 19 over over ADP iajs-1818 45 20 ℤ. ℤ. PROPN iajs-1818 45 21 ( ( PUNCT iajs-1818 45 22 6 6 NUM iajs-1818 45 23 ) ) PUNCT iajs-1818 45 24 every every DET iajs-1818 45 25 integral integral ADJ iajs-1818 45 26 domain domain NOUN iajs-1818 45 27 is be AUX iajs-1818 45 28 coclosed coclose VERB iajs-1818 45 29 rickart rickart NOUN iajs-1818 45 30 . . PUNCT iajs-1818 46 1 proof proof NOUN iajs-1818 46 2 . . PUNCT iajs-1818 47 1 when when SCONJ iajs-1818 47 2 𝑅 𝑅 PROPN iajs-1818 47 3 as as ADP iajs-1818 47 4 in in ADP iajs-1818 47 5 assumption assumption NOUN iajs-1818 47 6 , , PUNCT iajs-1818 47 7 implies imply VERB iajs-1818 47 8 𝑅 𝑅 PROPN iajs-1818 47 9 is be AUX iajs-1818 47 10 commutative commutative ADJ iajs-1818 47 11 implies imply VERB iajs-1818 47 12 that that SCONJ iajs-1818 47 13 for for ADP iajs-1818 47 14 each each DET iajs-1818 47 15 𝑎 𝑎 PRON iajs-1818 47 16 ∈ ∈ PROPN iajs-1818 47 17 𝑅 𝑅 PROPN iajs-1818 47 18 and and CCONJ iajs-1818 47 19 𝑓 𝑓 PRON iajs-1818 47 20 ∈ ∈ NOUN iajs-1818 47 21 end end NOUN iajs-1818 47 22 𝑅 𝑅 PROPN iajs-1818 47 23 ≅ ≅ PROPN iajs-1818 47 24 𝑅 𝑅 PROPN iajs-1818 47 25 , , PUNCT iajs-1818 47 26 we we PRON iajs-1818 47 27 can can AUX iajs-1818 47 28 define define VERB iajs-1818 47 29 𝑓 𝑓 DET iajs-1818 47 30 ∶ ∶ PROPN iajs-1818 47 31 𝑅 𝑅 PROPN iajs-1818 47 32 ⟶ ⟶ NOUN iajs-1818 47 33 𝑅 𝑅 PROPN iajs-1818 47 34 by by ADP iajs-1818 47 35 𝑓 𝑓 PRON iajs-1818 47 36 𝑟 𝑟 DET iajs-1818 47 37 𝑟𝑎 𝑟𝑎 PROPN iajs-1818 47 38 for for ADP iajs-1818 47 39 each each DET iajs-1818 47 40 𝑟 𝑟 DET iajs-1818 47 41 ∈ ∈ NOUN iajs-1818 47 42 𝑅. 𝑅. NOUN iajs-1818 47 43 it it PRON iajs-1818 47 44 follows follow VERB iajs-1818 47 45 that that DET iajs-1818 47 46 ker ker NOUN iajs-1818 47 47 𝑓 𝑓 PRON iajs-1818 47 48 𝑟 𝑟 X iajs-1818 47 49 ∈ ∈ PROPN iajs-1818 47 50 𝑅| 𝑅| PROPN iajs-1818 47 51 𝑓 𝑓 ADP iajs-1818 47 52 𝑟 𝑟 NOUN iajs-1818 47 53 0 0 NUM iajs-1818 47 54 𝑟 𝑟 NOUN iajs-1818 47 55 ∈ ∈ NOUN iajs-1818 47 56 𝑅| 𝑅| PROPN iajs-1818 47 57 𝑟𝑎 𝑟𝑎 PRON iajs-1818 47 58 0 0 NUM iajs-1818 47 59 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-1818 47 60 𝑎 𝑎 NOUN iajs-1818 47 61 0 0 NUM iajs-1818 47 62 is be AUX iajs-1818 47 63 coclosed coclose VERB iajs-1818 47 64 in in ADP iajs-1818 47 65 𝑅. 𝑅. NOUN iajs-1818 47 66 ( ( PUNCT iajs-1818 47 67 7 7 NUM iajs-1818 47 68 ) ) PUNCT iajs-1818 47 69 the the DET iajs-1818 47 70 reverse reverse NOUN iajs-1818 47 71 of of ADP iajs-1818 47 72 remark remark NOUN iajs-1818 47 73 ( ( PUNCT iajs-1818 47 74 7 7 X iajs-1818 47 75 ) ) PUNCT iajs-1818 47 76 need need AUX iajs-1818 47 77 not not PART iajs-1818 47 78 be be AUX iajs-1818 47 79 true true ADJ iajs-1818 47 80 generally generally ADV iajs-1818 47 81 . . PUNCT iajs-1818 48 1 for for ADP iajs-1818 48 2 example example NOUN iajs-1818 48 3 , , PUNCT iajs-1818 48 4 the the DET iajs-1818 48 5 ring ring NOUN iajs-1818 48 6 ℤ ℤ PROPN iajs-1818 48 7 is be AUX iajs-1818 48 8 coclosed coclose VERB iajs-1818 48 9 rickart rickart NOUN iajs-1818 48 10 but but CCONJ iajs-1818 48 11 it it PRON iajs-1818 48 12 is be AUX iajs-1818 48 13 not not PART iajs-1818 48 14 integral integral ADJ iajs-1818 48 15 domain domain NOUN iajs-1818 48 16 . . PUNCT iajs-1818 49 1 proposition proposition NOUN iajs-1818 49 2 under under ADP iajs-1818 49 3 an an DET iajs-1818 49 4 isomorphism isomorphism NOUN iajs-1818 49 5 , , PUNCT iajs-1818 49 6 the the DET iajs-1818 49 7 coclosed coclose VERB iajs-1818 49 8 rickart rickart NOUN iajs-1818 49 9 property property NOUN iajs-1818 49 10 is be AUX iajs-1818 49 11 moved move VERB iajs-1818 49 12 . . PUNCT iajs-1818 50 1 proof proof NOUN iajs-1818 50 2 . . PUNCT iajs-1818 51 1 let let VERB iajs-1818 51 2 𝑀 𝑀 PROPN iajs-1818 51 3 and and CCONJ iajs-1818 51 4 𝑀 𝑀 PROPN iajs-1818 51 5 be be VERB iajs-1818 51 6 modules module NOUN iajs-1818 51 7 over over ADP iajs-1818 51 8 𝑅 𝑅 PROPN iajs-1818 51 9 , , PUNCT iajs-1818 51 10 where where SCONJ iajs-1818 51 11 𝑀 𝑀 PROPN iajs-1818 51 12 is be AUX iajs-1818 51 13 coclosed coclose VERB iajs-1818 51 14 rickart rickart NOUN iajs-1818 51 15 and and CCONJ iajs-1818 51 16 𝑓 𝑓 DET iajs-1818 51 17 ∶ ∶ NOUN iajs-1818 51 18 𝑀 𝑀 PROPN iajs-1818 51 19 → → SYM iajs-1818 51 20 𝑀 𝑀 PROPN iajs-1818 51 21 an an DET iajs-1818 51 22 isomorphism isomorphism NOUN iajs-1818 51 23 . . PUNCT iajs-1818 52 1 let let VERB iajs-1818 52 2 𝑔 𝑔 PROPN iajs-1818 52 3   PROPN iajs-1818 52 4 end end VERB iajs-1818 52 5 𝑀 𝑀 PROPN iajs-1818 52 6 , , PUNCT iajs-1818 52 7 we we PRON iajs-1818 52 8 prove prove VERB iajs-1818 52 9 that that SCONJ iajs-1818 52 10 ker ker PROPN iajs-1818 52 11 𝑔 𝑔 PROPN iajs-1818 52 12 is be AUX iajs-1818 52 13 a a DET iajs-1818 52 14 coclosed coclose VERB iajs-1818 52 15 submodule submodule NOUN iajs-1818 52 16 in in ADP iajs-1818 52 17 𝑀 𝑀 PROPN iajs-1818 52 18 . . PUNCT iajs-1818 53 1 let let VERB iajs-1818 53 2 𝐾 𝐾 PRON iajs-1818 53 3 be be AUX iajs-1818 53 4 any any DET iajs-1818 53 5 proper proper ADJ iajs-1818 53 6 submodule submodule NOUN iajs-1818 53 7 such such ADJ iajs-1818 53 8 that that SCONJ iajs-1818 53 9 𝐾 𝐾 PROPN iajs-1818 53 10 ⊂ ⊂ PROPN iajs-1818 53 11 ker ker PROPN iajs-1818 53 12 𝑔 𝑔 PROPN iajs-1818 53 13 implies imply VERB iajs-1818 53 14 that that SCONJ iajs-1818 53 15 𝑓 𝑓 X iajs-1818 53 16 ( ( PUNCT iajs-1818 53 17 𝐾 𝐾 PROPN iajs-1818 53 18 ⊂ ⊂ PROPN iajs-1818 53 19 𝑓 𝑓 DET iajs-1818 53 20 ker ker NOUN iajs-1818 53 21 𝑔 𝑔 X iajs-1818 53 22 . . PUNCT iajs-1818 54 1 but but CCONJ iajs-1818 54 2 𝑓 𝑓 PRON iajs-1818 54 3 ( ( PUNCT iajs-1818 54 4 ker ker PROPN iajs-1818 54 5 𝑔 𝑔 PROPN iajs-1818 54 6 ker ker PROPN iajs-1818 54 7 𝑓 𝑓 PRON iajs-1818 54 8 𝑔𝑓 𝑔𝑓 NOUN iajs-1818 54 9 , , PUNCT iajs-1818 54 10 to to PART iajs-1818 54 11 show show VERB iajs-1818 54 12 this this PRON iajs-1818 54 13 . . PUNCT iajs-1818 55 1 let let VERB iajs-1818 55 2 x x PRON iajs-1818 55 3   NOUN iajs-1818 55 4 𝑓 𝑓 X iajs-1818 55 5 ( ( PUNCT iajs-1818 55 6 ker ker PROPN iajs-1818 55 7 𝑔 𝑔 PROPN iajs-1818 55 8 ) ) PUNCT iajs-1818 55 9 , , PUNCT iajs-1818 55 10 𝑥 𝑥 PROPN iajs-1818 56 1 𝑓 𝑓 PRON iajs-1818 56 2 𝑦 𝑦 NUM iajs-1818 56 3 and and CCONJ iajs-1818 56 4 𝑦 𝑦 NOUN iajs-1818 56 5   NOUN iajs-1818 56 6 ker ker PROPN iajs-1818 57 1 𝑔 𝑔 PROPN iajs-1818 57 2 implies imply VERB iajs-1818 57 3 that that SCONJ iajs-1818 57 4 𝑓 𝑓 DET iajs-1818 57 5 𝑥 𝑥 PROPN iajs-1818 57 6 𝑦 𝑦 NOUN iajs-1818 57 7 and and CCONJ iajs-1818 57 8 𝑔 𝑔 PROPN iajs-1818 57 9 𝑦 𝑦 NOUN iajs-1818 57 10 0 0 NUM iajs-1818 57 11 . . PUNCT iajs-1818 58 1 then then ADV iajs-1818 58 2 𝑓 𝑓 DET iajs-1818 58 3 𝑔 𝑔 PROPN iajs-1818 58 4 𝑓 𝑓 PRON iajs-1818 58 5 𝑥 𝑥 X iajs-1818 58 6 𝑓 𝑓 PRON iajs-1818 58 7 𝑔 𝑔 PROPN iajs-1818 58 8 𝑦 𝑦 NOUN iajs-1818 58 9 0 0 NUM iajs-1818 58 10 , , PUNCT iajs-1818 58 11 implies imply VERB iajs-1818 58 12 𝑥 𝑥 PROPN iajs-1818 58 13   PROPN iajs-1818 58 14 ker ker NOUN iajs-1818 58 15 𝑓 𝑓 DET iajs-1818 58 16 𝑔𝑓 𝑔𝑓 NOUN iajs-1818 58 17 ) ) PUNCT iajs-1818 58 18 . . PUNCT iajs-1818 59 1 for for ADP iajs-1818 59 2 the the DET iajs-1818 59 3 other other ADJ iajs-1818 59 4 , , PUNCT iajs-1818 59 5 let let VERB iajs-1818 59 6 𝑥 𝑥 PRON iajs-1818 59 7   PROPN iajs-1818 59 8 ker ker NOUN iajs-1818 59 9 𝑓 𝑓 DET iajs-1818 59 10 𝑔𝑓 𝑔𝑓 NOUN iajs-1818 59 11 ) ) PUNCT iajs-1818 59 12 , , PUNCT iajs-1818 59 13 𝑓 𝑓 PRON iajs-1818 59 14 𝑔 𝑔 PROPN iajs-1818 59 15 f f X iajs-1818 59 16 ( ( PUNCT iajs-1818 59 17 x x SYM iajs-1818 59 18 ) ) PUNCT iajs-1818 59 19 0 0 NUM iajs-1818 60 1 and and CCONJ iajs-1818 60 2 so so ADV iajs-1818 60 3 𝑓 𝑓 PRON iajs-1818 60 4 𝑥 𝑥 PROPN iajs-1818 60 5   NOUN iajs-1818 60 6 ker ker NOUN iajs-1818 61 1 𝑓 𝑓 DET iajs-1818 61 2 𝑔 𝑔 NOUN iajs-1818 61 3 ) ) PUNCT iajs-1818 61 4 . . PUNCT iajs-1818 62 1 one one PRON iajs-1818 62 2 can can AUX iajs-1818 62 3 easily easily ADV iajs-1818 62 4 see see VERB iajs-1818 62 5 that that DET iajs-1818 62 6 ker ker NOUN iajs-1818 62 7 𝑓 𝑓 DET iajs-1818 62 8 𝑔 𝑔 PROPN iajs-1818 62 9 ker ker PROPN iajs-1818 62 10 𝑔 𝑔 NOUN iajs-1818 62 11 , , PUNCT iajs-1818 62 12 hence hence ADV iajs-1818 62 13 𝑓 𝑓 DET iajs-1818 62 14 𝑥 𝑥 PROPN iajs-1818 62 15   PROPN iajs-1818 62 16 ker ker PROPN iajs-1818 63 1 𝑔 𝑔 NOUN iajs-1818 63 2 , , PUNCT iajs-1818 63 3 𝑥 𝑥 DET iajs-1818 63 4   NOUN iajs-1818 63 5 𝑓 𝑓 DET iajs-1818 63 6 ker ker NOUN iajs-1818 63 7 𝑔 𝑔 NOUN iajs-1818 63 8 ) ) PUNCT iajs-1818 63 9 , , PUNCT iajs-1818 63 10 it it PRON iajs-1818 63 11 follows follow VERB iajs-1818 63 12 that that SCONJ iajs-1818 63 13 𝑓 𝑓 PRON iajs-1818 63 14 ( ( PUNCT iajs-1818 63 15 ker ker NOUN iajs-1818 63 16 𝑔 𝑔 PROPN iajs-1818 63 17 ) ) PUNCT iajs-1818 64 1 = = VERB iajs-1818 65 1 ker ker NOUN iajs-1818 65 2 𝑓 𝑓 DET iajs-1818 65 3 𝑔𝑓 𝑔𝑓 NOUN iajs-1818 65 4 ) ) PUNCT iajs-1818 65 5 . . PUNCT iajs-1818 66 1 this this PRON iajs-1818 66 2 means mean VERB iajs-1818 66 3 that that SCONJ iajs-1818 66 4 𝑓 𝑓 PROPN iajs-1818 66 5 ( ( PUNCT iajs-1818 66 6 k k PROPN iajs-1818 66 7 ) ) PUNCT iajs-1818 67 1 ⊂ ⊂ PROPN iajs-1818 68 1 ker ker PROPN iajs-1818 69 1 ( ( PUNCT iajs-1818 69 2 𝑓 𝑓 DET iajs-1818 69 3 𝑔𝑓 𝑔𝑓 NOUN iajs-1818 69 4 ) ) PUNCT iajs-1818 69 5 , , PUNCT iajs-1818 69 6 but but CCONJ iajs-1818 69 7 𝑓 𝑓 DET iajs-1818 69 8 𝑔𝑓 𝑔𝑓 ADP iajs-1818 69 9   PROPN iajs-1818 69 10 𝑓 𝑓 DET iajs-1818 69 11 ∈ ∈ PROPN iajs-1818 69 12 end end NOUN iajs-1818 69 13 ( ( PUNCT iajs-1818 69 14 𝑀 𝑀 PROPN iajs-1818 69 15 ) ) PUNCT iajs-1818 69 16 and and CCONJ iajs-1818 69 17 𝑀 𝑀 PROPN iajs-1818 69 18 coclosed coclose VERB iajs-1818 69 19 rickart rickart NOUN iajs-1818 69 20 then then ADV iajs-1818 69 21 ker ker PROPN iajs-1818 70 1 ( ( PUNCT iajs-1818 70 2 𝑓 𝑓 DET iajs-1818 70 3 𝑔𝑓 𝑔𝑓 NOUN iajs-1818 70 4 ) ) PUNCT iajs-1818 70 5 is be AUX iajs-1818 70 6 a a DET iajs-1818 70 7 coclosed coclose VERB iajs-1818 70 8 , , PUNCT iajs-1818 70 9 there there PRON iajs-1818 70 10 exists exist VERB iajs-1818 70 11 a a DET iajs-1818 70 12 submodule submodule NOUN iajs-1818 70 13 𝑁 𝑁 PROPN iajs-1818 70 14 of of ADP iajs-1818 70 15 𝑀 𝑀 PROPN iajs-1818 70 16 , , PUNCT iajs-1818 70 17 𝑓 𝑓 PROPN iajs-1818 70 18 ( ( PUNCT iajs-1818 70 19 ker ker NOUN iajs-1818 70 20 𝑔 𝑔 PROPN iajs-1818 70 21 ) ) PUNCT iajs-1818 71 1 + + CCONJ iajs-1818 71 2 𝑁 𝑁 PROPN iajs-1818 71 3 = = SYM iajs-1818 71 4 𝑀 𝑀 PROPN iajs-1818 71 5 but but CCONJ iajs-1818 71 6 𝑓 𝑓 PRON iajs-1818 71 7 ( ( PUNCT iajs-1818 71 8 𝐾 𝐾 PROPN iajs-1818 71 9 ) ) PUNCT iajs-1818 72 1 + + CCONJ iajs-1818 72 2 𝑁 𝑁 PROPN iajs-1818 72 3 𝑀 𝑀 PROPN iajs-1818 72 4 . . PUNCT iajs-1818 73 1 this this PRON iajs-1818 73 2 means mean VERB iajs-1818 73 3 that that SCONJ iajs-1818 73 4 ker ker PROPN iajs-1818 74 1 𝑔 𝑔 PROPN iajs-1818 75 1 + + PROPN iajs-1818 75 2 𝑓 𝑓 DET iajs-1818 75 3 𝑁 𝑁 PROPN iajs-1818 75 4 = = PUNCT iajs-1818 75 5 𝑀 𝑀 PROPN iajs-1818 75 6 but but CCONJ iajs-1818 75 7 𝐾 𝐾 PROPN iajs-1818 75 8 + + CCONJ iajs-1818 75 9 𝑓 𝑓 DET iajs-1818 75 10 𝑁 𝑁 PROPN iajs-1818 75 11 𝑀 𝑀 PROPN iajs-1818 76 1 , , PUNCT iajs-1818 76 2 it it PRON iajs-1818 76 3 follows follow VERB iajs-1818 76 4 that that SCONJ iajs-1818 76 5 ker ker PROPN iajs-1818 76 6 𝑔 𝑔 PROPN iajs-1818 76 7 is be AUX iajs-1818 76 8 a a DET iajs-1818 76 9 coclosed coclose VERB iajs-1818 76 10 submodule submodule NOUN iajs-1818 76 11 in in ADP iajs-1818 76 12 𝑀 𝑀 PROPN iajs-1818 76 13 . . PUNCT iajs-1818 77 1 from from ADP iajs-1818 77 2 a a DET iajs-1818 77 3 module module NOUN iajs-1818 77 4 to to ADP iajs-1818 77 5 any any PRON iajs-1818 77 6 of of ADP iajs-1818 77 7 its its PRON iajs-1818 77 8 submodules submodule NOUN iajs-1818 77 9 or or CCONJ iajs-1818 77 10 inversely inversely ADV iajs-1818 77 11 , , PUNCT iajs-1818 77 12 coclosed coclose VERB iajs-1818 77 13 rickart rickart NOUN iajs-1818 77 14 property property NOUN iajs-1818 77 15 does do AUX iajs-1818 77 16 not not PART iajs-1818 77 17 always always ADV iajs-1818 77 18 passed pass VERB iajs-1818 77 19 in in ADP iajs-1818 77 20 general general ADJ iajs-1818 77 21 as as SCONJ iajs-1818 77 22 follows follow VERB iajs-1818 77 23 examples example NOUN iajs-1818 77 24 if if SCONJ iajs-1818 77 25 𝑁 𝑁 PROPN iajs-1818 77 26 is be AUX iajs-1818 77 27 a a DET iajs-1818 77 28 submodule submodule NOUN iajs-1818 77 29 of of ADP iajs-1818 77 30 a a DET iajs-1818 77 31 coclosed coclose VERB iajs-1818 77 32 rickart rickart NOUN iajs-1818 77 33 module module NOUN iajs-1818 77 34 𝑀 𝑀 PROPN iajs-1818 77 35 over over ADP iajs-1818 77 36 𝑅 𝑅 PROPN iajs-1818 77 37 , , PUNCT iajs-1818 77 38 implies imply VERB iajs-1818 77 39 𝑁 𝑁 PROPN iajs-1818 77 40 may may AUX iajs-1818 77 41 not not PART iajs-1818 77 42 be be AUX iajs-1818 77 43 coclosed coclose VERB iajs-1818 77 44 rickart rickart NOUN iajs-1818 77 45 . . PUNCT iajs-1818 78 1 study study NOUN iajs-1818 78 2 𝑀 𝑀 PROPN iajs-1818 78 3 ℚ ℚ PROPN iajs-1818 78 4   VERB iajs-1818 78 5 ℤ ℤ PROPN iajs-1818 78 6 as as ADP iajs-1818 78 7 ℤ-module ℤ-module PROPN iajs-1818 78 8 . . PUNCT iajs-1818 78 9 by by ADP iajs-1818 78 10 [ [ X iajs-1818 78 11 9 9 NUM iajs-1818 78 12 , , PUNCT iajs-1818 78 13 ex ex X iajs-1818 78 14 2.5 2.5 NUM iajs-1818 78 15 ] ] PUNCT iajs-1818 78 16 , , PUNCT iajs-1818 78 17 𝑀 𝑀 PROPN iajs-1818 78 18 is be AUX iajs-1818 78 19 rickart rickart NOUN iajs-1818 78 20 , , PUNCT iajs-1818 78 21 yield yield VERB iajs-1818 78 22 𝑀 𝑀 PROPN iajs-1818 78 23 is be AUX iajs-1818 78 24 coclosed coclose VERB iajs-1818 78 25 rickart rickart NOUN iajs-1818 78 26 . . PUNCT iajs-1818 79 1 now now ADV iajs-1818 79 2 consider consider VERB iajs-1818 79 3 𝑁 𝑁 PROPN iajs-1818 79 4 ℤ ℤ PROPN iajs-1818 79 5   ADJ iajs-1818 79 6 ℤ ℤ PROPN iajs-1818 79 7 a a DET iajs-1818 79 8 submodule submodule NOUN iajs-1818 79 9 of of ADP iajs-1818 79 10 𝑀. 𝑀. PROPN iajs-1818 79 11 see see VERB iajs-1818 79 12 that that SCONJ iajs-1818 79 13 𝑁 𝑁 PROPN iajs-1818 79 14 may may AUX iajs-1818 79 15 not not PART iajs-1818 79 16 be be AUX iajs-1818 79 17 a a DET iajs-1818 79 18 coclosed coclosed ADJ iajs-1818 79 19 rickart rickart NOUN iajs-1818 79 20 since since SCONJ iajs-1818 79 21 there there PRON iajs-1818 79 22 is be VERB iajs-1818 79 23 𝑓 𝑓 DET iajs-1818 79 24 ∶ ∶ NOUN iajs-1818 79 25 𝑁 𝑁 PROPN iajs-1818 79 26 → → SYM iajs-1818 79 27 𝑁 𝑁 PROPN iajs-1818 79 28 defined define VERB iajs-1818 79 29 by by ADP iajs-1818 79 30 𝑓 𝑓 DET iajs-1818 79 31 𝑚 𝑚 NOUN iajs-1818 79 32 , , PUNCT iajs-1818 79 33 𝑛 𝑛 PROPN iajs-1818 79 34 0 0 NUM iajs-1818 79 35 , , PUNCT iajs-1818 79 36 𝑚 𝑚 PROPN iajs-1818 79 37 where where SCONJ iajs-1818 79 38 𝑚 𝑚 NOUN iajs-1818 79 39 , , PUNCT iajs-1818 79 40 𝑛 𝑛 DET iajs-1818 79 41   NOUN iajs-1818 79 42 ℤ. ℤ. PROPN iajs-1818 79 43 thus thus ADV iajs-1818 79 44 ker ker VERB iajs-1818 79 45 𝑓 𝑓 PRON iajs-1818 79 46 𝑚 𝑚 NOUN iajs-1818 79 47 , , PUNCT iajs-1818 79 48 𝑛 𝑛 DET iajs-1818 79 49 ∈ ∈ NOUN iajs-1818 79 50 ℤ ℤ PROPN iajs-1818 79 51   NOUN iajs-1818 79 52 ℤ ℤ PROPN iajs-1818 80 1 | | ADV iajs-1818 80 2 𝑓 𝑓 NOUN iajs-1818 80 3 𝑚 𝑚 NOUN iajs-1818 80 4 , , PUNCT iajs-1818 80 5 𝑛 𝑛 PROPN iajs-1818 80 6 = = SYM iajs-1818 80 7 0 0 NUM iajs-1818 80 8 , , PUNCT iajs-1818 80 9 0 0 NUM iajs-1818 80 10 𝑚 𝑚 NOUN iajs-1818 80 11 , , PUNCT iajs-1818 80 12 𝑛 𝑛 VERB iajs-1818 80 13 | | NOUN iajs-1818 80 14 𝑚 𝑚 X iajs-1818 80 15 0 0 PUNCT iajs-1818 80 16 } } PUNCT iajs-1818 80 17 = = SYM iajs-1818 81 1 2ℤ 2ℤ NOUN iajs-1818 81 2   ADJ iajs-1818 81 3 ℤ ℤ PROPN iajs-1818 81 4 is be AUX iajs-1818 81 5 not not PART iajs-1818 81 6 a a DET iajs-1818 81 7 coclosed coclose VERB iajs-1818 81 8 submodule submodule NOUN iajs-1818 81 9 in in ADP iajs-1818 81 10 ℤ ℤ PROPN iajs-1818 81 11   NOUN iajs-1818 81 12 ℤ ℤ PROPN iajs-1818 81 13 , , PUNCT iajs-1818 81 14 because because SCONJ iajs-1818 81 15 2ℤ 2ℤ PROPN iajs-1818 81 16 ≅ ≅ PROPN iajs-1818 81 17 ℤ ℤ PROPN iajs-1818 81 18 ≅ ≅ PROPN iajs-1818 81 19 ℤ ℤ PROPN iajs-1818 81 20   NOUN iajs-1818 81 21 ℤ ℤ PROPN iajs-1818 81 22 ℤ ℤ PROPN iajs-1818 81 23   ADJ iajs-1818 81 24 ℤ ℤ PROPN iajs-1818 81 25 ≪ ≪ PUNCT iajs-1818 81 26 ℤ ℤ PROPN iajs-1818 81 27   ADJ iajs-1818 81 28 ℤ ℤ PROPN iajs-1818 81 29 ℤ ℤ PROPN iajs-1818 81 30   NOUN iajs-1818 81 31 ℤ ℤ PROPN iajs-1818 81 32 ≅ ≅ NOUN iajs-1818 81 33 ℤ ℤ PROPN iajs-1818 81 34 . . PUNCT iajs-1818 82 1 ihsciconf ihsciconf PROPN iajs-1818 82 2 2017 2017 NUM iajs-1818 82 3 special special ADJ iajs-1818 82 4 issue issue NOUN iajs-1818 82 5 ibn ibn PROPN iajs-1818 82 6 al al PROPN iajs-1818 82 7 - - PUNCT iajs-1818 82 8 haitham haitham PROPN iajs-1818 82 9 journal journal PROPN iajs-1818 82 10 for for ADP iajs-1818 82 11 pure pure ADJ iajs-1818 82 12 and and CCONJ iajs-1818 82 13 applied apply VERB iajs-1818 82 14 science science NOUN iajs-1818 82 15 https://doi.org/ https://doi.org/ NOUN iajs-1818 82 16 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 82 17 for for ADP iajs-1818 82 18 more more ADJ iajs-1818 82 19 information information NOUN iajs-1818 82 20 about about ADP iajs-1818 82 21 the the DET iajs-1818 82 22 conference conference NOUN iajs-1818 82 23 please please INTJ iajs-1818 82 24 visit visit VERB iajs-1818 82 25 the the DET iajs-1818 82 26 websites website NOUN iajs-1818 82 27 : : PUNCT iajs-1818 82 28 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 82 29     SPACE iajs-1818 82 30 www.ihsciconf.org www.ihsciconf.org NOUN iajs-1818 82 31       SPACE iajs-1818 82 32 mathematics|455 mathematics|455 PROPN iajs-1818 82 33         SPACE iajs-1818 82 34 ( ( PUNCT iajs-1818 82 35 1 1 NUM iajs-1818 82 36 ) ) PUNCT iajs-1818 82 37 if if SCONJ iajs-1818 82 38 each each DET iajs-1818 82 39 proper proper ADJ iajs-1818 82 40 submodule submodule NOUN iajs-1818 82 41 of of ADP iajs-1818 82 42 𝑀 𝑀 PROPN iajs-1818 82 43 is be AUX iajs-1818 82 44 a a DET iajs-1818 82 45 coclosed coclosed ADJ iajs-1818 82 46 rickart rickart NOUN iajs-1818 82 47 , , PUNCT iajs-1818 82 48 implies imply VERB iajs-1818 82 49 𝑀 𝑀 PROPN iajs-1818 82 50 may may AUX iajs-1818 82 51 not not PART iajs-1818 82 52 be be AUX iajs-1818 82 53 coclosed coclose VERB iajs-1818 82 54 rickart rickart NOUN iajs-1818 82 55 . . PUNCT iajs-1818 83 1 as as ADP iajs-1818 83 2 the the DET iajs-1818 83 3 next next ADJ iajs-1818 83 4 , , PUNCT iajs-1818 83 5 ℤ4 ℤ4 NOUN iajs-1818 83 6 as as ADP iajs-1818 83 7 ℤ-module ℤ-module PROPN iajs-1818 83 8 in in ADP iajs-1818 83 9 which which PRON iajs-1818 83 10 every every DET iajs-1818 83 11 proper proper ADJ iajs-1818 83 12 submodule submodule NOUN iajs-1818 83 13 is be AUX iajs-1818 83 14 semisimple semisimple NOUN iajs-1818 83 15 module module NOUN iajs-1818 83 16 and and CCONJ iajs-1818 83 17 hence hence ADV iajs-1818 83 18 they they PRON iajs-1818 83 19 are be AUX iajs-1818 83 20 coclosed coclose VERB iajs-1818 83 21 rickart rickart NOUN iajs-1818 83 22 modules module NOUN iajs-1818 83 23 . . PUNCT iajs-1818 84 1 but but CCONJ iajs-1818 84 2 ℤ4 ℤ4 PROPN iajs-1818 84 3 is be AUX iajs-1818 84 4 not not PART iajs-1818 84 5 coclosed coclose VERB iajs-1818 84 6 rickart rickart NOUN iajs-1818 84 7 . . PUNCT iajs-1818 85 1 we we PRON iajs-1818 85 2 record record VERB iajs-1818 85 3 the the DET iajs-1818 85 4 next next ADJ iajs-1818 85 5 from from ADP iajs-1818 85 6 [ [ X iajs-1818 85 7 1 1 NUM iajs-1818 85 8 ] ] PUNCT iajs-1818 85 9 . . PUNCT iajs-1818 86 1 lemma lemma PROPN iajs-1818 86 2 let let VERB iajs-1818 86 3 𝑁 𝑁 PROPN iajs-1818 86 4 , , PUNCT iajs-1818 86 5 𝐻 𝐻 PROPN iajs-1818 86 6 be be VERB iajs-1818 86 7 submodules submodule NOUN iajs-1818 86 8 of of ADP iajs-1818 86 9 an an DET iajs-1818 86 10 𝑅-module 𝑅-module PROPN iajs-1818 86 11 𝑀 𝑀 PROPN iajs-1818 86 12 and and CCONJ iajs-1818 86 13 𝐾 𝐾 PROPN iajs-1818 87 1 ⊂ ⊂ PROPN iajs-1818 87 2 𝑁. 𝑁. PROPN iajs-1818 87 3 if if SCONJ iajs-1818 87 4 𝐾 𝐾 PROPN iajs-1818 87 5 is be AUX iajs-1818 87 6 coclosed coclose VERB iajs-1818 87 7 in in ADP iajs-1818 87 8 𝑀 𝑀 PROPN iajs-1818 87 9 , , PUNCT iajs-1818 87 10 implies imply VERB iajs-1818 87 11 𝐾 𝐾 PROPN iajs-1818 87 12 is be AUX iajs-1818 87 13 coclosed coclose VERB iajs-1818 87 14 in in ADP iajs-1818 87 15 𝑁 𝑁 PROPN iajs-1818 87 16 and and CCONJ iajs-1818 87 17 the the DET iajs-1818 87 18 reverse reverse NOUN iajs-1818 87 19 is be AUX iajs-1818 87 20 hold hold ADJ iajs-1818 87 21 if if SCONJ iajs-1818 87 22 𝑁 𝑁 PROPN iajs-1818 87 23 is be AUX iajs-1818 87 24 coclosed coclose VERB iajs-1818 87 25 in in ADP iajs-1818 87 26 𝑀. 𝑀. NOUN iajs-1818 87 27 proposition proposition NOUN iajs-1818 87 28 . . PUNCT iajs-1818 88 1 each each DET iajs-1818 88 2 direct direct ADJ iajs-1818 88 3 summand summand NOUN iajs-1818 88 4 of of ADP iajs-1818 88 5 a a DET iajs-1818 88 6 coclosed coclose VERB iajs-1818 88 7 rickart rickart NOUN iajs-1818 88 8 module module NOUN iajs-1818 88 9 is be AUX iajs-1818 88 10 coclosed coclose VERB iajs-1818 88 11 rickart rickart NOUN iajs-1818 88 12 . . PUNCT iajs-1818 89 1 proof proof NOUN iajs-1818 89 2 . . PUNCT iajs-1818 90 1 let let VERB iajs-1818 90 2 𝑀 𝑀 PRON iajs-1818 90 3 be be AUX iajs-1818 90 4 a a DET iajs-1818 90 5 coclosed coclosed ADJ iajs-1818 90 6 rickart rickart NOUN iajs-1818 90 7 module module NOUN iajs-1818 90 8 over over ADP iajs-1818 90 9 𝑅 𝑅 PROPN iajs-1818 90 10 and and CCONJ iajs-1818 90 11 𝐴 𝐴 PROPN iajs-1818 90 12 a a DET iajs-1818 90 13 direct direct ADJ iajs-1818 90 14 summand summand NOUN iajs-1818 90 15 of of ADP iajs-1818 90 16 𝑀 𝑀 PROPN iajs-1818 90 17 , , PUNCT iajs-1818 90 18 then then ADV iajs-1818 90 19 𝑀 𝑀 PROPN iajs-1818 90 20 𝐴𝐵 𝐴𝐵 VERB iajs-1818 90 21 for for ADP iajs-1818 90 22 some some DET iajs-1818 90 23 submodule submodule NOUN iajs-1818 90 24 𝐵 𝐵 PROPN iajs-1818 90 25 of of ADP iajs-1818 90 26 𝑀. 𝑀. PROPN iajs-1818 90 27 let let VERB iajs-1818 90 28 𝑓 𝑓 DET iajs-1818 90 29 ∈ ∈ PROPN iajs-1818 90 30 end end NOUN iajs-1818 90 31 𝐴 𝐴 PROPN iajs-1818 90 32 , , PUNCT iajs-1818 90 33 then then ADV iajs-1818 90 34 we we PRON iajs-1818 90 35 have have VERB iajs-1818 90 36 the the DET iajs-1818 90 37 following follow VERB iajs-1818 90 38 𝑀 𝑀 PROPN iajs-1818 90 39 → → SYM iajs-1818 90 40 𝐴 𝐴 PROPN iajs-1818 90 41 → → SYM iajs-1818 90 42 𝐴 𝐴 PROPN iajs-1818 90 43 → → SYM iajs-1818 90 44 𝑀. 𝑀. PROPN iajs-1818 90 45 say say VERB iajs-1818 90 46 𝑔 𝑔 PROPN iajs-1818 90 47 𝑖 𝑖 ADP iajs-1818 90 48 𝑓 𝑓 DET iajs-1818 90 49   NOUN iajs-1818 90 50 , , PUNCT iajs-1818 90 51 𝑔 𝑔 PROPN iajs-1818 90 52 ∈ ∈ PROPN iajs-1818 90 53 endr endr PROPN iajs-1818 90 54 𝑀 𝑀 PROPN iajs-1818 90 55 . . PUNCT iajs-1818 91 1 therefore therefore ADV iajs-1818 91 2 ker ker PROPN iajs-1818 91 3 𝑔 𝑔 PROPN iajs-1818 91 4 is be AUX iajs-1818 91 5 a a DET iajs-1818 91 6 coclosed coclose VERB iajs-1818 91 7 in in ADP iajs-1818 91 8 𝑀. 𝑀. PROPN iajs-1818 91 9 see see NOUN iajs-1818 91 10 ker ker PROPN iajs-1818 91 11 𝑔 𝑔 PROPN iajs-1818 91 12 ker ker PROPN iajs-1818 92 1 𝑓 𝑓 DET iajs-1818 92 2 𝐵. 𝐵. NOUN iajs-1818 92 3 to to PART iajs-1818 92 4 show show VERB iajs-1818 92 5 this this PRON iajs-1818 92 6 , , PUNCT iajs-1818 92 7 let let VERB iajs-1818 92 8 𝑚 𝑚 X iajs-1818 92 9 ∈ ∈ PROPN iajs-1818 92 10 ker ker NOUN iajs-1818 92 11 𝑓 𝑓 PROPN iajs-1818 92 12 𝐵 𝐵 PROPN iajs-1818 92 13 , , PUNCT iajs-1818 92 14 𝑚 𝑚 ADP iajs-1818 92 15 𝑥 𝑥 X iajs-1818 92 16 𝑦 𝑦 NOUN iajs-1818 92 17 where where SCONJ iajs-1818 92 18 𝑥 𝑥 DET iajs-1818 92 19 ∈ ∈ PROPN iajs-1818 92 20 ker ker NOUN iajs-1818 92 21 𝑓 𝑓 X iajs-1818 92 22 and and CCONJ iajs-1818 92 23 𝑦 𝑦 PROPN iajs-1818 92 24 ∈ ∈ NOUN iajs-1818 92 25 𝐵. 𝐵. NOUN iajs-1818 92 26 hence hence ADV iajs-1818 92 27 𝑔 𝑔 PROPN iajs-1818 92 28 𝑚 𝑚 PROPN iajs-1818 92 29 𝑔 𝑔 PROPN iajs-1818 92 30 𝑥 𝑥 X iajs-1818 92 31 𝑦 𝑦 PROPN iajs-1818 92 32 𝑖 𝑖 X iajs-1818 92 33 𝑓 𝑓 PRON iajs-1818 92 34   ADP iajs-1818 92 35 𝑥 𝑥 PUNCT iajs-1818 92 36 𝑦 𝑦 SYM iajs-1818 92 37 𝑖 𝑖 X iajs-1818 92 38 𝑓 𝑓 NOUN iajs-1818 92 39 𝑥 𝑥 X iajs-1818 92 40 𝑓 𝑓 PRON iajs-1818 92 41 𝑥 𝑥 NOUN iajs-1818 92 42 0 0 X iajs-1818 92 43 . . PUNCT iajs-1818 93 1 therefore therefore ADV iajs-1818 93 2 𝑚 𝑚 X iajs-1818 93 3 ∈ ∈ PROPN iajs-1818 93 4 ker ker PROPN iajs-1818 94 1 𝑔 𝑔 PROPN iajs-1818 94 2 implies imply VERB iajs-1818 94 3 that that SCONJ iajs-1818 94 4 ker ker PROPN iajs-1818 94 5 𝑓 𝑓 DET iajs-1818 94 6 𝐵 𝐵 NOUN iajs-1818 94 7   PROPN iajs-1818 94 8 ker ker PROPN iajs-1818 94 9 𝑔. 𝑔. PROPN iajs-1818 94 10 conversely conversely ADV iajs-1818 94 11 , , PUNCT iajs-1818 94 12 let let VERB iajs-1818 94 13 𝑚 𝑚 X iajs-1818 94 14 ∈ ∈ PROPN iajs-1818 94 15 ker ker PROPN iajs-1818 95 1 𝑔 𝑔 PROPN iajs-1818 95 2   PROPN iajs-1818 95 3 𝑀 𝑀 PROPN iajs-1818 95 4 𝐴𝐵. 𝐴𝐵. PROPN iajs-1818 95 5 let let VERB iajs-1818 95 6 𝑚 𝑚 INTJ iajs-1818 95 7 𝑥 𝑥 VERB iajs-1818 95 8 𝑦 𝑦 NOUN iajs-1818 95 9 , , PUNCT iajs-1818 95 10 𝑥 𝑥 PROPN iajs-1818 95 11   PROPN iajs-1818 95 12 𝐴 𝐴 PROPN iajs-1818 95 13 and and CCONJ iajs-1818 95 14 𝑦 𝑦 PRON iajs-1818 95 15   PROPN iajs-1818 95 16 𝐵 𝐵 PROPN iajs-1818 95 17 , , PUNCT iajs-1818 95 18 then then ADV iajs-1818 95 19 𝑔 𝑔 PROPN iajs-1818 95 20 𝑚 𝑚 PROPN iajs-1818 95 21 𝑔 𝑔 PROPN iajs-1818 95 22 𝑥 𝑥 X iajs-1818 95 23 𝑦 𝑦 PROPN iajs-1818 95 24 𝑖 𝑖 X iajs-1818 95 25 𝑓 𝑓 PRON iajs-1818 95 26   ADP iajs-1818 95 27 𝑥 𝑥 PRON iajs-1818 95 28 𝑦 𝑦 NOUN iajs-1818 95 29 0 0 NUM iajs-1818 95 30 . . PUNCT iajs-1818 96 1 thus thus ADV iajs-1818 96 2 𝑖 𝑖 ADP iajs-1818 96 3 𝑓 𝑓 ADV iajs-1818 96 4 𝑥 𝑥 NOUN iajs-1818 96 5 0 0 NUM iajs-1818 96 6 and and CCONJ iajs-1818 96 7 hence hence ADV iajs-1818 96 8 𝑓 𝑓 ADV iajs-1818 96 9 𝑥 𝑥 NOUN iajs-1818 96 10 0 0 X iajs-1818 96 11 . . PUNCT iajs-1818 97 1 that that PRON iajs-1818 97 2 is be AUX iajs-1818 97 3 𝑥 𝑥 DET iajs-1818 97 4 ∈ ∈ PROPN iajs-1818 97 5 ker ker NOUN iajs-1818 98 1 𝑓 𝑓 PRON iajs-1818 98 2 and and CCONJ iajs-1818 98 3 since since SCONJ iajs-1818 98 4 𝑦 𝑦 PROPN iajs-1818 98 5 ∈ ∈ PROPN iajs-1818 98 6 𝐵. 𝐵. NOUN iajs-1818 98 7 thus thus ADV iajs-1818 98 8 𝑚 𝑚 ADP iajs-1818 98 9 𝑥 𝑥 PUNCT iajs-1818 98 10 𝑦 𝑦 NUM iajs-1818 98 11 ∈ ∈ NOUN iajs-1818 98 12 ker ker NOUN iajs-1818 98 13 𝑓 𝑓 DET iajs-1818 98 14 𝐵 𝐵 NOUN iajs-1818 98 15 , , PUNCT iajs-1818 98 16 ker ker PROPN iajs-1818 98 17 𝑔 𝑔 PROPN iajs-1818 98 18   PROPN iajs-1818 98 19 ker ker PROPN iajs-1818 98 20 𝑓 𝑓 DET iajs-1818 98 21 𝐵. 𝐵. PROPN iajs-1818 98 22 that that PRON iajs-1818 98 23 is be AUX iajs-1818 98 24 , , PUNCT iajs-1818 98 25 ker ker PROPN iajs-1818 98 26 𝑔 𝑔 PROPN iajs-1818 98 27 ker ker PROPN iajs-1818 99 1 𝑓 𝑓 PRON iajs-1818 99 2 𝐵. 𝐵. PROPN iajs-1818 99 3 clearly clearly ADV iajs-1818 99 4 ker ker VERB iajs-1818 99 5 𝑓 𝑓 DET iajs-1818 99 6 ⋂ ⋂ PROPN iajs-1818 99 7 𝐵 𝐵 PROPN iajs-1818 99 8 0 0 NUM iajs-1818 99 9 . . PUNCT iajs-1818 100 1 therefore therefore ADV iajs-1818 100 2 ker ker PROPN iajs-1818 100 3 𝑔 𝑔 PROPN iajs-1818 100 4 ker ker PROPN iajs-1818 100 5 𝑓 𝑓 DET iajs-1818 100 6   ADJ iajs-1818 100 7 𝐵 𝐵 NOUN iajs-1818 100 8 and and CCONJ iajs-1818 100 9 hence hence ADV iajs-1818 100 10 ker ker VERB iajs-1818 100 11 𝑓 𝑓 PRON iajs-1818 100 12 is be AUX iajs-1818 100 13 a a DET iajs-1818 100 14 coclosed coclose VERB iajs-1818 100 15 submodule submodule NOUN iajs-1818 100 16 in in ADP iajs-1818 100 17 ker ker PROPN iajs-1818 100 18 𝑔. 𝑔. PROPN iajs-1818 101 1 but but CCONJ iajs-1818 101 2 ker ker PROPN iajs-1818 101 3 𝑔 𝑔 PROPN iajs-1818 101 4 is be AUX iajs-1818 101 5 coclosed coclose VERB iajs-1818 101 6 in in ADP iajs-1818 101 7 𝑀 𝑀 PROPN iajs-1818 101 8 , , PUNCT iajs-1818 101 9 then then ADV iajs-1818 101 10 ker ker VERB iajs-1818 101 11 𝑓 𝑓 PRON iajs-1818 101 12 is be AUX iajs-1818 101 13 coclosed coclose VERB iajs-1818 101 14 in in ADP iajs-1818 101 15 𝑀. 𝑀. PROPN iajs-1818 101 16 but but CCONJ iajs-1818 101 17 𝐴 𝐴 PROPN iajs-1818 101 18 is be AUX iajs-1818 101 19 containing contain VERB iajs-1818 101 20 ker ker PROPN iajs-1818 102 1 𝑓 𝑓 X iajs-1818 102 2 , , PUNCT iajs-1818 102 3 so so ADV iajs-1818 102 4 by by ADP iajs-1818 102 5 lemma lemma PROPN iajs-1818 102 6 2.5 2.5 NUM iajs-1818 102 7 , , PUNCT iajs-1818 102 8 ker ker NOUN iajs-1818 102 9 𝑓 𝑓 PRON iajs-1818 102 10 is be AUX iajs-1818 102 11 coclosed coclose VERB iajs-1818 102 12 in in ADP iajs-1818 102 13 𝐴. 𝐴. PROPN iajs-1818 102 14 therefore therefore ADV iajs-1818 102 15 𝐴 𝐴 PROPN iajs-1818 102 16 is be AUX iajs-1818 102 17 a a DET iajs-1818 102 18 coclosed coclosed ADJ iajs-1818 102 19 rickart rickart NOUN iajs-1818 102 20 𝑅-module 𝑅-module PROPN iajs-1818 102 21 . . PUNCT iajs-1818 103 1 remark remark VERB iajs-1818 103 2 the the DET iajs-1818 103 3 direct direct ADJ iajs-1818 103 4 sum sum NOUN iajs-1818 103 5 of of ADP iajs-1818 103 6 coclosed coclose VERB iajs-1818 103 7 rickart rickart NOUN iajs-1818 103 8 modules module NOUN iajs-1818 103 9 need need AUX iajs-1818 103 10 not not PART iajs-1818 103 11 be be AUX iajs-1818 103 12 coclosed coclose VERB iajs-1818 103 13 rickart rickart NOUN iajs-1818 103 14 , , PUNCT iajs-1818 103 15 as as SCONJ iajs-1818 103 16 follows follow VERB iajs-1818 103 17 . . PUNCT iajs-1818 104 1 ℤ ℤ PROPN iajs-1818 104 2   VERB iajs-1818 104 3 ℤ2 ℤ2 NOUN iajs-1818 104 4 as as SCONJ iajs-1818 104 5 ℤ-module ℤ-module PROPN iajs-1818 104 6 is be AUX iajs-1818 104 7 not not PART iajs-1818 104 8 coclosed coclose VERB iajs-1818 104 9 rickart rickart NOUN iajs-1818 104 10 while while SCONJ iajs-1818 104 11 each each PRON iajs-1818 104 12 of of ADP iajs-1818 104 13 ℤ ℤ PROPN iajs-1818 104 14 and and CCONJ iajs-1818 104 15 ℤ2 ℤ2 NOUN iajs-1818 104 16 is be AUX iajs-1818 104 17 a a DET iajs-1818 104 18 coclosed coclosed ADJ iajs-1818 104 19 rickart rickart NOUN iajs-1818 104 20 module module NOUN iajs-1818 104 21 . . PUNCT iajs-1818 105 1 recall recall VERB iajs-1818 105 2 that that SCONJ iajs-1818 105 3 a a DET iajs-1818 105 4 submodule submodule NOUN iajs-1818 105 5 𝑁 𝑁 PROPN iajs-1818 105 6 of of ADP iajs-1818 105 7 an an DET iajs-1818 105 8 𝑅-module 𝑅-module PROPN iajs-1818 105 9 𝑀 𝑀 PROPN iajs-1818 105 10 is be AUX iajs-1818 105 11 said say VERB iajs-1818 105 12 to to PART iajs-1818 105 13 be be AUX iajs-1818 105 14 fully fully ADV iajs-1818 105 15 invariant invariant ADJ iajs-1818 105 16 if if SCONJ iajs-1818 105 17 𝑓 𝑓 DET iajs-1818 105 18 𝑁 𝑁 PROPN iajs-1818 105 19 is be AUX iajs-1818 105 20 included include VERB iajs-1818 105 21 in in ADP iajs-1818 105 22 𝑁 𝑁 PROPN iajs-1818 105 23 for for ADP iajs-1818 105 24 each each DET iajs-1818 105 25 𝑓 𝑓 DET iajs-1818 105 26 ∈ ∈ PROPN iajs-1818 105 27 end end NOUN iajs-1818 105 28 𝑀 𝑀 PROPN iajs-1818 106 1 [ [ X iajs-1818 106 2 13 13 NUM iajs-1818 106 3 ] ] PUNCT iajs-1818 106 4 . . PUNCT iajs-1818 107 1 we we PRON iajs-1818 107 2 call call VERB iajs-1818 107 3 𝑀 𝑀 PROPN iajs-1818 107 4 is be AUX iajs-1818 107 5 duo duo NOUN iajs-1818 107 6 module module NOUN iajs-1818 107 7 if if SCONJ iajs-1818 107 8 every every DET iajs-1818 107 9 submodule submodule NOUN iajs-1818 107 10 in in ADP iajs-1818 107 11 𝑀 𝑀 PROPN iajs-1818 107 12 is be AUX iajs-1818 107 13 fully fully ADV iajs-1818 107 14 invariant invariant ADJ iajs-1818 107 15 [ [ PUNCT iajs-1818 107 16 13 13 NUM iajs-1818 107 17 ] ] PUNCT iajs-1818 107 18 . . PUNCT iajs-1818 108 1 proposition proposition NOUN iajs-1818 108 2 let let VERB iajs-1818 108 3 𝑀 𝑀 PROPN iajs-1818 108 4 ⊕ ⊕ PROPN iajs-1818 109 1 ∈ ∈ PROPN iajs-1818 109 2 𝑀 𝑀 PROPN iajs-1818 109 3 be be VERB iajs-1818 109 4 a a DET iajs-1818 109 5 duo duo NOUN iajs-1818 109 6 module module NOUN iajs-1818 109 7 over over ADP iajs-1818 109 8 𝑅. 𝑅. NOUN iajs-1818 109 9 𝑀 𝑀 NOUN iajs-1818 109 10 is be AUX iajs-1818 109 11 a a DET iajs-1818 109 12 coclosed coclose VERB iajs-1818 109 13 rickart rickart NOUN iajs-1818 109 14 module module NOUN iajs-1818 109 15 if if SCONJ iajs-1818 109 16 and and CCONJ iajs-1818 109 17 only only ADV iajs-1818 109 18 if if SCONJ iajs-1818 109 19 𝑀 𝑀 PROPN iajs-1818 109 20 is be AUX iajs-1818 109 21 a a DET iajs-1818 109 22 coclosed coclosed ADJ iajs-1818 109 23 rickart rickart NOUN iajs-1818 109 24 for for ADP iajs-1818 109 25 each each DET iajs-1818 109 26 𝑖 𝑖 PROPN iajs-1818 109 27 ∈ ∈ PROPN iajs-1818 109 28   NOUN iajs-1818 109 29 . . PUNCT iajs-1818 110 1 proof proof NOUN iajs-1818 110 2 . . PUNCT iajs-1818 111 1 first first ADJ iajs-1818 111 2 side side NOUN iajs-1818 111 3 is be AUX iajs-1818 111 4 already already ADV iajs-1818 111 5 from from ADP iajs-1818 111 6 proposition proposition NOUN iajs-1818 111 7 2.6 2.6 NUM iajs-1818 111 8 . . PUNCT iajs-1818 112 1 the the DET iajs-1818 112 2 other other ADJ iajs-1818 112 3 side side NOUN iajs-1818 112 4 , , PUNCT iajs-1818 112 5 assume assume VERB iajs-1818 112 6 𝑓 𝑓 PRON iajs-1818 112 7 𝑓 𝑓 DET iajs-1818 112 8 ∈ ∈ PROPN iajs-1818 112 9 end end NOUN iajs-1818 112 10 𝑀 𝑀 PROPN iajs-1818 112 11 where where SCONJ iajs-1818 112 12 𝑓 𝑓 DET iajs-1818 112 13 ∈ ∈ PROPN iajs-1818 112 14 hom hom X iajs-1818 112 15 𝑀 𝑀 PROPN iajs-1818 112 16 , , PUNCT iajs-1818 112 17 𝑀 𝑀 PROPN iajs-1818 112 18 . . PUNCT iajs-1818 113 1 since since SCONJ iajs-1818 113 2 𝑀 𝑀 PROPN iajs-1818 113 3 is be AUX iajs-1818 113 4 fully fully ADV iajs-1818 113 5 invariant invariant ADJ iajs-1818 113 6 in in ADP iajs-1818 113 7 𝑀 𝑀 PROPN iajs-1818 113 8 ⊕ ⊕ PROPN iajs-1818 113 9 ∈ ∈ PROPN iajs-1818 113 10 𝑀 𝑀 PROPN iajs-1818 113 11 , , PUNCT iajs-1818 113 12 implies imply VERB iajs-1818 113 13 hom hom ADV iajs-1818 113 14 𝑀 𝑀 PROPN iajs-1818 113 15 , , PUNCT iajs-1818 113 16 𝑀 𝑀 PROPN iajs-1818 113 17 ) ) PUNCT iajs-1818 113 18 0 0 PUNCT iajs-1818 114 1 for for ADP iajs-1818 114 2 each each DET iajs-1818 114 3 𝑖 𝑖 X iajs-1818 114 4 𝑗 𝑗 PROPN iajs-1818 114 5 [ [ X iajs-1818 114 6 13 13 NUM iajs-1818 114 7 , , PUNCT iajs-1818 114 8 lemma lemma PROPN iajs-1818 114 9 1.9 1.9 NUM iajs-1818 114 10 ] ] PUNCT iajs-1818 114 11 . . PUNCT iajs-1818 115 1 further far ADV iajs-1818 115 2 𝑓 𝑓 PRON iajs-1818 115 3 𝑀 𝑀 PROPN iajs-1818 115 4 ⊆ ⊆ NUM iajs-1818 115 5 𝑀 𝑀 PROPN iajs-1818 115 6 for for ADP iajs-1818 115 7 all all DET iajs-1818 115 8 𝑖 𝑖 SYM iajs-1818 115 9 ∈ ∈ PROPN iajs-1818 115 10   NOUN iajs-1818 115 11 , , PUNCT iajs-1818 115 12 this this PRON iajs-1818 115 13 implies imply VERB iajs-1818 115 14 that that SCONJ iajs-1818 115 15 ker𝑓 ker𝑓 PROPN iajs-1818 115 16 ⊕ ⊕ PROPN iajs-1818 115 17 ∈ ∈ PROPN iajs-1818 115 18 ker ker VERB iajs-1818 115 19 𝑓 𝑓 X iajs-1818 115 20 . . PUNCT iajs-1818 116 1 we we PRON iajs-1818 116 2 claim claim VERB iajs-1818 116 3 that that SCONJ iajs-1818 116 4 ker𝑓 ker𝑓 PROPN iajs-1818 116 5 is be AUX iajs-1818 116 6 a a DET iajs-1818 116 7 coclosed coclose VERB iajs-1818 116 8 submodule submodule NOUN iajs-1818 116 9 in in ADP iajs-1818 116 10 𝑀. 𝑀. PROPN iajs-1818 116 11 to to PART iajs-1818 116 12 show show VERB iajs-1818 116 13 this this PRON iajs-1818 116 14 , , PUNCT iajs-1818 116 15 let let VERB iajs-1818 116 16 𝐾 𝐾 PROPN iajs-1818 116 17 ⊂ ⊂ PROPN iajs-1818 116 18 ⊕ ⊕ PROPN iajs-1818 117 1 ∈ ∈ PROPN iajs-1818 117 2 ker ker VERB iajs-1818 117 3 𝑓 𝑓 PRON iajs-1818 117 4 be be AUX iajs-1818 117 5 a a DET iajs-1818 117 6 submodule submodule NOUN iajs-1818 117 7 of of ADP iajs-1818 117 8 𝑀 𝑀 PROPN iajs-1818 117 9 , , PUNCT iajs-1818 117 10 then then ADV iajs-1818 117 11 𝐾 𝐾 PROPN iajs-1818 117 12 is be AUX iajs-1818 117 13 a a DET iajs-1818 117 14 fully fully ADV iajs-1818 117 15 invariant invariant ADJ iajs-1818 117 16 of of ADP iajs-1818 117 17 𝑀. 𝑀. PROPN iajs-1818 117 18 by by ADP iajs-1818 117 19 [ [ X iajs-1818 117 20 7 7 NUM iajs-1818 117 21 , , PUNCT iajs-1818 117 22 result result VERB iajs-1818 117 23 2.1 2.1 NUM iajs-1818 117 24 ] ] PUNCT iajs-1818 117 25 , , PUNCT iajs-1818 117 26 𝐾 𝐾 PROPN iajs-1818 117 27 ⊕ ⊕ PROPN iajs-1818 117 28 ∈ ∈ PROPN iajs-1818 118 1 𝐾 𝐾 PROPN iajs-1818 118 2 ∩ ∩ NOUN iajs-1818 118 3 𝑀 𝑀 PROPN iajs-1818 118 4 , , PUNCT iajs-1818 118 5 let let VERB iajs-1818 118 6 𝐾 𝐾 PROPN iajs-1818 118 7 𝐾 𝐾 PROPN iajs-1818 118 8 ∩ ∩ NOUN iajs-1818 118 9 𝑀 𝑀 PROPN iajs-1818 118 10 for for ADP iajs-1818 118 11 each each DET iajs-1818 118 12 𝑖 𝑖 SYM iajs-1818 118 13 ∈ ∈ PROPN iajs-1818 118 14 .easily .easily NOUN iajs-1818 118 15 to to PART iajs-1818 118 16 check check VERB iajs-1818 118 17 𝐾 𝐾 PROPN iajs-1818 118 18 ⊆ ⊆ NUM iajs-1818 118 19 ker ker NOUN iajs-1818 118 20 𝑓 𝑓 X iajs-1818 118 21 . . PUNCT iajs-1818 119 1 since since SCONJ iajs-1818 119 2 𝑀 𝑀 PROPN iajs-1818 119 3 is be AUX iajs-1818 119 4 coclosed coclose VERB iajs-1818 119 5 rickart rickart NOUN iajs-1818 119 6 implies imply VERB iajs-1818 119 7 that that SCONJ iajs-1818 119 8 ker ker PROPN iajs-1818 119 9 𝑓 𝑓 PRON iajs-1818 119 10 is be AUX iajs-1818 119 11 a a DET iajs-1818 119 12 coclosed coclosed ADJ iajs-1818 119 13 submodule submodule NOUN iajs-1818 119 14 of of ADP iajs-1818 119 15 𝑀 𝑀 PROPN iajs-1818 119 16 for for ADP iajs-1818 119 17 all all PRON iajs-1818 119 18 𝑖 𝑖 SYM iajs-1818 119 19 ∈ ∈ NOUN iajs-1818 119 20 . . VERB iajs-1818 119 21 thus thus ADV iajs-1818 119 22 there there PRON iajs-1818 119 23 is be VERB iajs-1818 119 24 a a DET iajs-1818 119 25 submodule submodule NOUN iajs-1818 119 26 𝑁 𝑁 PROPN iajs-1818 119 27 of of ADP iajs-1818 119 28 𝑀 𝑀 PROPN iajs-1818 119 29 , , PUNCT iajs-1818 119 30 ker ker X iajs-1818 119 31 𝑓 𝑓 DET iajs-1818 119 32 𝑁 𝑁 PROPN iajs-1818 119 33 𝑀 𝑀 PROPN iajs-1818 119 34 but but CCONJ iajs-1818 119 35 𝐾 𝐾 PROPN iajs-1818 119 36 𝑁 𝑁 PROPN iajs-1818 119 37 𝑀 𝑀 PROPN iajs-1818 119 38 . . PUNCT iajs-1818 120 1 this this PRON iajs-1818 120 2 implies imply VERB iajs-1818 120 3 that that SCONJ iajs-1818 120 4 ⊕ ⊕ PROPN iajs-1818 120 5 ∈ ∈ PROPN iajs-1818 120 6 ker ker VERB iajs-1818 121 1 𝑓 𝑓 ADV iajs-1818 121 2 ∑ ∑ DET iajs-1818 121 3 𝑁∈ 𝑁∈ PROPN iajs-1818 121 4 ⊕ ⊕ PROPN iajs-1818 121 5 ∈ ∈ PROPN iajs-1818 121 6 𝑀 𝑀 PROPN iajs-1818 121 7 but but CCONJ iajs-1818 121 8 ⊕ ⊕ PROPN iajs-1818 121 9 ∈ ∈ PROPN iajs-1818 122 1 𝐾 𝐾 PROPN iajs-1818 122 2 ∑ ∑ PROPN iajs-1818 122 3 𝑁∈ 𝑁∈ PROPN iajs-1818 122 4 ⊕ ⊕ PROPN iajs-1818 122 5 ∈ ∈ PROPN iajs-1818 122 6 𝑀 𝑀 PROPN iajs-1818 122 7 . . PUNCT iajs-1818 123 1 put put VERB iajs-1818 123 2 𝑁 𝑁 PROPN iajs-1818 123 3 ∑ ∑ PUNCT iajs-1818 123 4 𝑁∈ 𝑁∈ PUNCT iajs-1818 123 5 . . PUNCT iajs-1818 124 1 so so ADV iajs-1818 124 2 we we PRON iajs-1818 124 3 have have VERB iajs-1818 124 4 ker𝑓 ker𝑓 ADJ iajs-1818 124 5 𝑁 𝑁 PROPN iajs-1818 124 6 𝑀 𝑀 PROPN iajs-1818 124 7 but but CCONJ iajs-1818 124 8 𝐾 𝐾 PROPN iajs-1818 124 9 𝑁 𝑁 PROPN iajs-1818 124 10 𝑀 𝑀 PROPN iajs-1818 124 11 and and CCONJ iajs-1818 124 12 hence hence ADV iajs-1818 124 13 ker𝑓 ker𝑓 X iajs-1818 124 14 is be AUX iajs-1818 124 15 a a DET iajs-1818 124 16 coclosed coclosed ADJ iajs-1818 124 17 submodule submodule NOUN iajs-1818 124 18 of of ADP iajs-1818 124 19 𝑀. 𝑀. PROPN iajs-1818 124 20 therefore therefore ADV iajs-1818 124 21 𝑀 𝑀 PROPN iajs-1818 124 22 is be AUX iajs-1818 124 23 coclosed coclose VERB iajs-1818 124 24 rickart rickart NOUN iajs-1818 124 25 . . PUNCT iajs-1818 125 1 proposition proposition NOUN iajs-1818 125 2 2.9 2.9 NUM iajs-1818 125 3 . . PUNCT iajs-1818 126 1 let let VERB iajs-1818 126 2 𝑅 𝑅 NOUN iajs-1818 126 3 be be AUX iajs-1818 126 4 a a DET iajs-1818 126 5 ring ring NOUN iajs-1818 126 6 . . PUNCT iajs-1818 127 1 the the DET iajs-1818 127 2 next next ADJ iajs-1818 127 3 statements statement NOUN iajs-1818 127 4 are be AUX iajs-1818 127 5 equivalently equivalently ADV iajs-1818 127 6 ( ( PUNCT iajs-1818 127 7 1 1 X iajs-1818 127 8 ) ) PUNCT iajs-1818 127 9 𝑅 𝑅 NOUN iajs-1818 127 10 is be AUX iajs-1818 127 11 a a DET iajs-1818 127 12 coclosed coclose VERB iajs-1818 127 13 rickart rickart NOUN iajs-1818 127 14 𝑅-module 𝑅-module PROPN iajs-1818 127 15 for for ADP iajs-1818 127 16 any any DET iajs-1818 127 17 index index NOUN iajs-1818 127 18 set set VERB iajs-1818 127 19 . . NOUN iajs-1818 127 20 ihsciconf ihsciconf NOUN iajs-1818 127 21 2017 2017 NUM iajs-1818 127 22 special special ADJ iajs-1818 127 23 issue issue NOUN iajs-1818 127 24 ibn ibn PROPN iajs-1818 127 25 al al PROPN iajs-1818 127 26 - - PUNCT iajs-1818 127 27 haitham haitham PROPN iajs-1818 127 28 journal journal PROPN iajs-1818 127 29 for for ADP iajs-1818 127 30 pure pure ADJ iajs-1818 127 31 and and CCONJ iajs-1818 127 32 applied apply VERB iajs-1818 127 33 science science NOUN iajs-1818 127 34 https://doi.org/ https://doi.org/ NOUN iajs-1818 127 35 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 127 36 for for ADP iajs-1818 127 37 more more ADJ iajs-1818 127 38 information information NOUN iajs-1818 127 39 about about ADP iajs-1818 127 40 the the DET iajs-1818 127 41 conference conference NOUN iajs-1818 127 42 please please INTJ iajs-1818 127 43 visit visit VERB iajs-1818 127 44 the the DET iajs-1818 127 45 websites website NOUN iajs-1818 127 46 : : PUNCT iajs-1818 127 47 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 127 48     SPACE iajs-1818 127 49 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1818 127 50       SPACE iajs-1818 127 51 mathematics|456 mathematics|456 NOUN iajs-1818 127 52         SPACE iajs-1818 127 53 ( ( PUNCT iajs-1818 127 54 2 2 X iajs-1818 127 55 ) ) PUNCT iajs-1818 127 56 all all DET iajs-1818 127 57 projective projective ADJ iajs-1818 127 58 𝑅-modules 𝑅-modules PROPN iajs-1818 127 59 are be AUX iajs-1818 127 60 coclosed coclose VERB iajs-1818 127 61 rickart rickart NOUN iajs-1818 127 62 modules module NOUN iajs-1818 127 63 . . PUNCT iajs-1818 128 1 ( ( PUNCT iajs-1818 128 2 3 3 X iajs-1818 128 3 ) ) PUNCT iajs-1818 128 4 all all DET iajs-1818 128 5 free free ADJ iajs-1818 128 6 𝑅-modules 𝑅-modules PROPN iajs-1818 128 7 are be AUX iajs-1818 128 8 coclosed coclose VERB iajs-1818 128 9 rickart rickart NOUN iajs-1818 128 10 modules module NOUN iajs-1818 128 11 . . PUNCT iajs-1818 129 1 proof proof NOUN iajs-1818 129 2 . . PUNCT iajs-1818 130 1 1 1 NUM iajs-1818 130 2   NOUN iajs-1818 130 3 2 2 NUM iajs-1818 130 4 assume assume VERB iajs-1818 130 5 𝑀 𝑀 PROPN iajs-1818 130 6 is be AUX iajs-1818 130 7 projective projective ADJ iajs-1818 130 8 over over ADP iajs-1818 130 9 𝑅 𝑅 PROPN iajs-1818 130 10 , , PUNCT iajs-1818 130 11 we we PRON iajs-1818 130 12 get get VERB iajs-1818 130 13 a a DET iajs-1818 130 14 free free ADJ iajs-1818 130 15 module module NOUN iajs-1818 130 16 𝐹 𝐹 PROPN iajs-1818 130 17 over over ADP iajs-1818 130 18 𝑅 𝑅 PROPN iajs-1818 130 19 with with ADP iajs-1818 130 20 an an DET iajs-1818 130 21 epimorphism epimorphism NOUN iajs-1818 130 22 𝑓 𝑓 DET iajs-1818 130 23 ∶ ∶ NOUN iajs-1818 130 24 𝐹 𝐹 PROPN iajs-1818 130 25   NOUN iajs-1818 130 26 𝑀. 𝑀. PROPN iajs-1818 130 27 we we PRON iajs-1818 130 28 have have VERB iajs-1818 130 29 the the DET iajs-1818 130 30 short short ADJ iajs-1818 130 31 exact exact ADJ iajs-1818 130 32 sequence sequence NOUN iajs-1818 130 33 0 0 NUM iajs-1818 131 1 → → SYM iajs-1818 131 2 ker ker X iajs-1818 131 3 𝑓 𝑓 PRON iajs-1818 131 4 → → SYM iajs-1818 131 5 𝑅 𝑅 PROPN iajs-1818 131 6 → → SYM iajs-1818 131 7 𝑀 𝑀 PROPN iajs-1818 131 8 → → SYM iajs-1818 131 9 0 0 NUM iajs-1818 131 10 where where SCONJ iajs-1818 131 11 𝐹 𝐹 PROPN iajs-1818 131 12   PROPN iajs-1818 131 13 𝑅 𝑅 VERB iajs-1818 131 14 for for ADP iajs-1818 131 15 some some DET iajs-1818 131 16 index index NOUN iajs-1818 131 17 set set VERB iajs-1818 131 18 . . PROPN iajs-1818 131 19 but but CCONJ iajs-1818 131 20 𝑀 𝑀 PROPN iajs-1818 131 21 is be AUX iajs-1818 131 22 projective projective ADJ iajs-1818 131 23 then then ADV iajs-1818 131 24 the the DET iajs-1818 131 25 sequence sequence NOUN iajs-1818 131 26 splits split VERB iajs-1818 131 27 . . PUNCT iajs-1818 132 1 thus thus ADV iajs-1818 132 2 𝑅 𝑅 VERB iajs-1818 132 3   PRON iajs-1818 132 4 ker ker VERB iajs-1818 132 5 𝑓 𝑓 DET iajs-1818 132 6   ADJ iajs-1818 132 7 𝑀. 𝑀. PROPN iajs-1818 132 8 because because SCONJ iajs-1818 132 9 𝑅 𝑅 NOUN iajs-1818 132 10 is be AUX iajs-1818 132 11 a a DET iajs-1818 132 12 coclosed coclosed ADJ iajs-1818 132 13 rickart rickart NOUN iajs-1818 132 14 module module NOUN iajs-1818 132 15 , , PUNCT iajs-1818 132 16 therefore therefore ADV iajs-1818 132 17 from from ADP iajs-1818 132 18 proposition proposition NOUN iajs-1818 132 19 2.3 2.3 NUM iajs-1818 132 20 , , PUNCT iajs-1818 132 21 ker ker NOUN iajs-1818 132 22 𝑓 𝑓 DET iajs-1818 132 23   PROPN iajs-1818 132 24 𝑀 𝑀 PROPN iajs-1818 132 25 is be AUX iajs-1818 132 26 also also ADV iajs-1818 132 27 coclosed coclose VERB iajs-1818 132 28 rickart rickart NOUN iajs-1818 132 29 . . PUNCT iajs-1818 133 1 hence hence ADV iajs-1818 133 2 from from ADP iajs-1818 133 3 lemma lemma PROPN iajs-1818 133 4 2.5 2.5 NUM iajs-1818 133 5 , , PUNCT iajs-1818 133 6 𝑀 𝑀 PROPN iajs-1818 133 7 is be AUX iajs-1818 133 8 coclosed coclose VERB iajs-1818 133 9 rickart rickart NOUN iajs-1818 133 10 . . PUNCT iajs-1818 134 1 it it PRON iajs-1818 134 2 is be AUX iajs-1818 134 3 clear clear ADJ iajs-1818 134 4 2 2 NUM iajs-1818 134 5   NOUN iajs-1818 134 6 1 1 NUM iajs-1818 134 7 and and CCONJ iajs-1818 134 8 2 2 NUM iajs-1818 134 9   ADP iajs-1818 134 10 3 3 NUM iajs-1818 134 11 . . PUNCT iajs-1818 135 1 similarly similarly ADV iajs-1818 135 2 , , PUNCT iajs-1818 135 3 we we PRON iajs-1818 135 4 can can AUX iajs-1818 135 5 prove prove VERB iajs-1818 135 6 1 1 NUM iajs-1818 135 7   ADP iajs-1818 135 8 3 3 NUM iajs-1818 135 9 . . PUNCT iajs-1818 136 1 proposition proposition NOUN iajs-1818 136 2 let let VERB iajs-1818 136 3 𝑀 𝑀 PRON iajs-1818 136 4 be be AUX iajs-1818 136 5 a a DET iajs-1818 136 6 coclosed coclosed ADJ iajs-1818 136 7 simple simple ADJ iajs-1818 136 8 coclosed coclose VERB iajs-1818 136 9 rickart rickart NOUN iajs-1818 136 10 module module NOUN iajs-1818 136 11 which which PRON iajs-1818 136 12 has have VERB iajs-1818 136 13 a a DET iajs-1818 136 14 nonzero nonzero ADJ iajs-1818 136 15 maximal maximal ADJ iajs-1818 136 16 submodule submodule NOUN iajs-1818 136 17 𝑁 𝑁 PROPN iajs-1818 136 18 , , PUNCT iajs-1818 136 19 then then ADV iajs-1818 136 20 𝑀 𝑀 PROPN iajs-1818 136 21 ⊕ ⊕ PROPN iajs-1818 136 22 is be AUX iajs-1818 136 23 not not PART iajs-1818 136 24 a a DET iajs-1818 136 25 coclosed coclose VERB iajs-1818 136 26 rickart rickart NOUN iajs-1818 136 27 module module NOUN iajs-1818 136 28 . . PUNCT iajs-1818 137 1 proof proof NOUN iajs-1818 137 2 . . PUNCT iajs-1818 138 1 suppose suppose VERB iajs-1818 138 2 𝑀 𝑀 PROPN iajs-1818 138 3 ⊕ ⊕ PROPN iajs-1818 138 4 is be AUX iajs-1818 138 5 coclosed coclose VERB iajs-1818 138 6 rickart rickart NOUN iajs-1818 138 7 , , PUNCT iajs-1818 138 8 𝑓 𝑓 DET iajs-1818 138 9 ∈ ∈ NOUN iajs-1818 138 10 end end NOUN iajs-1818 138 11 𝑀 𝑀 PROPN iajs-1818 138 12 ⊕ ⊕ PROPN iajs-1818 138 13 defined define VERB iajs-1818 138 14 by by ADP iajs-1818 138 15 𝑓 𝑓 DET iajs-1818 138 16 𝑚 𝑚 NOUN iajs-1818 138 17 , , PUNCT iajs-1818 138 18 𝑛 𝑛 PROPN iajs-1818 138 19 0 0 NUM iajs-1818 138 20 , , PUNCT iajs-1818 138 21 𝑚 𝑚 X iajs-1818 138 22 for for ADP iajs-1818 138 23 all all DET iajs-1818 138 24 𝑚 𝑚 ADP iajs-1818 138 25 ∈ ∈ PROPN iajs-1818 138 26 𝑀 𝑀 PROPN iajs-1818 138 27 , , PUNCT iajs-1818 138 28 𝑛 𝑛 PRON iajs-1818 138 29 ∈ ∈ PROPN iajs-1818 138 30 . . PUNCT iajs-1818 139 1 then then ADV iajs-1818 139 2 ker ker VERB iajs-1818 139 3 𝑓 𝑓 PRON iajs-1818 139 4 𝑚 𝑚 NOUN iajs-1818 139 5 , , PUNCT iajs-1818 139 6 𝑛 𝑛 PROPN iajs-1818 139 7 ∈ ∈ PROPN iajs-1818 139 8 𝑀 𝑀 PROPN iajs-1818 139 9 ⊕ ⊕ PROPN iajs-1818 139 10 | | ADV iajs-1818 139 11 𝑓 𝑓 ADV iajs-1818 139 12 𝑚 𝑚 NOUN iajs-1818 139 13 , , PUNCT iajs-1818 139 14 𝑛 𝑛 PROPN iajs-1818 139 15 0 0 NUM iajs-1818 139 16 , , PUNCT iajs-1818 139 17 0 0 NUM iajs-1818 139 18 𝑚 𝑚 NOUN iajs-1818 139 19 , , PUNCT iajs-1818 139 20 𝑛 𝑛 PROPN iajs-1818 139 21 ∈ ∈ PROPN iajs-1818 139 22 𝑀 𝑀 PROPN iajs-1818 139 23 ⊕ ⊕ PROPN iajs-1818 139 24 𝑚 𝑚 ADP iajs-1818 139 25 𝑁 𝑁 PROPN iajs-1818 139 26 𝑁 𝑁 PROPN iajs-1818 139 27 𝑁 𝑁 PROPN iajs-1818 139 28 ⊕ ⊕ PROPN iajs-1818 139 29 is be AUX iajs-1818 139 30 a a DET iajs-1818 139 31 coclosed coclosed NOUN iajs-1818 139 32 of of ADP iajs-1818 139 33 𝑀 𝑀 PROPN iajs-1818 139 34 ⊕ ⊕ PROPN iajs-1818 139 35 . . PUNCT iajs-1818 140 1 but but CCONJ iajs-1818 140 2 𝑁 𝑁 PROPN iajs-1818 140 3 is be AUX iajs-1818 140 4 a a DET iajs-1818 140 5 coclosed coclosed NOUN iajs-1818 140 6 of of ADP iajs-1818 140 7 𝑁 𝑁 PROPN iajs-1818 140 8 ⊕ ⊕ PROPN iajs-1818 140 9 , , PUNCT iajs-1818 140 10 then then ADV iajs-1818 140 11 by by ADP iajs-1818 140 12 lemma lemma PROPN iajs-1818 140 13 2.5 2.5 NUM iajs-1818 140 14 , , PUNCT iajs-1818 140 15 𝑁 𝑁 PROPN iajs-1818 140 16 is be AUX iajs-1818 140 17 a a DET iajs-1818 140 18 coclosed coclosed NOUN iajs-1818 140 19 of of ADP iajs-1818 140 20 𝑀 𝑀 PROPN iajs-1818 140 21 ⊕ ⊕ PROPN iajs-1818 140 22 . . PUNCT iajs-1818 141 1 again again ADV iajs-1818 141 2 by by ADP iajs-1818 141 3 result result NOUN iajs-1818 141 4 2.5 2.5 NUM iajs-1818 141 5 , , PUNCT iajs-1818 141 6 𝑁 𝑁 PROPN iajs-1818 141 7 is be AUX iajs-1818 141 8 a a DET iajs-1818 141 9 coclosed coclosed NOUN iajs-1818 141 10 of of ADP iajs-1818 141 11 𝑀 𝑀 PROPN iajs-1818 141 12 , , PUNCT iajs-1818 141 13 this this PRON iajs-1818 141 14 implies imply VERB iajs-1818 141 15 a a DET iajs-1818 141 16 false false ADJ iajs-1818 141 17 because because SCONJ iajs-1818 141 18 𝑀 𝑀 PROPN iajs-1818 141 19 is be AUX iajs-1818 141 20 coclosed coclose VERB iajs-1818 141 21 simple simple ADJ iajs-1818 141 22 . . PUNCT iajs-1818 142 1 yeild yeild PROPN iajs-1818 142 2 𝑀 𝑀 PROPN iajs-1818 142 3 ⊕ ⊕ PROPN iajs-1818 142 4 is be AUX iajs-1818 142 5 not not PART iajs-1818 142 6 coclosed coclose VERB iajs-1818 142 7 rickart rickart NOUN iajs-1818 142 8 . . PUNCT iajs-1818 143 1 coclosed coclose VERB iajs-1818 143 2 rickart rickart NOUN iajs-1818 143 3 modules module NOUN iajs-1818 143 4 and and CCONJ iajs-1818 143 5 other other ADJ iajs-1818 143 6 topics topic NOUN iajs-1818 143 7 this this DET iajs-1818 143 8 part part NOUN iajs-1818 143 9 looks look VERB iajs-1818 143 10 for for ADP iajs-1818 143 11 any any DET iajs-1818 143 12 relationship relationship NOUN iajs-1818 143 13 between between ADP iajs-1818 143 14 coclosed coclose VERB iajs-1818 143 15 rickart rickart NOUN iajs-1818 143 16 modules module NOUN iajs-1818 143 17 and and CCONJ iajs-1818 143 18 other other ADJ iajs-1818 143 19 concepts concept NOUN iajs-1818 143 20 . . PUNCT iajs-1818 144 1 we we PRON iajs-1818 144 2 first first ADV iajs-1818 144 3 study study VERB iajs-1818 144 4 the the DET iajs-1818 144 5 next next ADJ iajs-1818 144 6 condition condition NOUN iajs-1818 144 7 ( ( PUNCT iajs-1818 144 8 * * NOUN iajs-1818 144 9 ) ) PUNCT iajs-1818 144 10 for for ADP iajs-1818 144 11 an an DET iajs-1818 144 12 𝑅-module 𝑅-module PROPN iajs-1818 144 13 : : PUNCT iajs-1818 144 14 for for ADP iajs-1818 144 15 any any DET iajs-1818 144 16 submodule submodule NOUN iajs-1818 144 17 𝑁 𝑁 PROPN iajs-1818 144 18 of of ADP iajs-1818 144 19 𝑀 𝑀 PROPN iajs-1818 144 20 for for ADP iajs-1818 144 21 which which PRON iajs-1818 144 22 ≅ ≅ NOUN iajs-1818 144 23 𝐻 𝐻 VERB iajs-1818 144 24 where where SCONJ iajs-1818 144 25 𝐻 𝐻 PROPN iajs-1818 144 26 is be AUX iajs-1818 144 27 a a DET iajs-1818 144 28 summand summand NOUN iajs-1818 144 29 of of ADP iajs-1818 144 30 𝑀 𝑀 PROPN iajs-1818 144 31 , , PUNCT iajs-1818 144 32 yeild yeild PROPN iajs-1818 144 33 𝑁 𝑁 PROPN iajs-1818 144 34 is be AUX iajs-1818 144 35 a a DET iajs-1818 144 36 coclosed coclosed ADJ iajs-1818 144 37 submodule submodule NOUN iajs-1818 144 38 of of ADP iajs-1818 144 39 𝑀. 𝑀. PROPN iajs-1818 144 40 proposition proposition NOUN iajs-1818 144 41 . . PUNCT iajs-1818 145 1 every every DET iajs-1818 145 2 coclosed coclose VERB iajs-1818 145 3 rickart rickart NOUN iajs-1818 145 4 module module NOUN iajs-1818 145 5 satisfies satisfy VERB iajs-1818 145 6 the the DET iajs-1818 145 7 condition condition NOUN iajs-1818 145 8 ( ( PUNCT iajs-1818 145 9 * * NOUN iajs-1818 145 10 ) ) PUNCT iajs-1818 145 11 . . PUNCT iajs-1818 146 1 proof proof NOUN iajs-1818 146 2 . . PUNCT iajs-1818 147 1 when when SCONJ iajs-1818 147 2 𝑁 𝑁 PROPN iajs-1818 147 3 is be AUX iajs-1818 147 4 a a DET iajs-1818 147 5 submodule submodule NOUN iajs-1818 147 6 of of ADP iajs-1818 147 7 a a DET iajs-1818 147 8 coclosed coclosed ADJ iajs-1818 147 9 rickart rickart NOUN iajs-1818 147 10 𝑀 𝑀 PROPN iajs-1818 147 11 over over ADP iajs-1818 147 12 𝑅 𝑅 PROPN iajs-1818 147 13 with with ADP iajs-1818 147 14 ≅ ≅ PROPN iajs-1818 147 15 𝐻 𝐻 PROPN iajs-1818 147 16 where where SCONJ iajs-1818 147 17 𝐻 𝐻 PROPN iajs-1818 147 18 is be AUX iajs-1818 147 19 a a DET iajs-1818 147 20 summand summand NOUN iajs-1818 147 21 in in ADP iajs-1818 147 22 𝑀. 𝑀. PROPN iajs-1818 147 23 hence hence ADV iajs-1818 147 24 𝑓 𝑓 DET iajs-1818 147 25 ∶ ∶ NOUN iajs-1818 147 26 → → PUNCT iajs-1818 147 27 𝐻 𝐻 NOUN iajs-1818 147 28 is be AUX iajs-1818 147 29 an an DET iajs-1818 147 30 isomorphism isomorphism NOUN iajs-1818 147 31 , , PUNCT iajs-1818 147 32 implies imply VERB iajs-1818 147 33 𝑀 𝑀 PROPN iajs-1818 147 34 → → SYM iajs-1818 147 35 → → SYM iajs-1818 147 36 𝐻 𝐻 PROPN iajs-1818 147 37 → → SYM iajs-1818 147 38 𝑀. 𝑀. PROPN iajs-1818 147 39 then then ADV iajs-1818 147 40 𝑖𝑓𝜋 𝑖𝑓𝜋 X iajs-1818 147 41 ∈ ∈ PROPN iajs-1818 147 42 end end NOUN iajs-1818 148 1 𝑀 𝑀 PROPN iajs-1818 148 2 ker ker NOUN iajs-1818 149 1 𝑖𝑓𝜋 𝑖𝑓𝜋 VERB iajs-1818 149 2 𝜋 𝜋 PRON iajs-1818 150 1 ker ker NOUN iajs-1818 150 2 𝑖𝑓 𝑖𝑓 INTJ iajs-1818 151 1 𝜋 𝜋 NOUN iajs-1818 151 2 0 0 NUM iajs-1818 151 3 𝑁. 𝑁. PROPN iajs-1818 151 4 because because SCONJ iajs-1818 151 5 𝑀 𝑀 PROPN iajs-1818 151 6 is be AUX iajs-1818 151 7 coclosed coclose VERB iajs-1818 151 8 rickart rickart NOUN iajs-1818 151 9 , , PUNCT iajs-1818 151 10 then then ADV iajs-1818 151 11 𝑁 𝑁 PROPN iajs-1818 151 12 = = PUNCT iajs-1818 151 13 ker ker PROPN iajs-1818 151 14 𝑖𝑓𝜋 𝑖𝑓𝜋 PROPN iajs-1818 151 15 is be AUX iajs-1818 151 16 coclosed coclose VERB iajs-1818 151 17 in in ADP iajs-1818 151 18 𝑀. 𝑀. PROPN iajs-1818 151 19 corollary corollary NOUN iajs-1818 151 20 when when SCONJ iajs-1818 151 21 𝑀 𝑀 PROPN iajs-1818 151 22 is be AUX iajs-1818 151 23 a a DET iajs-1818 151 24 coclosed coclose VERB iajs-1818 151 25 rickart rickart NOUN iajs-1818 151 26 module module NOUN iajs-1818 151 27 over over ADP iajs-1818 151 28 𝑅 𝑅 PROPN iajs-1818 151 29 with with ADP iajs-1818 151 30 the the DET iajs-1818 151 31 condition condition NOUN iajs-1818 151 32 ( ( PUNCT iajs-1818 151 33 * * PUNCT iajs-1818 151 34 ) ) PUNCT iajs-1818 151 35 for for ADP iajs-1818 151 36 each each DET iajs-1818 151 37 submodule submodule NOUN iajs-1818 151 38 𝑁 𝑁 PROPN iajs-1818 151 39 of of ADP iajs-1818 151 40 𝑀 𝑀 PROPN iajs-1818 151 41 , , PUNCT iajs-1818 151 42 therefore therefore ADV iajs-1818 151 43 𝑀 𝑀 PROPN iajs-1818 151 44 is be AUX iajs-1818 151 45 a a DET iajs-1818 151 46 cosemisimple cosemisimple NOUN iajs-1818 151 47 . . PUNCT iajs-1818 152 1 proof proof NOUN iajs-1818 152 2 . . PUNCT iajs-1818 153 1 obvious obvious ADJ iajs-1818 153 2 by by ADP iajs-1818 153 3 proposition proposition NOUN iajs-1818 153 4 3.1 3.1 NUM iajs-1818 153 5 . . PUNCT iajs-1818 154 1 the the DET iajs-1818 154 2 next next ADJ iajs-1818 154 3 result result NOUN iajs-1818 154 4 is be AUX iajs-1818 154 5 to to PART iajs-1818 154 6 find find VERB iajs-1818 154 7 a a DET iajs-1818 154 8 certain certain ADJ iajs-1818 154 9 situation situation NOUN iajs-1818 154 10 under under ADP iajs-1818 154 11 which which PRON iajs-1818 154 12 the the DET iajs-1818 154 13 reverse reverse NOUN iajs-1818 154 14 of of ADP iajs-1818 154 15 proposition proposition NOUN iajs-1818 154 16 3.1 3.1 NUM iajs-1818 154 17 is be AUX iajs-1818 154 18 hold hold NOUN iajs-1818 154 19 . . PUNCT iajs-1818 155 1 proposition proposition NOUN iajs-1818 155 2 if if SCONJ iajs-1818 155 3 𝑀 𝑀 PROPN iajs-1818 155 4 a a DET iajs-1818 155 5 module module NOUN iajs-1818 155 6 has have VERB iajs-1818 155 7 the the DET iajs-1818 155 8 condition condition NOUN iajs-1818 155 9 ( ( PUNCT iajs-1818 155 10 * * PUNCT iajs-1818 155 11 ) ) PUNCT iajs-1818 155 12 with with ADP iajs-1818 155 13 im𝑓 im𝑓 NOUN iajs-1818 155 14 isomorphic isomorphic ADJ iajs-1818 155 15 to to ADP iajs-1818 155 16 a a DET iajs-1818 155 17 summand summand NOUN iajs-1818 155 18 of of ADP iajs-1818 155 19 𝑀 𝑀 PROPN iajs-1818 155 20 for for ADP iajs-1818 155 21 all all DET iajs-1818 155 22 𝑓 𝑓 DET iajs-1818 155 23 ∈ ∈ NOUN iajs-1818 155 24 end end NOUN iajs-1818 155 25 𝑀 𝑀 PROPN iajs-1818 155 26 , , PUNCT iajs-1818 155 27 implies imply VERB iajs-1818 155 28 𝑀 𝑀 PROPN iajs-1818 155 29 is be AUX iajs-1818 155 30 coclosed coclose VERB iajs-1818 155 31 rickart rickart NOUN iajs-1818 155 32 . . PUNCT iajs-1818 156 1 proof proof NOUN iajs-1818 156 2 . . PUNCT iajs-1818 157 1 when when SCONJ iajs-1818 157 2 𝑓 𝑓 DET iajs-1818 157 3 ∈ ∈ PROPN iajs-1818 157 4 end end NOUN iajs-1818 157 5 𝑀 𝑀 PROPN iajs-1818 157 6 , , PUNCT iajs-1818 157 7 from from ADP iajs-1818 157 8 assumption assumption NOUN iajs-1818 157 9 , , PUNCT iajs-1818 157 10 im𝑓 im𝑓 NOUN iajs-1818 157 11 ≅ ≅ PROPN iajs-1818 157 12 𝐻 𝐻 PROPN iajs-1818 157 13 where where SCONJ iajs-1818 157 14 𝐻 𝐻 PROPN iajs-1818 157 15 is be AUX iajs-1818 157 16 a a DET iajs-1818 157 17 direct direct ADJ iajs-1818 157 18 summand summand NOUN iajs-1818 157 19 of of ADP iajs-1818 157 20 𝑀 𝑀 PROPN iajs-1818 157 21 , , PUNCT iajs-1818 157 22 implies imply VERB iajs-1818 157 23 that that SCONJ iajs-1818 157 24 ≅ ≅ PROPN iajs-1818 157 25 𝐻. 𝐻. PROPN iajs-1818 157 26 so so ADV iajs-1818 157 27 by by ADP iajs-1818 157 28 the the DET iajs-1818 157 29 condition condition NOUN iajs-1818 157 30 ( ( PUNCT iajs-1818 157 31 * * NOUN iajs-1818 157 32 ) ) PUNCT iajs-1818 157 33 , , PUNCT iajs-1818 157 34 ker ker X iajs-1818 157 35 𝑓 𝑓 PRON iajs-1818 157 36 is be AUX iajs-1818 157 37 coclosed coclose VERB iajs-1818 157 38 of of ADP iajs-1818 157 39 𝑀 𝑀 PROPN iajs-1818 157 40 , , PUNCT iajs-1818 157 41 as as SCONJ iajs-1818 157 42 asserted assert VERB iajs-1818 157 43 . . PUNCT iajs-1818 158 1 ihsciconf ihsciconf PROPN iajs-1818 158 2 2017 2017 NUM iajs-1818 158 3 special special ADJ iajs-1818 158 4 issue issue NOUN iajs-1818 158 5 ibn ibn PROPN iajs-1818 158 6 al al PROPN iajs-1818 158 7 - - PUNCT iajs-1818 158 8 haitham haitham PROPN iajs-1818 158 9 journal journal PROPN iajs-1818 158 10 for for ADP iajs-1818 158 11 pure pure ADJ iajs-1818 158 12 and and CCONJ iajs-1818 158 13 applied apply VERB iajs-1818 158 14 science science NOUN iajs-1818 158 15 https://doi.org/ https://doi.org/ NOUN iajs-1818 158 16 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 158 17 for for ADP iajs-1818 158 18 more more ADJ iajs-1818 158 19 information information NOUN iajs-1818 158 20 about about ADP iajs-1818 158 21 the the DET iajs-1818 158 22 conference conference NOUN iajs-1818 158 23 please please INTJ iajs-1818 158 24 visit visit VERB iajs-1818 158 25 the the DET iajs-1818 158 26 websites website NOUN iajs-1818 158 27 : : PUNCT iajs-1818 158 28 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 158 29     SPACE iajs-1818 158 30 www.ihsciconf.org www.ihsciconf.org NOUN iajs-1818 158 31       SPACE iajs-1818 158 32 mathematics|457 mathematics|457 NOUN iajs-1818 158 33         SPACE iajs-1818 158 34 let let VERB iajs-1818 158 35 𝑀 𝑀 PRON iajs-1818 158 36 be be AUX iajs-1818 158 37 a a DET iajs-1818 158 38 module module NOUN iajs-1818 158 39 over over ADP iajs-1818 158 40 𝑅. 𝑅. NOUN iajs-1818 158 41 we we PRON iajs-1818 158 42 record record VERB iajs-1818 158 43 that that PRON iajs-1818 158 44 , , PUNCT iajs-1818 158 45 𝑀 𝑀 PROPN iajs-1818 158 46 lifiting lifite VERB iajs-1818 158 47 when when SCONJ iajs-1818 158 48 each each DET iajs-1818 158 49 submodule submodule NOUN iajs-1818 158 50 𝑁 𝑁 PROPN iajs-1818 158 51 of of ADP iajs-1818 158 52 𝑀 𝑀 PROPN iajs-1818 158 53 , , PUNCT iajs-1818 158 54 there there PRON iajs-1818 158 55 is be VERB iajs-1818 158 56 a a DET iajs-1818 158 57 direct direct ADJ iajs-1818 158 58 summand summand NOUN iajs-1818 158 59 𝐾 𝐾 PROPN iajs-1818 158 60 of of ADP iajs-1818 158 61 𝑀 𝑀 PROPN iajs-1818 158 62 such such ADJ iajs-1818 158 63 that that SCONJ iajs-1818 158 64 𝐾 𝐾 PROPN iajs-1818 158 65 ⊆ ⊆ NUM iajs-1818 158 66 𝑁 𝑁 PROPN iajs-1818 158 67 and and CCONJ iajs-1818 158 68 ≪ ≪ ADJ iajs-1818 158 69 [ [ X iajs-1818 158 70 1 1 NUM iajs-1818 158 71 ] ] PUNCT iajs-1818 158 72 . . PUNCT iajs-1818 159 1 proposition proposition NOUN iajs-1818 159 2 every every DET iajs-1818 159 3 lifiting lifite VERB iajs-1818 159 4 coclosed coclose VERB iajs-1818 159 5 rickart rickart NOUN iajs-1818 159 6 module module NOUN iajs-1818 159 7 is be AUX iajs-1818 159 8 rickart rickart NOUN iajs-1818 159 9 module module NOUN iajs-1818 159 10 . . PUNCT iajs-1818 160 1 proof proof NOUN iajs-1818 160 2 . . PUNCT iajs-1818 161 1 when when SCONJ iajs-1818 161 2 𝑀 𝑀 PROPN iajs-1818 161 3 be be VERB iajs-1818 161 4 a a DET iajs-1818 161 5 lifiting lifite VERB iajs-1818 161 6 coclosed coclose VERB iajs-1818 161 7 rickart rickart NOUN iajs-1818 161 8 over over ADP iajs-1818 161 9 𝑅 𝑅 PROPN iajs-1818 161 10 and and CCONJ iajs-1818 161 11 𝑓 𝑓 DET iajs-1818 161 12 ∈ ∈ PROPN iajs-1818 161 13 end end NOUN iajs-1818 161 14 𝑀 𝑀 PROPN iajs-1818 161 15 . . PUNCT iajs-1818 162 1 because because SCONJ iajs-1818 162 2 𝑀 𝑀 PROPN iajs-1818 162 3 is be AUX iajs-1818 162 4 lifting lift VERB iajs-1818 162 5 , , PUNCT iajs-1818 162 6 there there PRON iajs-1818 162 7 exists exist VERB iajs-1818 162 8 a a DET iajs-1818 162 9 summand summand NOUN iajs-1818 162 10 𝐾 𝐾 PROPN iajs-1818 162 11 of of ADP iajs-1818 162 12 𝑀 𝑀 PROPN iajs-1818 162 13 , , PUNCT iajs-1818 162 14 𝐾 𝐾 PROPN iajs-1818 162 15 ⊆ ⊆ NUM iajs-1818 162 16 ker ker NOUN iajs-1818 162 17 𝑓 𝑓 X iajs-1818 162 18 with with ADP iajs-1818 162 19 ≪ ≪ NOUN iajs-1818 162 20 . . PUNCT iajs-1818 163 1 but but CCONJ iajs-1818 163 2 𝑀 𝑀 PROPN iajs-1818 163 3 is be AUX iajs-1818 163 4 coclosed coclose VERB iajs-1818 163 5 rickart rickart NOUN iajs-1818 163 6 , , PUNCT iajs-1818 163 7 implies imply VERB iajs-1818 163 8 ker ker PROPN iajs-1818 163 9 𝑓 𝑓 PRON iajs-1818 163 10 is be AUX iajs-1818 163 11 a a DET iajs-1818 163 12 coclosed coclose VERB iajs-1818 163 13 in in ADP iajs-1818 163 14 𝑀 𝑀 PROPN iajs-1818 163 15 , , PUNCT iajs-1818 163 16 hence hence ADV iajs-1818 163 17 ker ker VERB iajs-1818 163 18 𝑓 𝑓 DET iajs-1818 163 19 𝐾 𝐾 PROPN iajs-1818 163 20 , , PUNCT iajs-1818 163 21 as as SCONJ iajs-1818 163 22 desired desire VERB iajs-1818 163 23 . . PUNCT iajs-1818 164 1 examples example NOUN iajs-1818 164 2 investigate investigate VERB iajs-1818 164 3 ℤ ℤ PROPN iajs-1818 164 4   PROPN iajs-1818 164 5 ℤ ℤ PROPN iajs-1818 164 6 is be AUX iajs-1818 164 7 lifiting lifite VERB iajs-1818 164 8 as as ADP iajs-1818 164 9 ℤ ℤ NOUN iajs-1818 164 10 module module NOUN iajs-1818 164 11 . . PUNCT iajs-1818 165 1 but but CCONJ iajs-1818 165 2 ℤ ℤ PROPN iajs-1818 165 3   PROPN iajs-1818 165 4 ℤ ℤ PROPN iajs-1818 165 5 is be AUX iajs-1818 165 6 not not PART iajs-1818 165 7 coclosed coclose VERB iajs-1818 165 8 rickart rickart NOUN iajs-1818 165 9 as as ADP iajs-1818 165 10 ℤ ℤ NOUN iajs-1818 165 11 module module NOUN iajs-1818 165 12 , , PUNCT iajs-1818 165 13 if if SCONJ iajs-1818 165 14 otherwise otherwise ADV iajs-1818 165 15 , , PUNCT iajs-1818 165 16 then then ADV iajs-1818 165 17 by by ADP iajs-1818 165 18     SPACE iajs-1818 165 19 proposition proposition NOUN iajs-1818 165 20 2.5 2.5 NUM iajs-1818 165 21 , , PUNCT iajs-1818 165 22 each each DET iajs-1818 165 23 summand summand NOUN iajs-1818 165 24 of of ADP iajs-1818 165 25 ℤ ℤ PROPN iajs-1818 165 26   PROPN iajs-1818 165 27 ℤ ℤ PROPN iajs-1818 165 28 is be AUX iajs-1818 165 29 coclosed coclose VERB iajs-1818 165 30 rickart rickart NOUN iajs-1818 165 31 , , PUNCT iajs-1818 165 32 whereas whereas SCONJ iajs-1818 165 33 this this PRON iajs-1818 165 34 contradicts contradict VERB iajs-1818 165 35 that that SCONJ iajs-1818 165 36 ℤ ℤ PROPN iajs-1818 165 37   VERB iajs-1818 165 38 0 0 NUM iajs-1818 166 1 ≅ ≅ PROPN iajs-1818 166 2 ℤ ℤ PROPN iajs-1818 166 3 is be AUX iajs-1818 166 4 not not PART iajs-1818 166 5 coclosed coclose VERB iajs-1818 166 6 rickart rickart NOUN iajs-1818 166 7 as as ADP iajs-1818 166 8 ℤ ℤ NOUN iajs-1818 166 9 module module NOUN iajs-1818 166 10 . . PUNCT iajs-1818 167 1 ( ( PUNCT iajs-1818 167 2 1 1 X iajs-1818 167 3 ) ) PUNCT iajs-1818 167 4 ℤ ℤ PROPN iajs-1818 167 5 as as SCONJ iajs-1818 167 6 ℤ ℤ NOUN iajs-1818 167 7 -module -module NOUN iajs-1818 167 8 is be AUX iajs-1818 167 9 coclosed coclose VERB iajs-1818 167 10 rickart rickart NOUN iajs-1818 167 11 while while SCONJ iajs-1818 167 12 not not PART iajs-1818 167 13 lifiting lifite VERB iajs-1818 167 14 . . PUNCT iajs-1818 168 1 a a DET iajs-1818 168 2 module module NOUN iajs-1818 168 3 𝑀 𝑀 PROPN iajs-1818 168 4 is be AUX iajs-1818 168 5 called call VERB iajs-1818 168 6 hopfian hopfian ADJ iajs-1818 168 7 if if SCONJ iajs-1818 168 8 every every DET iajs-1818 168 9 epimorphism epimorphism NOUN iajs-1818 168 10 𝑓 𝑓 DET iajs-1818 168 11 ∈ ∈ PROPN iajs-1818 168 12 end end NOUN iajs-1818 168 13 𝑀 𝑀 PROPN iajs-1818 168 14 is be AUX iajs-1818 168 15 an an DET iajs-1818 168 16 isomorphism isomorphism NOUN iajs-1818 168 17 [ [ X iajs-1818 168 18 1 1 NUM iajs-1818 168 19 ] ] PUNCT iajs-1818 168 20 . . PUNCT iajs-1818 169 1 proposition proposition NOUN iajs-1818 169 2 when when SCONJ iajs-1818 169 3 𝑀 𝑀 PROPN iajs-1818 169 4 is be AUX iajs-1818 169 5 a a DET iajs-1818 169 6 coclosed coclosed ADJ iajs-1818 169 7 rickart rickart NOUN iajs-1818 169 8 coclosed coclose VERB iajs-1818 169 9 simple simple ADJ iajs-1818 169 10 module module NOUN iajs-1818 169 11 over over ADP iajs-1818 169 12 𝑅 𝑅 PROPN iajs-1818 169 13 implies imply VERB iajs-1818 169 14 𝑀 𝑀 PROPN iajs-1818 169 15 is be AUX iajs-1818 169 16 hopfian hopfian ADJ iajs-1818 169 17 . . PUNCT iajs-1818 170 1 proof proof NOUN iajs-1818 170 2 . . PUNCT iajs-1818 171 1 when when SCONJ iajs-1818 171 2 0 0 NUM iajs-1818 171 3 𝑓 𝑓 DET iajs-1818 171 4 ∈ ∈ NOUN iajs-1818 171 5 end end NOUN iajs-1818 171 6 𝑀 𝑀 PROPN iajs-1818 171 7 is be AUX iajs-1818 171 8 an an DET iajs-1818 171 9 epimorphism epimorphism NOUN iajs-1818 171 10 . . PUNCT iajs-1818 172 1 because because SCONJ iajs-1818 172 2 𝑀 𝑀 PROPN iajs-1818 172 3 is be AUX iajs-1818 172 4 coclosed coclose VERB iajs-1818 172 5 rickart rickart NOUN iajs-1818 172 6 , , PUNCT iajs-1818 172 7 implies imply VERB iajs-1818 172 8 ker ker PROPN iajs-1818 173 1 𝑓 𝑓 PRON iajs-1818 173 2 is be AUX iajs-1818 173 3 a a DET iajs-1818 173 4 coclosed coclosed NOUN iajs-1818 173 5 of of ADP iajs-1818 173 6 𝑀. 𝑀. PROPN iajs-1818 173 7 while while SCONJ iajs-1818 173 8 𝑀 𝑀 PROPN iajs-1818 173 9 is be AUX iajs-1818 173 10 coclosed coclose VERB iajs-1818 173 11 simple simple ADJ iajs-1818 173 12 and and CCONJ iajs-1818 173 13 𝑓 𝑓 DET iajs-1818 173 14 0 0 NUM iajs-1818 173 15 , , PUNCT iajs-1818 173 16 it it PRON iajs-1818 173 17 follows follow VERB iajs-1818 173 18 that that DET iajs-1818 173 19 ker ker NOUN iajs-1818 173 20 𝑓 𝑓 DET iajs-1818 173 21 0 0 NUM iajs-1818 173 22 , , PUNCT iajs-1818 173 23 and and CCONJ iajs-1818 173 24 hence hence ADV iajs-1818 173 25 𝑀 𝑀 PROPN iajs-1818 173 26 is be AUX iajs-1818 173 27 hopfian hopfian ADJ iajs-1818 173 28 . . PUNCT iajs-1818 174 1 record record NOUN iajs-1818 174 2 that that PRON iajs-1818 174 3 , , PUNCT iajs-1818 174 4 let let VERB iajs-1818 174 5 𝑀 𝑀 PRON iajs-1818 174 6 be be AUX iajs-1818 174 7 a a DET iajs-1818 174 8 module module NOUN iajs-1818 174 9 over over ADP iajs-1818 174 10 𝑅 𝑅 PROPN iajs-1818 174 11 , , PUNCT iajs-1818 174 12 we we PRON iajs-1818 174 13 call call VERB iajs-1818 174 14 it it PRON iajs-1818 174 15 scalar scalar ADV iajs-1818 174 16 when when SCONJ iajs-1818 174 17 each each DET iajs-1818 174 18 𝑓 𝑓 PROPN iajs-1818 174 19 ∈ ∈ PROPN iajs-1818 174 20 end end NOUN iajs-1818 174 21 𝑀 𝑀 PROPN iajs-1818 174 22 , , PUNCT iajs-1818 174 23 there there PRON iajs-1818 174 24 is be VERB iajs-1818 174 25 𝑎 𝑎 PRON iajs-1818 174 26 ∈ ∈ PROPN iajs-1818 174 27 𝑅 𝑅 PROPN iajs-1818 174 28 with with ADP iajs-1818 174 29 𝑓 𝑓 DET iajs-1818 174 30 𝑚 𝑚 NOUN iajs-1818 174 31 𝑚𝑎 𝑚𝑎 ADV iajs-1818 174 32 for for ADP iajs-1818 174 33 each each DET iajs-1818 174 34 𝑚 𝑚 PROPN iajs-1818 174 35 ∈ ∈ PROPN iajs-1818 174 36 𝑀 𝑀 PROPN iajs-1818 175 1 [ [ X iajs-1818 175 2 14 14 NUM iajs-1818 175 3 ] ] PUNCT iajs-1818 175 4 . . PUNCT iajs-1818 176 1 proposition proposition NOUN iajs-1818 176 2 when when SCONJ iajs-1818 176 3 𝑀 𝑀 PROPN iajs-1818 176 4 is be AUX iajs-1818 176 5 a a DET iajs-1818 176 6 scalar scalar ADJ iajs-1818 176 7 coclosed coclose VERB iajs-1818 176 8 rickart rickart NOUN iajs-1818 176 9 module module NOUN iajs-1818 176 10 , , PUNCT iajs-1818 176 11 yield yield NOUN iajs-1818 176 12 for for ADP iajs-1818 176 13 each each PRON iajs-1818 176 14 0 0 NUM iajs-1818 177 1 𝑓 𝑓 DET iajs-1818 177 2 ∈ ∈ PROPN iajs-1818 177 3 end end NOUN iajs-1818 177 4 𝑀 𝑀 PROPN iajs-1818 177 5 , , PUNCT iajs-1818 177 6 there there PRON iajs-1818 177 7 is be VERB iajs-1818 177 8 0 0 NUM iajs-1818 177 9 𝑎 𝑎 PROPN iajs-1818 177 10 ∈ ∈ PROPN iajs-1818 177 11 𝑅 𝑅 PROPN iajs-1818 177 12 with with ADP iajs-1818 177 13 𝑟 𝑟 NOUN iajs-1818 177 14 𝑎 𝑎 NOUN iajs-1818 177 15 is be AUX iajs-1818 177 16 coclosed coclose VERB iajs-1818 177 17 of of ADP iajs-1818 177 18 𝑀. 𝑀. NOUN iajs-1818 177 19 proof proof NOUN iajs-1818 177 20 . . PUNCT iajs-1818 178 1 when when SCONJ iajs-1818 178 2 0 0 NUM iajs-1818 178 3 𝑓 𝑓 PRON iajs-1818 178 4 ∈ ∈ PROPN iajs-1818 178 5 end end NOUN iajs-1818 178 6 𝑀 𝑀 PROPN iajs-1818 178 7 and and CCONJ iajs-1818 178 8 𝑀 𝑀 PROPN iajs-1818 178 9 a a DET iajs-1818 178 10 scalar scalar ADJ iajs-1818 178 11 𝑅-module 𝑅-module PROPN iajs-1818 178 12 yield yield NOUN iajs-1818 178 13 there there PRON iajs-1818 178 14 is be VERB iajs-1818 178 15 𝑎 𝑎 PRON iajs-1818 178 16 ∈ ∈ PROPN iajs-1818 178 17 𝑅 𝑅 PROPN iajs-1818 178 18 , , PUNCT iajs-1818 178 19 𝑓 𝑓 DET iajs-1818 178 20 𝑚 𝑚 NOUN iajs-1818 178 21 𝑚𝑎 𝑚𝑎 ADV iajs-1818 178 22 for for ADP iajs-1818 178 23 every every DET iajs-1818 178 24 𝑚 𝑚 PROPN iajs-1818 178 25 ∈ ∈ PROPN iajs-1818 178 26 𝑀. 𝑀. PROPN iajs-1818 178 27 but but CCONJ iajs-1818 178 28 𝑀 𝑀 PROPN iajs-1818 178 29 is be AUX iajs-1818 178 30 coclosed coclose VERB iajs-1818 178 31 rickart rickart NOUN iajs-1818 178 32 , , PUNCT iajs-1818 178 33 therefore therefore ADV iajs-1818 178 34 𝑟 𝑟 DET iajs-1818 178 35 𝑎 𝑎 NOUN iajs-1818 178 36 is be AUX iajs-1818 178 37 a a DET iajs-1818 178 38 coclosed coclosed NOUN iajs-1818 178 39 of of ADP iajs-1818 178 40 𝑀. 𝑀. PROPN iajs-1818 178 41 corollary corollary NOUN iajs-1818 178 42 . . PUNCT iajs-1818 179 1 if if SCONJ iajs-1818 179 2 𝑅 𝑅 PROPN iajs-1818 179 3 is be AUX iajs-1818 179 4 a a DET iajs-1818 179 5 commutative commutative ADJ iajs-1818 179 6 via via ADP iajs-1818 179 7 unity unity NOUN iajs-1818 179 8 coclosed coclose VERB iajs-1818 179 9 rickart rickart NOUN iajs-1818 179 10 over over ADP iajs-1818 179 11 𝑅 𝑅 PROPN iajs-1818 179 12 , , PUNCT iajs-1818 179 13 it it PRON iajs-1818 179 14 follows follow VERB iajs-1818 179 15 for for ADP iajs-1818 179 16 any any DET iajs-1818 179 17 0 0 NUM iajs-1818 179 18 𝑓 𝑓 PRON iajs-1818 179 19 ∈ ∈ PROPN iajs-1818 179 20 end end NOUN iajs-1818 179 21 𝑅 𝑅 PROPN iajs-1818 179 22 , , PUNCT iajs-1818 179 23 there there PRON iajs-1818 179 24 is be VERB iajs-1818 179 25 0 0 NUM iajs-1818 179 26 𝑎 𝑎 PROPN iajs-1818 179 27 ∈ ∈ PROPN iajs-1818 179 28 𝑅 𝑅 PROPN iajs-1818 179 29 and and CCONJ iajs-1818 179 30 𝑟 𝑟 DET iajs-1818 179 31 𝑎 𝑎 NOUN iajs-1818 179 32 is be AUX iajs-1818 179 33 a a DET iajs-1818 179 34 coclosed coclosed ADJ iajs-1818 179 35 submodule submodule NOUN iajs-1818 179 36 of of ADP iajs-1818 179 37 𝑅. 𝑅. ADJ iajs-1818 179 38 proof proof NOUN iajs-1818 179 39 . . PUNCT iajs-1818 180 1 due due ADP iajs-1818 180 2 to to ADP iajs-1818 180 3 every every DET iajs-1818 180 4 commutative commutative ADJ iajs-1818 180 5 ring ring NOUN iajs-1818 180 6 𝑅 𝑅 PROPN iajs-1818 180 7 is be AUX iajs-1818 180 8 a a DET iajs-1818 180 9 scalar scalar NOUN iajs-1818 180 10 , , PUNCT iajs-1818 180 11 then then ADV iajs-1818 180 12 result result NOUN iajs-1818 180 13 is be AUX iajs-1818 180 14 already already ADV iajs-1818 180 15 obtained obtain VERB iajs-1818 180 16 by by ADP iajs-1818 180 17 proposition proposition NOUN iajs-1818 180 18 3.7 3.7 NUM iajs-1818 180 19 . . PUNCT iajs-1818 181 1 proposition proposition NOUN iajs-1818 181 2 assume assume VERB iajs-1818 181 3 𝑀 𝑀 PROPN iajs-1818 181 4 is be AUX iajs-1818 181 5 a a DET iajs-1818 181 6 scalar scalar ADJ iajs-1818 181 7 faithful faithful NOUN iajs-1818 181 8 over over ADP iajs-1818 181 9 𝑅 𝑅 PROPN iajs-1818 181 10 , , PUNCT iajs-1818 181 11 implies imply VERB iajs-1818 181 12 𝑅 𝑅 PROPN iajs-1818 181 13 is be AUX iajs-1818 181 14 coclosed coclose VERB iajs-1818 181 15 rickart rickart NOUN iajs-1818 181 16 if if SCONJ iajs-1818 181 17 only only ADV iajs-1818 181 18 if if SCONJ iajs-1818 181 19 end end NOUN iajs-1818 181 20 𝑅 𝑅 PROPN iajs-1818 181 21 is be AUX iajs-1818 181 22 coclosed coclose VERB iajs-1818 181 23 rickart rickart NOUN iajs-1818 181 24 . . PUNCT iajs-1818 182 1 proof proof NOUN iajs-1818 182 2 . . PUNCT iajs-1818 183 1 because because SCONJ iajs-1818 183 2 𝑀 𝑀 PROPN iajs-1818 183 3 is be AUX iajs-1818 183 4 a a DET iajs-1818 183 5 scalar scalar ADJ iajs-1818 183 6 module module NOUN iajs-1818 183 7 over over ADP iajs-1818 183 8 𝑅 𝑅 PROPN iajs-1818 183 9 , , PUNCT iajs-1818 183 10 then then ADV iajs-1818 183 11 by by ADP iajs-1818 183 12 [ [ X iajs-1818 183 13 14 14 NUM iajs-1818 183 14 , , PUNCT iajs-1818 183 15 lemma lemma PROPN iajs-1818 183 16 6.2 6.2 NUM iajs-1818 183 17 ] ] PUNCT iajs-1818 183 18 , , PUNCT iajs-1818 183 19 end end VERB iajs-1818 183 20 𝑅 𝑅 PROPN iajs-1818 183 21 ≅ ≅ PROPN iajs-1818 183 22 . . PUNCT iajs-1818 184 1 but but CCONJ iajs-1818 184 2 𝑀 𝑀 PROPN iajs-1818 184 3 is be AUX iajs-1818 184 4 faithful faithful ADJ iajs-1818 184 5 implies imply VERB iajs-1818 184 6 that that SCONJ iajs-1818 184 7 𝑅 𝑅 PROPN iajs-1818 184 8 ≅ ≅ PROPN iajs-1818 184 9 end end PROPN iajs-1818 184 10 𝑅 𝑅 PROPN iajs-1818 184 11 , , PUNCT iajs-1818 184 12 as as SCONJ iajs-1818 184 13 required require VERB iajs-1818 184 14 . . PUNCT iajs-1818 185 1 ihsciconf ihsciconf PROPN iajs-1818 185 2 2017 2017 NUM iajs-1818 185 3 special special ADJ iajs-1818 185 4 issue issue NOUN iajs-1818 185 5 ibn ibn PROPN iajs-1818 185 6 al al PROPN iajs-1818 185 7 - - PUNCT iajs-1818 185 8 haitham haitham PROPN iajs-1818 185 9 journal journal PROPN iajs-1818 185 10 for for ADP iajs-1818 185 11 pure pure ADJ iajs-1818 185 12 and and CCONJ iajs-1818 185 13 applied apply VERB iajs-1818 185 14 science science NOUN iajs-1818 185 15 https://doi.org/ https://doi.org/ NOUN iajs-1818 185 16 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 185 17 for for ADP iajs-1818 185 18 more more ADJ iajs-1818 185 19 information information NOUN iajs-1818 185 20 about about ADP iajs-1818 185 21 the the DET iajs-1818 185 22 conference conference NOUN iajs-1818 185 23 please please INTJ iajs-1818 185 24 visit visit VERB iajs-1818 185 25 the the DET iajs-1818 185 26 websites website NOUN iajs-1818 185 27 : : PUNCT iajs-1818 185 28 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 185 29     SPACE iajs-1818 185 30 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1818 185 31       SPACE iajs-1818 185 32 mathematics|458 mathematics|458 NOUN iajs-1818 185 33         SPACE iajs-1818 185 34 remarks remark NOUN iajs-1818 185 35 and and CCONJ iajs-1818 185 36 examples example NOUN iajs-1818 185 37 ( ( PUNCT iajs-1818 185 38 1 1 X iajs-1818 185 39 ) ) PUNCT iajs-1818 185 40 the the DET iajs-1818 185 41 factor factor NOUN iajs-1818 185 42 module module NOUN iajs-1818 185 43 of of ADP iajs-1818 185 44 a a DET iajs-1818 185 45 coclosed coclose VERB iajs-1818 185 46 rickart rickart NOUN iajs-1818 185 47 module module NOUN iajs-1818 185 48 may may AUX iajs-1818 185 49 not not PART iajs-1818 185 50 be be AUX iajs-1818 185 51 coclosed coclose VERB iajs-1818 185 52 rickart rickart NOUN iajs-1818 185 53 . . PUNCT iajs-1818 186 1 study study VERB iajs-1818 186 2 ℤ ℤ PROPN iajs-1818 186 3 as as SCONJ iajs-1818 186 4 ℤ ℤ NOUN iajs-1818 186 5 -module -module NOUN iajs-1818 186 6 is be AUX iajs-1818 186 7 coclosed coclose VERB iajs-1818 186 8 rickart rickart NOUN iajs-1818 186 9 , , PUNCT iajs-1818 186 10 whereas whereas SCONJ iajs-1818 186 11 ≅ ≅ PROPN iajs-1818 186 12 𝑍 𝑍 PROPN iajs-1818 186 13 is be AUX iajs-1818 186 14 not not PART iajs-1818 186 15 a a DET iajs-1818 186 16 coclosed coclosed ADJ iajs-1818 186 17 rickart rickart NOUN iajs-1818 186 18 . . PUNCT iajs-1818 187 1 ( ( PUNCT iajs-1818 187 2 2 2 X iajs-1818 187 3 ) ) PUNCT iajs-1818 187 4 when when SCONJ iajs-1818 187 5 𝑀 𝑀 PROPN iajs-1818 187 6 is be AUX iajs-1818 187 7 a a DET iajs-1818 187 8 coclosed coclosed ADJ iajs-1818 187 9 rickart rickart NOUN iajs-1818 187 10 module module NOUN iajs-1818 187 11 , , PUNCT iajs-1818 187 12 yeild yeild PROPN iajs-1818 187 13 is be AUX iajs-1818 187 14 coclosed coclose VERB iajs-1818 187 15 rickart rickart NOUN iajs-1818 187 16 for for ADP iajs-1818 187 17 each each DET iajs-1818 187 18 summand summand NOUN iajs-1818 187 19 𝑁 𝑁 PROPN iajs-1818 187 20 of of ADP iajs-1818 187 21 𝑀. 𝑀. PROPN iajs-1818 187 22 ( ( PUNCT iajs-1818 187 23 3 3 NUM iajs-1818 187 24 ) ) PUNCT iajs-1818 187 25 if if SCONJ iajs-1818 187 26 𝑁 𝑁 PROPN iajs-1818 187 27 is be AUX iajs-1818 187 28 a a DET iajs-1818 187 29 submodule submodule NOUN iajs-1818 187 30 of of ADP iajs-1818 187 31 𝑀. 𝑀. PROPN iajs-1818 187 32 when when SCONJ iajs-1818 187 33 𝑁 𝑁 PROPN iajs-1818 187 34 and and CCONJ iajs-1818 187 35 are be AUX iajs-1818 187 36 coclosed coclose VERB iajs-1818 187 37 rickart rickart NOUN iajs-1818 187 38 modules module NOUN iajs-1818 187 39 implies imply VERB iajs-1818 187 40 𝑀 𝑀 PROPN iajs-1818 187 41 need need AUX iajs-1818 187 42 not not PART iajs-1818 187 43 be be AUX iajs-1818 187 44 coclosed coclose VERB iajs-1818 187 45 rickart rickart NOUN iajs-1818 187 46 . . PUNCT iajs-1818 188 1 study study NOUN iajs-1818 188 2 𝑀 𝑀 PROPN iajs-1818 189 1 = = PROPN iajs-1818 189 2 ℤ ℤ PROPN iajs-1818 189 3 ℤ2 ℤ2 PROPN iajs-1818 189 4 over over ADP iajs-1818 189 5 ℤ ℤ PROPN iajs-1818 189 6 and and CCONJ iajs-1818 189 7 𝑁 𝑁 PROPN iajs-1818 189 8 0 0 NUM iajs-1818 189 9   ADJ iajs-1818 189 10 ℤ2 ℤ2 PROPN iajs-1818 189 11 . . PUNCT iajs-1818 190 1 thus thus ADV iajs-1818 190 2 𝑁 𝑁 PROPN iajs-1818 190 3 and and CCONJ iajs-1818 190 4 ≅ ≅ PROPN iajs-1818 190 5 ℤ ℤ PROPN iajs-1818 190 6 are be AUX iajs-1818 190 7 coclosed coclose VERB iajs-1818 190 8 rickart rickart NOUN iajs-1818 190 9 ℤ-modules ℤ-modules PROPN iajs-1818 190 10 while while SCONJ iajs-1818 190 11 𝑀 𝑀 PROPN iajs-1818 190 12 is be AUX iajs-1818 190 13 not not PART iajs-1818 190 14 coclosed coclose VERB iajs-1818 190 15 rickart rickart NOUN iajs-1818 190 16 . . PUNCT iajs-1818 191 1 relatively relatively ADV iajs-1818 191 2 coclosed coclose VERB iajs-1818 191 3 rickart rickart NOUN iajs-1818 191 4 modules module NOUN iajs-1818 191 5 in in ADP iajs-1818 191 6 this this DET iajs-1818 191 7 part part NOUN iajs-1818 192 1 , , PUNCT iajs-1818 192 2 we we PRON iajs-1818 192 3 study study VERB iajs-1818 192 4 relatively relatively ADV iajs-1818 192 5 coclosed coclosed ADJ iajs-1818 192 6 rickart rickart NOUN iajs-1818 192 7 modules module NOUN iajs-1818 192 8 . . PUNCT iajs-1818 193 1 main main ADJ iajs-1818 193 2 results result NOUN iajs-1818 193 3 of of ADP iajs-1818 193 4 this this DET iajs-1818 193 5 type type NOUN iajs-1818 193 6 of of ADP iajs-1818 193 7 modules module NOUN iajs-1818 193 8 are be AUX iajs-1818 193 9 investigated investigate VERB iajs-1818 193 10 . . PUNCT iajs-1818 193 11     SPACE iajs-1818 194 1 the the DET iajs-1818 194 2 family family NOUN iajs-1818 194 3 of of ADP iajs-1818 194 4 rings ring NOUN iajs-1818 194 5 𝑅 𝑅 PROPN iajs-1818 194 6 for for ADP iajs-1818 194 7 which which PRON iajs-1818 194 8 each each DET iajs-1818 194 9 right right ADJ iajs-1818 194 10 𝑅-module 𝑅-module PROPN iajs-1818 194 11 is be AUX iajs-1818 194 12 relatively relatively ADV iajs-1818 194 13 coclosed coclose VERB iajs-1818 194 14 rickart rickart NOUN iajs-1818 194 15 is be AUX iajs-1818 194 16 right right ADJ iajs-1818 194 17 cosemisimple cosemisimple NOUN iajs-1818 194 18 rings ring NOUN iajs-1818 194 19 . . PUNCT iajs-1818 195 1 our our PRON iajs-1818 195 2 concern concern NOUN iajs-1818 195 3 is be AUX iajs-1818 195 4 : : PUNCT iajs-1818 195 5 when when SCONJ iajs-1818 195 6 do do AUX iajs-1818 195 7 modules module NOUN iajs-1818 195 8 have have VERB iajs-1818 195 9 the the DET iajs-1818 195 10 relatively relatively ADV iajs-1818 195 11 coclosed coclose VERB iajs-1818 195 12 rickart rickart NOUN iajs-1818 195 13 property property NOUN iajs-1818 195 14 . . PUNCT iajs-1818 196 1 definition definition NOUN iajs-1818 196 2 . . PUNCT iajs-1818 197 1 let let VERB iajs-1818 197 2 𝑀 𝑀 PROPN iajs-1818 197 3 and and CCONJ iajs-1818 197 4 𝑁 𝑁 PROPN iajs-1818 197 5 be be AUX iajs-1818 197 6 𝑅-modules 𝑅-modules PROPN iajs-1818 197 7 . . PUNCT iajs-1818 198 1 𝑀 𝑀 PROPN iajs-1818 198 2 is be AUX iajs-1818 198 3 called call VERB iajs-1818 198 4 relatively relatively ADV iajs-1818 198 5 coclosed coclosed ADJ iajs-1818 198 6 rickart rickart NOUN iajs-1818 198 7 to to ADP iajs-1818 198 8 𝑁 𝑁 PROPN iajs-1818 198 9 if if SCONJ iajs-1818 198 10 for for ADP iajs-1818 198 11 every every DET iajs-1818 198 12 𝑓 𝑓 PROPN iajs-1818 198 13 ∈ ∈ PROPN iajs-1818 198 14 hom hom X iajs-1818 198 15 𝑀 𝑀 PROPN iajs-1818 198 16 , , PUNCT iajs-1818 198 17 𝑁 𝑁 PROPN iajs-1818 198 18 , , PUNCT iajs-1818 198 19 ker ker NOUN iajs-1818 198 20 𝑓 𝑓 PRON iajs-1818 198 21 is be AUX iajs-1818 198 22 a a DET iajs-1818 198 23 coclosed coclosed ADJ iajs-1818 198 24 submodule submodule NOUN iajs-1818 198 25 of of ADP iajs-1818 198 26 𝑀. 𝑀. PROPN iajs-1818 198 27 as as ADP iajs-1818 198 28 special special ADJ iajs-1818 198 29 case case NOUN iajs-1818 198 30 , , PUNCT iajs-1818 198 31 𝑀 𝑀 PROPN iajs-1818 198 32 is be AUX iajs-1818 198 33 coclosed coclose VERB iajs-1818 198 34 rickart rickart NOUN iajs-1818 198 35 if if SCONJ iajs-1818 199 1 and and CCONJ iajs-1818 199 2 only only ADV iajs-1818 199 3 if if SCONJ iajs-1818 199 4 𝑀 𝑀 PROPN iajs-1818 199 5 is be AUX iajs-1818 199 6 relatively relatively ADV iajs-1818 199 7 coclosed coclose VERB iajs-1818 199 8 rickart rickart NOUN iajs-1818 199 9 to to ADP iajs-1818 199 10 𝑀. 𝑀. NOUN iajs-1818 199 11 remarks remark NOUN iajs-1818 199 12 with with ADP iajs-1818 199 13 examples example NOUN iajs-1818 199 14 ( ( PUNCT iajs-1818 199 15 1 1 X iajs-1818 199 16 ) ) PUNCT iajs-1818 199 17 obviously obviously ADV iajs-1818 199 18 every every DET iajs-1818 199 19 cosemisimple cosemisimple NOUN iajs-1818 199 20 𝑅-module 𝑅-module PROPN iajs-1818 199 21 𝑀 𝑀 PROPN iajs-1818 199 22 is be AUX iajs-1818 199 23 relatively relatively ADV iajs-1818 199 24 coclosed coclose VERB iajs-1818 199 25 rickart rickart NOUN iajs-1818 199 26 to to ADP iajs-1818 199 27 any any DET iajs-1818 199 28 𝑅module 𝑅module PROPN iajs-1818 199 29 𝑁. 𝑁. PROPN iajs-1818 199 30 ( ( PUNCT iajs-1818 199 31 2 2 NUM iajs-1818 199 32 ) ) PUNCT iajs-1818 199 33 let let VERB iajs-1818 199 34 𝑀 𝑀 PRON iajs-1818 199 35 and and CCONJ iajs-1818 199 36 𝑁 𝑁 PROPN iajs-1818 199 37 be be AUX iajs-1818 199 38 𝑅-modules 𝑅-modules PROPN iajs-1818 199 39 . . PUNCT iajs-1818 200 1 if if SCONJ iajs-1818 200 2 𝑀 𝑀 PROPN iajs-1818 200 3 is be AUX iajs-1818 200 4 relatively relatively ADV iajs-1818 200 5 coclosed coclose VERB iajs-1818 200 6 rickart rickart NOUN iajs-1818 200 7 to to ADP iajs-1818 200 8 𝑁 𝑁 PROPN iajs-1818 200 9 , , PUNCT iajs-1818 200 10 then then ADV iajs-1818 200 11 𝑁 𝑁 PROPN iajs-1818 200 12 need need AUX iajs-1818 200 13 not not PART iajs-1818 200 14 be be AUX iajs-1818 200 15 relatively relatively ADV iajs-1818 200 16 coclosed coclose VERB iajs-1818 200 17 rickart rickart NOUN iajs-1818 200 18 to to ADP iajs-1818 200 19 𝑀. 𝑀. PROPN iajs-1818 200 20 for for ADP iajs-1818 200 21 example example NOUN iajs-1818 200 22 , , PUNCT iajs-1818 200 23 let let VERB iajs-1818 200 24 ℤn ℤn VERB iajs-1818 200 25 and and CCONJ iajs-1818 200 26 ℤ ℤ PROPN iajs-1818 200 27 as as ADP iajs-1818 200 28 ℤ-modules ℤ-modules PROPN iajs-1818 200 29 . . PUNCT iajs-1818 201 1 then then ADV iajs-1818 201 2 ℤn ℤn PROPN iajs-1818 201 3 is be AUX iajs-1818 201 4 relatively relatively ADV iajs-1818 201 5 coclosed coclose VERB iajs-1818 201 6 rickart rickart NOUN iajs-1818 201 7 to to ADP iajs-1818 201 8 ℤ ℤ PROPN iajs-1818 201 9 for for ADP iajs-1818 201 10 each each DET iajs-1818 201 11 positive positive ADJ iajs-1818 201 12 integer integer NOUN iajs-1818 201 13 𝑛 𝑛 ADP iajs-1818 201 14 greater great ADJ iajs-1818 201 15 than than ADP iajs-1818 201 16 one one NUM iajs-1818 201 17 , , PUNCT iajs-1818 201 18 in in ADP iajs-1818 201 19 fact fact NOUN iajs-1818 201 20 homℤ homℤ VERB iajs-1818 201 21 ℤ ℤ PROPN iajs-1818 201 22 , , PUNCT iajs-1818 201 23 ℤ ℤ PROPN iajs-1818 201 24 0 0 PROPN iajs-1818 201 25 . . PUNCT iajs-1818 202 1 on on ADP iajs-1818 202 2 the the DET iajs-1818 202 3 other other ADJ iajs-1818 202 4 hand hand NOUN iajs-1818 202 5 , , PUNCT iajs-1818 202 6 ℤ ℤ PROPN iajs-1818 202 7 is be AUX iajs-1818 202 8 not not PART iajs-1818 202 9 relatively relatively ADV iajs-1818 202 10 coclosed coclose VERB iajs-1818 202 11 rickart rickart NOUN iajs-1818 202 12 to to PART iajs-1818 202 13 ℤn ℤn VERB iajs-1818 202 14 , , PUNCT iajs-1818 202 15 since since SCONJ iajs-1818 202 16 there there PRON iajs-1818 202 17 exists exist VERB iajs-1818 202 18 the the DET iajs-1818 202 19 natural natural ADJ iajs-1818 202 20 homomorphism homomorphism NOUN iajs-1818 202 21 𝑓 𝑓 DET iajs-1818 202 22 ∈ ∈ PROPN iajs-1818 202 23 homℤ homℤ VERB iajs-1818 202 24 ℤ ℤ PROPN iajs-1818 202 25 , , PUNCT iajs-1818 202 26 ℤ ℤ PROPN iajs-1818 202 27 defined define VERB iajs-1818 202 28 by by ADP iajs-1818 202 29 𝑓 𝑓 PRON iajs-1818 202 30 𝑚 𝑚 PROPN iajs-1818 202 31 𝑚 𝑚 PRON iajs-1818 202 32 for for ADP iajs-1818 202 33 each each DET iajs-1818 202 34 ∈ ∈ PROPN iajs-1818 202 35 ℤ ℤ PROPN iajs-1818 202 36 , , PUNCT iajs-1818 202 37 for for ADP iajs-1818 202 38 some some DET iajs-1818 202 39 𝑛 𝑛 DET iajs-1818 202 40 1 1 NUM iajs-1818 202 41 , , PUNCT iajs-1818 202 42 ker ker NOUN iajs-1818 202 43 𝑓 𝑓 DET iajs-1818 202 44 𝑛ℤ 𝑛ℤ PROPN iajs-1818 202 45 is be AUX iajs-1818 202 46 not not PART iajs-1818 202 47 coclosed coclose VERB iajs-1818 202 48 in in ADP iajs-1818 202 49 ℤ ℤ PROPN iajs-1818 202 50 . . PUNCT iajs-1818 203 1 ( ( PUNCT iajs-1818 203 2 3 3 X iajs-1818 203 3 ) ) PUNCT iajs-1818 203 4 when when SCONJ iajs-1818 203 5 𝑀 𝑀 PROPN iajs-1818 203 6 is be AUX iajs-1818 203 7 a a DET iajs-1818 203 8 coclosed coclose VERB iajs-1818 203 9 rickart rickart NOUN iajs-1818 203 10 module module NOUN iajs-1818 203 11 over over ADP iajs-1818 203 12 𝑅 𝑅 PROPN iajs-1818 203 13 , , PUNCT iajs-1818 203 14 implies imply VERB iajs-1818 203 15 𝑀 𝑀 PROPN iajs-1818 203 16 needs needs AUX iajs-1818 203 17 not not PART iajs-1818 203 18 be be AUX iajs-1818 203 19 relatively relatively ADV iajs-1818 203 20 coclosed coclose VERB iajs-1818 203 21 rickart rickart NOUN iajs-1818 203 22 to to ADP iajs-1818 203 23 an an DET iajs-1818 203 24 𝑅-module 𝑅-module PROPN iajs-1818 203 25 𝑁. 𝑁. PROPN iajs-1818 203 26 for for ADP iajs-1818 203 27 example example NOUN iajs-1818 203 28 , , PUNCT iajs-1818 203 29 ℤ ℤ PROPN iajs-1818 203 30 as as ADP iajs-1818 203 31 ℤ-module ℤ-module PROPN iajs-1818 203 32 is be AUX iajs-1818 203 33 coclosed coclose VERB iajs-1818 203 34 rickart rickart NOUN iajs-1818 203 35 . . PUNCT iajs-1818 204 1 whereas whereas SCONJ iajs-1818 204 2 ℤ ℤ PROPN iajs-1818 204 3 is be AUX iajs-1818 204 4 not not PART iajs-1818 204 5 relatively relatively ADV iajs-1818 204 6 coclosed coclose VERB iajs-1818 204 7 rickart rickart NOUN iajs-1818 204 8 to to PART iajs-1818 204 9 ℤn ℤn VERB iajs-1818 204 10 as as ADP iajs-1818 204 11 ℤ-module ℤ-module PROPN iajs-1818 204 12 for for ADP iajs-1818 204 13 any any DET iajs-1818 204 14 𝑛 𝑛 DET iajs-1818 204 15 1 1 NUM iajs-1818 204 16 . . PUNCT iajs-1818 205 1 ( ( PUNCT iajs-1818 205 2 4 4 X iajs-1818 205 3 ) ) PUNCT iajs-1818 205 4 if if SCONJ iajs-1818 205 5 𝑀 𝑀 PROPN iajs-1818 205 6 is be AUX iajs-1818 205 7 relatively relatively ADV iajs-1818 205 8 coclosed coclose VERB iajs-1818 205 9 rickart rickart NOUN iajs-1818 205 10 to to ADP iajs-1818 205 11 an an DET iajs-1818 205 12 𝑅-module 𝑅-module PROPN iajs-1818 205 13 𝑁 𝑁 PROPN iajs-1818 205 14 , , PUNCT iajs-1818 205 15 then then ADV iajs-1818 205 16 𝑀 𝑀 PROPN iajs-1818 205 17 may may AUX iajs-1818 205 18 not not PART iajs-1818 205 19 be be AUX iajs-1818 205 20 coclosed coclose VERB iajs-1818 205 21 rickart rickart NOUN iajs-1818 205 22 . . PUNCT iajs-1818 206 1 for for ADP iajs-1818 206 2 example example NOUN iajs-1818 206 3 , , PUNCT iajs-1818 206 4 consider consider VERB iajs-1818 206 5 the the DET iajs-1818 206 6 ℤ-module ℤ-module PROPN iajs-1818 206 7 ℤ ℤ PROPN iajs-1818 206 8 is be AUX iajs-1818 206 9 relatively relatively ADV iajs-1818 206 10 coclosed coclose VERB iajs-1818 206 11 rickart rickart NOUN iajs-1818 206 12 to to ADP iajs-1818 206 13 the the DET iajs-1818 206 14 ℤ-module ℤ-module PROPN iajs-1818 206 15 ℤ ℤ PROPN iajs-1818 206 16 , , PUNCT iajs-1818 206 17 because because SCONJ iajs-1818 206 18 homℤ homℤ VERB iajs-1818 206 19 ℤ ℤ PROPN iajs-1818 206 20 , , PUNCT iajs-1818 206 21 ℤ ℤ PROPN iajs-1818 206 22 0 0 PROPN iajs-1818 206 23 . . PUNCT iajs-1818 207 1 but but CCONJ iajs-1818 207 2 ℤ ℤ PROPN iajs-1818 207 3 is be AUX iajs-1818 207 4 not not PART iajs-1818 207 5 coclosed coclose VERB iajs-1818 207 6 rickart rickart NOUN iajs-1818 207 7 . . PUNCT iajs-1818 208 1 ( ( PUNCT iajs-1818 208 2 5 5 NUM iajs-1818 208 3 ) ) PUNCT iajs-1818 208 4 when when SCONJ iajs-1818 208 5 𝑀 𝑀 PROPN iajs-1818 208 6 is be AUX iajs-1818 208 7 a a DET iajs-1818 208 8 coclosed coclosed ADJ iajs-1818 208 9 simple simple ADJ iajs-1818 208 10 or or CCONJ iajs-1818 208 11 quasi quasi ADJ iajs-1818 208 12 - - NOUN iajs-1818 208 13 dedekind dedekind ADJ iajs-1818 208 14 over over ADP iajs-1818 208 15 𝑅 𝑅 PROPN iajs-1818 208 16 , , PUNCT iajs-1818 208 17 yeild yeild PROPN iajs-1818 208 18 𝑀 𝑀 PROPN iajs-1818 208 19 need need AUX iajs-1818 208 20 not not PART iajs-1818 208 21 be be AUX iajs-1818 208 22 relatively relatively ADV iajs-1818 208 23 coclosed coclose VERB iajs-1818 208 24 rickart rickart NOUN iajs-1818 208 25 to to ADP iajs-1818 208 26 a a DET iajs-1818 208 27 module module NOUN iajs-1818 208 28 𝑁 𝑁 NOUN iajs-1818 208 29 over over ADV iajs-1818 208 30 𝑅. 𝑅. NOUN iajs-1818 208 31 discuss discuss VERB iajs-1818 208 32 ℤ ℤ PROPN iajs-1818 208 33 over over ADP iajs-1818 208 34 ℤ ℤ PROPN iajs-1818 208 35 is be AUX iajs-1818 208 36 coclosed coclose VERB iajs-1818 208 37 simple simple ADJ iajs-1818 208 38 and and CCONJ iajs-1818 208 39 quasi quasi ADJ iajs-1818 208 40 - - ADJ iajs-1818 208 41 dedekind dedekind ADJ iajs-1818 208 42 while while SCONJ iajs-1818 208 43 not not PART iajs-1818 208 44 relatively relatively ADV iajs-1818 208 45 coclosed coclose VERB iajs-1818 208 46 rickart rickart NOUN iajs-1818 208 47 to to ADP iajs-1818 208 48 the the DET iajs-1818 208 49 ℤ-module ℤ-module PROPN iajs-1818 208 50 ℤn ℤn PROPN iajs-1818 208 51 , , PUNCT iajs-1818 208 52 for for ADP iajs-1818 208 53 each each DET iajs-1818 208 54 𝑛 𝑛 DET iajs-1818 208 55 1 1 NUM iajs-1818 208 56 . . PUNCT iajs-1818 209 1 theorem theorem NOUN iajs-1818 209 2 let let VERB iajs-1818 209 3 𝑀 𝑀 PROPN iajs-1818 209 4 and and CCONJ iajs-1818 209 5 𝑁 𝑁 PROPN iajs-1818 209 6 be be AUX iajs-1818 209 7 𝑅-modules 𝑅-modules PROPN iajs-1818 209 8 . . PUNCT iajs-1818 210 1 the the DET iajs-1818 210 2 next next ADJ iajs-1818 210 3 statements statement NOUN iajs-1818 210 4 are be AUX iajs-1818 210 5 equivalent equivalent ADJ iajs-1818 210 6 ( ( PUNCT iajs-1818 210 7 1 1 NUM iajs-1818 210 8 ) ) PUNCT iajs-1818 210 9 𝑀 𝑀 PROPN iajs-1818 210 10 is be AUX iajs-1818 210 11 relatively relatively ADV iajs-1818 210 12 coclosed coclose VERB iajs-1818 210 13 rickart rickart NOUN iajs-1818 210 14 to to ADP iajs-1818 210 15 𝑁. 𝑁. PROPN iajs-1818 210 16 ( ( PUNCT iajs-1818 210 17 2 2 NUM iajs-1818 210 18 ) ) PUNCT iajs-1818 210 19 each each DET iajs-1818 210 20 submodule submodule NOUN iajs-1818 210 21 𝐵 𝐵 NOUN iajs-1818 210 22 of of ADP iajs-1818 210 23 𝑁 𝑁 PROPN iajs-1818 210 24 and and CCONJ iajs-1818 210 25 each each DET iajs-1818 210 26 summand summand NOUN iajs-1818 210 27 𝐴 𝐴 PROPN iajs-1818 210 28 of of ADP iajs-1818 210 29 𝑀 𝑀 PROPN iajs-1818 210 30 is be AUX iajs-1818 210 31 relatively relatively ADV iajs-1818 210 32 coclosed coclose VERB iajs-1818 210 33 rickart rickart NOUN iajs-1818 210 34 to to ADP iajs-1818 210 35 𝐵. 𝐵. PROPN iajs-1818 210 36 ( ( PUNCT iajs-1818 210 37 3 3 NUM iajs-1818 210 38 ) ) PUNCT iajs-1818 210 39 each each DET iajs-1818 210 40 direct direct ADJ iajs-1818 210 41 𝐴 𝐴 PROPN iajs-1818 210 42 of of ADP iajs-1818 210 43 𝑀 𝑀 PROPN iajs-1818 210 44 and and CCONJ iajs-1818 210 45 for for ADP iajs-1818 210 46 each each DET iajs-1818 210 47 coclosed coclose VERB iajs-1818 210 48 𝐵 𝐵 NOUN iajs-1818 210 49 of of ADP iajs-1818 210 50 𝑁 𝑁 PROPN iajs-1818 210 51 and and CCONJ iajs-1818 210 52 each each DET iajs-1818 210 53 𝑓 𝑓 PRON iajs-1818 210 54 ∈ ∈ PROPN iajs-1818 210 55 hom hom X iajs-1818 210 56 𝑀 𝑀 PROPN iajs-1818 210 57 , , PUNCT iajs-1818 210 58 𝐵 𝐵 PROPN iajs-1818 210 59 , , PUNCT iajs-1818 210 60 ker ker PROPN iajs-1818 210 61 𝑓| 𝑓| PROPN iajs-1818 210 62 is be AUX iajs-1818 210 63 a a DET iajs-1818 210 64 coclosed coclose VERB iajs-1818 210 65 in in ADP iajs-1818 210 66 𝐴. 𝐴. PROPN iajs-1818 210 67 ihsciconf ihsciconf NOUN iajs-1818 210 68 2017 2017 NUM iajs-1818 210 69 special special ADJ iajs-1818 210 70 issue issue NOUN iajs-1818 210 71 ibn ibn PROPN iajs-1818 210 72 al al PROPN iajs-1818 210 73 - - PUNCT iajs-1818 210 74 haitham haitham PROPN iajs-1818 210 75 journal journal PROPN iajs-1818 210 76 for for ADP iajs-1818 210 77 pure pure ADJ iajs-1818 210 78 and and CCONJ iajs-1818 210 79 applied apply VERB iajs-1818 210 80 science science NOUN iajs-1818 210 81 https://doi.org/ https://doi.org/ NOUN iajs-1818 210 82 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 210 83 for for ADP iajs-1818 210 84 more more ADJ iajs-1818 210 85 information information NOUN iajs-1818 210 86 about about ADP iajs-1818 210 87 the the DET iajs-1818 210 88 conference conference NOUN iajs-1818 210 89 please please INTJ iajs-1818 210 90 visit visit VERB iajs-1818 210 91 the the DET iajs-1818 210 92 websites website NOUN iajs-1818 210 93 : : PUNCT iajs-1818 210 94 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 210 95     SPACE iajs-1818 210 96 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1818 210 97       SPACE iajs-1818 210 98 mathematics|459 mathematics|459 NOUN iajs-1818 210 99         SPACE iajs-1818 210 100 proof proof NOUN iajs-1818 210 101 . . PUNCT iajs-1818 211 1 1 1 NUM iajs-1818 211 2   NOUN iajs-1818 211 3 2 2 NUM iajs-1818 211 4 assume assume VERB iajs-1818 211 5 𝑀 𝑀 PROPN iajs-1818 211 6 is be AUX iajs-1818 211 7 relatively relatively ADV iajs-1818 211 8 coclosed coclose VERB iajs-1818 211 9 rickart rickart NOUN iajs-1818 211 10 to to ADP iajs-1818 211 11 𝑁. 𝑁. PROPN iajs-1818 211 12 when when SCONJ iajs-1818 211 13 𝐴 𝐴 PROPN iajs-1818 211 14 is be AUX iajs-1818 211 15 a a DET iajs-1818 211 16 direct direct ADJ iajs-1818 211 17 summand summand NOUN iajs-1818 211 18 of of ADP iajs-1818 211 19 𝑀 𝑀 PROPN iajs-1818 211 20 and and CCONJ iajs-1818 211 21 𝐵 𝐵 PROPN iajs-1818 211 22 of of ADP iajs-1818 211 23 𝑁. 𝑁. PROPN iajs-1818 211 24 let let VERB iajs-1818 211 25 𝑓 𝑓 PRON iajs-1818 211 26 ∈ ∈ PROPN iajs-1818 211 27 hom hom X iajs-1818 211 28 𝐴 𝐴 PROPN iajs-1818 211 29 , , PUNCT iajs-1818 211 30 𝐵 𝐵 PROPN iajs-1818 211 31 . . PUNCT iajs-1818 212 1 study study VERB iajs-1818 212 2 the the DET iajs-1818 212 3 next next ADJ iajs-1818 212 4 𝑀 𝑀 PROPN iajs-1818 212 5 𝐴 𝐴 PROPN iajs-1818 212 6 ⊕ ⊕ PROPN iajs-1818 212 7 𝐻 𝐻 PROPN iajs-1818 212 8 → → SYM iajs-1818 212 9 𝐴 𝐴 PROPN iajs-1818 212 10 → → SYM iajs-1818 212 11 𝐵 𝐵 NOUN iajs-1818 212 12 → → SYM iajs-1818 212 13 𝑁 𝑁 PROPN iajs-1818 212 14 where where SCONJ iajs-1818 212 15 𝐻 𝐻 PROPN iajs-1818 212 16 is be AUX iajs-1818 212 17 a a DET iajs-1818 212 18 submodule submodule NOUN iajs-1818 212 19 of of ADP iajs-1818 212 20 𝑀. 𝑀. PROPN iajs-1818 212 21 say say VERB iajs-1818 212 22 𝑔 𝑔 PROPN iajs-1818 212 23 𝑖 𝑖 ADP iajs-1818 212 24 𝑓 𝑓 X iajs-1818 212 25   X iajs-1818 212 26 ∈ ∈ PROPN iajs-1818 212 27 hom hom X iajs-1818 212 28 𝑀 𝑀 PROPN iajs-1818 212 29 , , PUNCT iajs-1818 212 30 𝑁 𝑁 PROPN iajs-1818 212 31 . . PUNCT iajs-1818 213 1 this this PRON iajs-1818 213 2 implies imply VERB iajs-1818 213 3 that that SCONJ iajs-1818 213 4 ker ker PROPN iajs-1818 213 5 𝑔 𝑔 PROPN iajs-1818 213 6 is be AUX iajs-1818 213 7 a a DET iajs-1818 213 8 coclosed coclose VERB iajs-1818 213 9 in in ADP iajs-1818 213 10 𝑀. 𝑀. NOUN iajs-1818 213 11 by by ADP iajs-1818 213 12 similar similar ADJ iajs-1818 213 13 steps step NOUN iajs-1818 213 14 of of ADP iajs-1818 213 15 result result NOUN iajs-1818 213 16 2.6 2.6 NUM iajs-1818 213 17 , , PUNCT iajs-1818 213 18 we we PRON iajs-1818 213 19 gain gain VERB iajs-1818 213 20 ker ker NOUN iajs-1818 214 1 𝑓 𝑓 PRON iajs-1818 214 2 is be AUX iajs-1818 214 3 a a DET iajs-1818 214 4 coclosed coclosed ADJ iajs-1818 214 5 submodule submodule NOUN iajs-1818 214 6 in in ADP iajs-1818 214 7 𝐴. 𝐴. PROPN iajs-1818 214 8 therefore therefore ADV iajs-1818 214 9 𝐴 𝐴 PROPN iajs-1818 214 10 is be AUX iajs-1818 214 11 relatively relatively ADV iajs-1818 214 12 coclosed coclose VERB iajs-1818 214 13 rickart rickart NOUN iajs-1818 214 14 to to ADP iajs-1818 214 15 𝐵. 𝐵. PROPN iajs-1818 214 16 2 2 NUM iajs-1818 214 17   NOUN iajs-1818 214 18 3 3 NUM iajs-1818 214 19 let let VERB iajs-1818 214 20 𝐵 𝐵 PRON iajs-1818 214 21 be be AUX iajs-1818 214 22 a a DET iajs-1818 214 23 coclosed coclosed ADJ iajs-1818 214 24 submodule submodule NOUN iajs-1818 214 25 in in ADP iajs-1818 214 26 𝑁 𝑁 PROPN iajs-1818 214 27 and and CCONJ iajs-1818 214 28 𝐴 𝐴 PROPN iajs-1818 214 29 a a DET iajs-1818 214 30 direct direct ADJ iajs-1818 214 31 summand summand NOUN iajs-1818 214 32 of of ADP iajs-1818 214 33 𝑀. 𝑀. PROPN iajs-1818 214 34 let let VERB iajs-1818 214 35 𝑓 𝑓 PRON iajs-1818 214 36 ∈ ∈ PROPN iajs-1818 214 37 hom hom X iajs-1818 214 38 𝑀 𝑀 PROPN iajs-1818 214 39 , , PUNCT iajs-1818 214 40 𝐵 𝐵 PROPN iajs-1818 214 41 , , PUNCT iajs-1818 214 42 then then ADV iajs-1818 214 43 𝑓| 𝑓| PROPN iajs-1818 214 44 ∈ ∈ PROPN iajs-1818 214 45 hom hom X iajs-1818 214 46 𝐴 𝐴 PROPN iajs-1818 214 47 , , PUNCT iajs-1818 214 48 𝐵 𝐵 PROPN iajs-1818 214 49 . . PUNCT iajs-1818 215 1 since since SCONJ iajs-1818 215 2 𝐴 𝐴 PROPN iajs-1818 215 3 is be AUX iajs-1818 215 4 relatively relatively ADV iajs-1818 215 5 coclosed coclose VERB iajs-1818 215 6 rickart rickart NOUN iajs-1818 215 7 to to ADP iajs-1818 215 8 𝐵 𝐵 PROPN iajs-1818 215 9 , , PUNCT iajs-1818 215 10 implies imply VERB iajs-1818 215 11 ker𝑓| ker𝑓| NOUN iajs-1818 215 12 is be AUX iajs-1818 215 13 a a DET iajs-1818 215 14 closed closed NOUN iajs-1818 215 15 of of ADP iajs-1818 215 16 𝐴. 𝐴. PROPN iajs-1818 215 17 3 3 NUM iajs-1818 215 18   NOUN iajs-1818 215 19 1 1 NUM iajs-1818 215 20 by by ADP iajs-1818 215 21 taking take VERB iajs-1818 215 22 𝐴 𝐴 PROPN iajs-1818 215 23 𝑀 𝑀 PROPN iajs-1818 215 24 and and CCONJ iajs-1818 215 25 𝐵 𝐵 PROPN iajs-1818 215 26 𝑁. 𝑁. PROPN iajs-1818 215 27 the the DET iajs-1818 215 28 next next ADJ iajs-1818 215 29 two two NUM iajs-1818 215 30 lemmas lemma NOUN iajs-1818 215 31 are be AUX iajs-1818 215 32 proved prove VERB iajs-1818 215 33 in in ADP iajs-1818 215 34 [ [ X iajs-1818 215 35 4 4 NUM iajs-1818 215 36 ] ] PUNCT iajs-1818 215 37 . . PUNCT iajs-1818 216 1 lemma lemma PROPN iajs-1818 216 2 a a DET iajs-1818 216 3 module module NOUN iajs-1818 216 4 𝑀 𝑀 PROPN iajs-1818 216 5 has have VERB iajs-1818 216 6 the the DET iajs-1818 216 7 ccip ccip PROPN iajs-1818 216 8 iff iff PROPN iajs-1818 216 9 every every DET iajs-1818 216 10 coclosed coclose VERB iajs-1818 216 11 submodule submodule NOUN iajs-1818 216 12 in in ADP iajs-1818 216 13 𝑀 𝑀 PROPN iajs-1818 216 14 gets get VERB iajs-1818 216 15 the the DET iajs-1818 216 16 ccip ccip NOUN iajs-1818 216 17 . . PUNCT iajs-1818 217 1 lemma lemma PROPN iajs-1818 217 2 when when SCONJ iajs-1818 217 3 𝑀 𝑀 PROPN iajs-1818 217 4 be be VERB iajs-1818 217 5 a a DET iajs-1818 217 6 module module NOUN iajs-1818 217 7 over over ADP iajs-1818 217 8 𝑅 𝑅 PROPN iajs-1818 217 9 with with ADP iajs-1818 217 10 the the DET iajs-1818 217 11 ccip ccip NOUN iajs-1818 217 12 , , PUNCT iajs-1818 217 13 yeild yeild PROPN iajs-1818 217 14 each each DET iajs-1818 217 15 decomposition decomposition NOUN iajs-1818 217 16 𝑀 𝑀 PROPN iajs-1818 217 17 𝐴 𝐴 PROPN iajs-1818 217 18 𝐵 𝐵 PROPN iajs-1818 217 19 and and CCONJ iajs-1818 217 20 for for ADP iajs-1818 217 21 every every DET iajs-1818 217 22 𝑓 𝑓 PROPN iajs-1818 217 23 ∈ ∈ PROPN iajs-1818 217 24 hom hom X iajs-1818 217 25 𝐴 𝐴 PROPN iajs-1818 217 26 , , PUNCT iajs-1818 217 27 𝐵 𝐵 PROPN iajs-1818 217 28 , , PUNCT iajs-1818 217 29 ker ker NOUN iajs-1818 217 30 𝑓 𝑓 PRON iajs-1818 217 31 is be AUX iajs-1818 217 32 a a DET iajs-1818 217 33 coclosed coclosed ADJ iajs-1818 217 34 submodule submodule NOUN iajs-1818 217 35 of of ADP iajs-1818 217 36 𝑀. 𝑀. PROPN iajs-1818 217 37 we we PRON iajs-1818 217 38 give give VERB iajs-1818 217 39 the the DET iajs-1818 217 40 following follow VERB iajs-1818 217 41 results result NOUN iajs-1818 217 42 proposition proposition NOUN iajs-1818 217 43 under under ADP iajs-1818 217 44 an an DET iajs-1818 217 45 isomorphism isomorphism NOUN iajs-1818 217 46 , , PUNCT iajs-1818 217 47 ccip ccip PROPN iajs-1818 217 48 is be AUX iajs-1818 217 49 transformed transform VERB iajs-1818 217 50 . . PUNCT iajs-1818 218 1 proof proof NOUN iajs-1818 218 2 . . PUNCT iajs-1818 219 1 when when SCONJ iajs-1818 219 2 𝑀 𝑀 PROPN iajs-1818 219 3 and and CCONJ iajs-1818 219 4 𝑀 𝑀 PROPN iajs-1818 219 5 are be AUX iajs-1818 219 6 modules module NOUN iajs-1818 219 7 over over ADP iajs-1818 219 8 𝑅 𝑅 PROPN iajs-1818 219 9 with with ADP iajs-1818 219 10 𝑀 𝑀 PROPN iajs-1818 219 11 has have VERB iajs-1818 219 12 the the DET iajs-1818 219 13 ccip ccip PROPN iajs-1818 219 14 and and CCONJ iajs-1818 219 15 𝑓 𝑓 DET iajs-1818 219 16 ∶ ∶ NOUN iajs-1818 219 17 𝑀 𝑀 PROPN iajs-1818 219 18 → → SYM iajs-1818 219 19 𝑀 𝑀 PROPN iajs-1818 219 20 an an DET iajs-1818 219 21 isomorphism isomorphism NOUN iajs-1818 219 22 . . PUNCT iajs-1818 220 1 assume assume PROPN iajs-1818 220 2 𝐴 𝐴 PROPN iajs-1818 220 3 , , PUNCT iajs-1818 220 4 𝐵 𝐵 PROPN iajs-1818 220 5 are be AUX iajs-1818 220 6 coclosed coclose VERB iajs-1818 220 7 of of ADP iajs-1818 220 8 𝑀 𝑀 PROPN iajs-1818 220 9 . . PUNCT iajs-1818 221 1 to to PART iajs-1818 221 2 show show VERB iajs-1818 221 3 𝐴 𝐴 PROPN iajs-1818 221 4 ∩ ∩ NOUN iajs-1818 221 5 𝐵 𝐵 NOUN iajs-1818 221 6 is be AUX iajs-1818 221 7 again again ADV iajs-1818 221 8 coclosed coclose VERB iajs-1818 221 9 of of ADP iajs-1818 221 10 𝑀 𝑀 PROPN iajs-1818 221 11 . . PUNCT iajs-1818 222 1 then then ADV iajs-1818 222 2 there there PRON iajs-1818 222 3 is be VERB iajs-1818 222 4 submodules submodules PROPN iajs-1818 222 5 𝐶 𝐶 PROPN iajs-1818 222 6 , , PUNCT iajs-1818 222 7 𝐷 𝐷 PROPN iajs-1818 222 8 of of ADP iajs-1818 222 9 𝑀 𝑀 PROPN iajs-1818 222 10 and and CCONJ iajs-1818 222 11 𝑓 𝑓 DET iajs-1818 222 12 𝐶 𝐶 PROPN iajs-1818 222 13 𝐴 𝐴 PROPN iajs-1818 222 14 , , PUNCT iajs-1818 222 15 𝑓 𝑓 DET iajs-1818 222 16 𝐷 𝐷 NOUN iajs-1818 222 17 𝐵 𝐵 PROPN iajs-1818 222 18 implies imply VERB iajs-1818 222 19 𝐶 𝐶 PROPN iajs-1818 222 20 𝑓 𝑓 PROPN iajs-1818 222 21 𝐴 𝐴 PROPN iajs-1818 222 22 , , PUNCT iajs-1818 222 23 𝐷 𝐷 PROPN iajs-1818 222 24 𝑓 𝑓 DET iajs-1818 222 25 𝐵 𝐵 NOUN iajs-1818 222 26 . . PUNCT iajs-1818 223 1 it it PRON iajs-1818 223 2 is be AUX iajs-1818 223 3 easy easy ADJ iajs-1818 223 4 to to PART iajs-1818 223 5 check check VERB iajs-1818 223 6 that that SCONJ iajs-1818 223 7 𝐶 𝐶 PROPN iajs-1818 223 8 𝑓 𝑓 PROPN iajs-1818 223 9 𝐴 𝐴 PROPN iajs-1818 223 10 , , PUNCT iajs-1818 223 11 𝐷 𝐷 PROPN iajs-1818 223 12 𝑓 𝑓 DET iajs-1818 223 13 𝐵 𝐵 NOUN iajs-1818 223 14 are be AUX iajs-1818 223 15 coclosed coclose VERB iajs-1818 223 16 submodules submodule NOUN iajs-1818 223 17 of of ADP iajs-1818 223 18 𝑀 𝑀 PROPN iajs-1818 223 19 . . PUNCT iajs-1818 224 1 it it PRON iajs-1818 224 2 follows follow VERB iajs-1818 224 3 that that SCONJ iajs-1818 224 4 𝐶 𝐶 PROPN iajs-1818 224 5 ∩ ∩ PROPN iajs-1818 224 6 𝐷 𝐷 PROPN iajs-1818 224 7 𝑓 𝑓 PROPN iajs-1818 224 8 𝐴 𝐴 PROPN iajs-1818 224 9 ∩ ∩ NOUN iajs-1818 224 10 𝑓 𝑓 PRON iajs-1818 224 11 𝐵 𝐵 NOUN iajs-1818 224 12 is be AUX iajs-1818 224 13 a a DET iajs-1818 224 14 coclosed coclosed NOUN iajs-1818 224 15 of of ADP iajs-1818 224 16 𝑀 𝑀 PROPN iajs-1818 224 17 . . PUNCT iajs-1818 225 1 it it PRON iajs-1818 225 2 is be AUX iajs-1818 225 3 not not PART iajs-1818 225 4 hard hard ADJ iajs-1818 225 5 to to PART iajs-1818 225 6 see see VERB iajs-1818 225 7 that that SCONJ iajs-1818 225 8 𝑓 𝑓 DET iajs-1818 225 9 𝐶 𝐶 PROPN iajs-1818 225 10 ∩ ∩ ADJ iajs-1818 225 11 𝐷 𝐷 PROPN iajs-1818 225 12 is be AUX iajs-1818 225 13 a a DET iajs-1818 225 14 coclosed coclosed ADJ iajs-1818 225 15 submodule submodule NOUN iajs-1818 225 16 of of ADP iajs-1818 225 17 𝑀 𝑀 PROPN iajs-1818 225 18 . . PUNCT iajs-1818 226 1 on on ADP iajs-1818 226 2 the the DET iajs-1818 226 3 other other ADJ iajs-1818 226 4 hand hand NOUN iajs-1818 226 5 , , PUNCT iajs-1818 226 6 𝑓 𝑓 DET iajs-1818 226 7 𝐶 𝐶 PROPN iajs-1818 226 8 ∩ ∩ ADJ iajs-1818 226 9 𝐷 𝐷 PROPN iajs-1818 226 10 𝐴 𝐴 PROPN iajs-1818 226 11 ∩ ∩ ADJ iajs-1818 226 12 𝐵 𝐵 NOUN iajs-1818 226 13 implies imply VERB iajs-1818 226 14 the the DET iajs-1818 226 15 result result NOUN iajs-1818 226 16 is be AUX iajs-1818 226 17 gained gain VERB iajs-1818 226 18 . . PUNCT iajs-1818 227 1 theorem theorem VERB iajs-1818 227 2 if if SCONJ iajs-1818 227 3 𝑀 𝑀 PROPN iajs-1818 227 4 is be AUX iajs-1818 227 5 a a DET iajs-1818 227 6 module module NOUN iajs-1818 227 7 over over ADP iajs-1818 227 8 𝑅 𝑅 PROPN iajs-1818 227 9 with with ADP iajs-1818 227 10 the the DET iajs-1818 227 11 ccip ccip PROPN iajs-1818 227 12 and and CCONJ iajs-1818 227 13 𝐴 𝐴 PROPN iajs-1818 227 14 𝐵 𝐵 PROPN iajs-1818 227 15 is be AUX iajs-1818 227 16 a a DET iajs-1818 227 17 coclosed coclosed NOUN iajs-1818 227 18 of of ADP iajs-1818 227 19 𝑀. 𝑀. PROPN iajs-1818 227 20 yeild yeild PROPN iajs-1818 227 21 𝐴 𝐴 PROPN iajs-1818 227 22 is be AUX iajs-1818 227 23 relatively relatively ADV iajs-1818 227 24 coclosed coclose VERB iajs-1818 227 25 rickart rickart NOUN iajs-1818 227 26 module module NOUN iajs-1818 227 27 to to ADP iajs-1818 227 28 𝐵. 𝐵. PROPN iajs-1818 227 29 proof proof NOUN iajs-1818 227 30 . . PUNCT iajs-1818 228 1 assume assume VERB iajs-1818 228 2 that that SCONJ iajs-1818 228 3 𝑀 𝑀 PROPN iajs-1818 228 4 has have VERB iajs-1818 228 5 the the DET iajs-1818 228 6 ccip ccip PROPN iajs-1818 228 7 . . PUNCT iajs-1818 229 1 then then ADV iajs-1818 229 2 by by ADP iajs-1818 229 3 lemma lemma PROPN iajs-1818 229 4 4.4 4.4 NUM iajs-1818 229 5 , , PUNCT iajs-1818 229 6 every every DET iajs-1818 229 7 coclosed coclose VERB iajs-1818 229 8 submodule submodule NOUN iajs-1818 229 9 of of ADP iajs-1818 229 10 𝑀 𝑀 PROPN iajs-1818 229 11 has have VERB iajs-1818 229 12 the the DET iajs-1818 229 13 ccip ccip PROPN iajs-1818 229 14 implies imply VERB iajs-1818 229 15 that that SCONJ iajs-1818 229 16 𝐴 𝐴 PROPN iajs-1818 229 17 𝐵 𝐵 PROPN iajs-1818 229 18 has have VERB iajs-1818 229 19 the the DET iajs-1818 229 20 ccip ccip PROPN iajs-1818 229 21 . . PUNCT iajs-1818 230 1 by by ADP iajs-1818 230 2 lemma lemma PROPN iajs-1818 230 3 4.5 4.5 NUM iajs-1818 230 4 , , PUNCT iajs-1818 230 5 for for ADP iajs-1818 230 6 every every DET iajs-1818 230 7 𝑓 𝑓 PROPN iajs-1818 230 8 ∈ ∈ PROPN iajs-1818 230 9 hom hom X iajs-1818 230 10 𝐴 𝐴 PROPN iajs-1818 230 11 , , PUNCT iajs-1818 230 12 𝐵 𝐵 PROPN iajs-1818 230 13 , , PUNCT iajs-1818 230 14 ker ker NOUN iajs-1818 230 15 𝑓 𝑓 PRON iajs-1818 230 16 is be AUX iajs-1818 230 17 a a DET iajs-1818 230 18 coclosed coclosed ADJ iajs-1818 230 19 submodule submodule NOUN iajs-1818 230 20 in in ADP iajs-1818 230 21 𝐴𝐵. 𝐴𝐵. PROPN iajs-1818 230 22 but but CCONJ iajs-1818 230 23 ker ker X iajs-1818 230 24 𝑓 𝑓 DET iajs-1818 230 25   PROPN iajs-1818 230 26 𝐴 𝐴 PROPN iajs-1818 230 27 , , PUNCT iajs-1818 230 28 then then ADV iajs-1818 230 29 by by ADP iajs-1818 230 30 lemma lemma PROPN iajs-1818 230 31 2.5 2.5 NUM iajs-1818 230 32 , , PUNCT iajs-1818 230 33 ker ker NOUN iajs-1818 230 34 𝑓 𝑓 PRON iajs-1818 230 35 is be AUX iajs-1818 230 36 a a DET iajs-1818 230 37 coclosed coclose VERB iajs-1818 230 38 in in ADP iajs-1818 230 39 𝐴. 𝐴. PROPN iajs-1818 230 40 therefore therefore ADV iajs-1818 230 41 𝐴 𝐴 PROPN iajs-1818 230 42 is be AUX iajs-1818 230 43 relatively relatively ADV iajs-1818 230 44 coclosed coclose VERB iajs-1818 230 45 rickart rickart NOUN iajs-1818 230 46 to to ADP iajs-1818 230 47 𝐵. 𝐵. PROPN iajs-1818 230 48 as as ADP iajs-1818 230 49 immediate immediate ADJ iajs-1818 230 50 consequences consequence NOUN iajs-1818 230 51 we we PRON iajs-1818 230 52 have have VERB iajs-1818 230 53 corollary corollary ADJ iajs-1818 230 54 when when SCONJ iajs-1818 230 55 𝑀 𝑀 PROPN iajs-1818 230 56 and and CCONJ iajs-1818 230 57 𝑁 𝑁 PROPN iajs-1818 230 58 be be VERB iajs-1818 230 59 modules module NOUN iajs-1818 230 60 over over ADP iajs-1818 230 61 𝑅. 𝑅. NOUN iajs-1818 230 62 if if SCONJ iajs-1818 230 63 𝑀 𝑀 PROPN iajs-1818 230 64 𝑁 𝑁 PROPN iajs-1818 230 65 has have VERB iajs-1818 230 66 the the DET iajs-1818 230 67 ccip ccip PROPN iajs-1818 230 68 , , PUNCT iajs-1818 230 69 yeild yeild PROPN iajs-1818 230 70 𝑀 𝑀 PROPN iajs-1818 230 71 is be AUX iajs-1818 230 72 relatively relatively ADV iajs-1818 230 73 coclosed coclose VERB iajs-1818 230 74 rickart rickart NOUN iajs-1818 230 75 to to ADP iajs-1818 230 76 𝑁. 𝑁. PROPN iajs-1818 230 77 corollary corollary NOUN iajs-1818 230 78 if if SCONJ iajs-1818 230 79 𝑀 𝑀 PROPN iajs-1818 230 80 be be VERB iajs-1818 230 81 a a DET iajs-1818 230 82 module module NOUN iajs-1818 230 83 over over ADP iajs-1818 230 84 . . PUNCT iajs-1818 231 1 if if SCONJ iajs-1818 231 2 𝑀 𝑀 PROPN iajs-1818 231 3   VERB iajs-1818 231 4 𝑀 𝑀 PROPN iajs-1818 231 5 gets get VERB iajs-1818 231 6 ccip ccip NOUN iajs-1818 231 7 , , PUNCT iajs-1818 231 8 implies imply VERB iajs-1818 231 9 𝑀 𝑀 PROPN iajs-1818 231 10 is be AUX iajs-1818 231 11 coclosed coclose VERB iajs-1818 231 12 rickart rickart NOUN iajs-1818 231 13 module module NOUN iajs-1818 231 14 . . PUNCT iajs-1818 232 1 remark remark VERB iajs-1818 232 2 the the DET iajs-1818 232 3 reverse reverse NOUN iajs-1818 232 4 of of ADP iajs-1818 232 5 corollary corollary ADJ iajs-1818 232 6 4.8 4.8 NUM iajs-1818 232 7 need need NOUN iajs-1818 232 8 not not PART iajs-1818 232 9 be be AUX iajs-1818 232 10 hold hold VERB iajs-1818 232 11 generally generally ADV iajs-1818 232 12 . . PUNCT iajs-1818 233 1 study study VERB iajs-1818 233 2 ℤ ℤ PROPN iajs-1818 233 3 as as SCONJ iajs-1818 233 4 ℤ ℤ NOUN iajs-1818 233 5 -module -module NOUN iajs-1818 233 6 is be AUX iajs-1818 233 7 cosemisimple cosemisimple NOUN iajs-1818 233 8 . . PUNCT iajs-1818 234 1 then then ADV iajs-1818 234 2 it it PRON iajs-1818 234 3 is be AUX iajs-1818 234 4 relatively relatively ADV iajs-1818 234 5 coclosed coclose VERB iajs-1818 234 6 rickart rickart NOUN iajs-1818 234 7 for for ADP iajs-1818 234 8 each each DET iajs-1818 234 9 𝑅-module 𝑅-module PROPN iajs-1818 234 10 𝑁. 𝑁. PROPN iajs-1818 234 11 let let VERB iajs-1818 234 12 ℤ ℤ PROPN iajs-1818 234 13 , , PUNCT iajs-1818 234 14 𝑀 𝑀 PROPN iajs-1818 234 15 = = PUNCT iajs-1818 235 1 ℤ ℤ PROPN iajs-1818 235 2 ℤ ℤ PROPN iajs-1818 235 3 be be AUX iajs-1818 235 4 modules module NOUN iajs-1818 235 5 over over ADP iajs-1818 235 6 ℤ. ℤ. PROPN iajs-1818 235 7 we we PRON iajs-1818 235 8 show show VERB iajs-1818 235 9 that that SCONJ iajs-1818 235 10 𝑀 𝑀 PROPN iajs-1818 235 11 does do AUX iajs-1818 235 12 not not PART iajs-1818 235 13 have have VERB iajs-1818 235 14 the the DET iajs-1818 235 15 ccip ccip PROPN iajs-1818 235 16 . . PUNCT iajs-1818 236 1 let let VERB iajs-1818 236 2 𝐴 𝐴 PROPN iajs-1818 236 3 1 1 NUM iajs-1818 236 4 , , PUNCT iajs-1818 236 5 0 0 NUM iajs-1818 236 6 ℤ ℤ PROPN iajs-1818 236 7 , , PUNCT iajs-1818 236 8 and and CCONJ iajs-1818 236 9 𝐵 𝐵 NOUN iajs-1818 236 10 1 1 NUM iajs-1818 236 11 , , PUNCT iajs-1818 236 12 1 1 NUM iajs-1818 236 13 ℤ ℤ PROPN iajs-1818 236 14 be be VERB iajs-1818 236 15 the the DET iajs-1818 236 16 submodules submodule NOUN iajs-1818 236 17 generated generate VERB iajs-1818 236 18 by by ADP iajs-1818 236 19 ( ( PUNCT iajs-1818 236 20 1 1 NUM iajs-1818 236 21 , , PUNCT iajs-1818 236 22 0 0 NUM iajs-1818 236 23 and and CCONJ iajs-1818 236 24 ( ( PUNCT iajs-1818 236 25 1 1 NUM iajs-1818 236 26 , , PUNCT iajs-1818 236 27 1 1 NUM iajs-1818 236 28 ) ) PUNCT iajs-1818 236 29 respectively respectively ADV iajs-1818 236 30 . . PUNCT iajs-1818 237 1 obviously obviously ADV iajs-1818 237 2 𝐴 𝐴 PROPN iajs-1818 237 3 and and CCONJ iajs-1818 237 4 𝐵 𝐵 NOUN iajs-1818 237 5 are be AUX iajs-1818 237 6 summands summand NOUN iajs-1818 237 7 of of ADP iajs-1818 237 8 𝑀. 𝑀. NOUN iajs-1818 237 9 then then ADV iajs-1818 237 10 , , PUNCT iajs-1818 237 11 𝐵 𝐵 NOUN iajs-1818 237 12 are be AUX iajs-1818 237 13 coclosed coclose VERB iajs-1818 237 14 in in ADP iajs-1818 237 15 𝑀. 𝑀. PROPN iajs-1818 237 16 but but CCONJ iajs-1818 237 17 𝐴⋂𝐵 𝐴⋂𝐵 NOUN iajs-1818 237 18 2ℤ 2ℤ NOUN iajs-1818 237 19   NOUN iajs-1818 237 20 0 0 NUM iajs-1818 237 21 is be AUX iajs-1818 237 22 not not PART iajs-1818 237 23 a a DET iajs-1818 237 24 coclosed coclose VERB iajs-1818 237 25 submodule submodule NOUN iajs-1818 237 26 in in ADP iajs-1818 237 27 𝑀 𝑀 PROPN iajs-1818 237 28 , , PUNCT iajs-1818 237 29 since since SCONJ iajs-1818 237 30 2 2 NUM iajs-1818 237 31   ADJ iajs-1818 237 32 0 0 NUM iajs-1818 238 1 ≅ ≅ X iajs-1818 238 2 ℤ ℤ PROPN iajs-1818 238 3   PROPN iajs-1818 238 4 ℤ ℤ PROPN iajs-1818 238 5   PROPN iajs-1818 238 6 ≪ ≪ PUNCT iajs-1818 238 7 ℤ ℤ PROPN iajs-1818 238 8 ℤ ℤ PROPN iajs-1818 238 9 ℤ ℤ PROPN iajs-1818 238 10   PROPN iajs-1818 238 11 ≅ ≅ PROPN iajs-1818 238 12 ℤ4 ℤ4 PROPN iajs-1818 238 13 ℤ2 ℤ2 PROPN iajs-1818 238 14 . . PUNCT iajs-1818 239 1 ihsciconf ihsciconf PROPN iajs-1818 239 2 2017 2017 NUM iajs-1818 239 3 special special ADJ iajs-1818 239 4 issue issue NOUN iajs-1818 239 5 ibn ibn PROPN iajs-1818 239 6 al al PROPN iajs-1818 239 7 - - PUNCT iajs-1818 239 8 haitham haitham PROPN iajs-1818 239 9 journal journal PROPN iajs-1818 239 10 for for ADP iajs-1818 239 11 pure pure ADJ iajs-1818 239 12 and and CCONJ iajs-1818 239 13 applied apply VERB iajs-1818 239 14 science science NOUN iajs-1818 239 15 https://doi.org/ https://doi.org/ NOUN iajs-1818 239 16 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 239 17 for for ADP iajs-1818 239 18 more more ADJ iajs-1818 239 19 information information NOUN iajs-1818 239 20 about about ADP iajs-1818 239 21 the the DET iajs-1818 239 22 conference conference NOUN iajs-1818 239 23 please please INTJ iajs-1818 239 24 visit visit VERB iajs-1818 239 25 the the DET iajs-1818 239 26 websites website NOUN iajs-1818 239 27 : : PUNCT iajs-1818 239 28 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 239 29     SPACE iajs-1818 239 30 www.ihsciconf.org www.ihsciconf.org ADJ iajs-1818 239 31       SPACE iajs-1818 239 32 mathematics|460 mathematics|460 ADJ iajs-1818 239 33         SPACE iajs-1818 239 34 proposition proposition NOUN iajs-1818 239 35 . . PUNCT iajs-1818 240 1 let let VERB iajs-1818 240 2 𝑀 𝑀 PROPN iajs-1818 240 3 ∈ ∈ NOUN iajs-1818 240 4 be be AUX iajs-1818 240 5 a a DET iajs-1818 240 6 family family NOUN iajs-1818 240 7 modules module NOUN iajs-1818 240 8 over over ADP iajs-1818 240 9 𝑅 𝑅 PROPN iajs-1818 240 10 and and CCONJ iajs-1818 240 11 𝑁 𝑁 ADP iajs-1818 240 12 an an DET iajs-1818 240 13 𝑅-module 𝑅-module PROPN iajs-1818 240 14 where where SCONJ iajs-1818 240 15   NOUN iajs-1818 240 16 1,2 1,2 NUM iajs-1818 240 17 , , PUNCT iajs-1818 240 18 … … PUNCT iajs-1818 240 19 , , PUNCT iajs-1818 240 20 𝑛 𝑛 PROPN iajs-1818 240 21 . . PUNCT iajs-1818 241 1 the the DET iajs-1818 241 2 next next ADJ iajs-1818 241 3 are be AUX iajs-1818 241 4 equivalent equivalent ADJ iajs-1818 241 5 ( ( PUNCT iajs-1818 241 6 1 1 NUM iajs-1818 241 7 ) ) PUNCT iajs-1818 241 8 if if SCONJ iajs-1818 241 9 𝑁 𝑁 PROPN iajs-1818 241 10 has have VERB iajs-1818 241 11 the the DET iajs-1818 241 12 ccip ccip NOUN iajs-1818 241 13 , , PUNCT iajs-1818 241 14 then then ADV iajs-1818 241 15 𝑁 𝑁 PROPN iajs-1818 241 16 is be AUX iajs-1818 241 17 relatively relatively ADV iajs-1818 241 18 coclosed coclose VERB iajs-1818 241 19 rickart rickart NOUN iajs-1818 241 20 to to ADP iajs-1818 241 21   PROPN iajs-1818 241 22 𝑀 𝑀 PROPN iajs-1818 241 23 . . PUNCT iajs-1818 242 1 ( ( PUNCT iajs-1818 242 2 2 2 X iajs-1818 242 3 ) ) PUNCT iajs-1818 242 4 𝑁 𝑁 NOUN iajs-1818 242 5 is be AUX iajs-1818 242 6 relatively relatively ADV iajs-1818 242 7 coclosed coclose VERB iajs-1818 242 8 rickart rickart NOUN iajs-1818 242 9 to to ADP iajs-1818 242 10 𝑀 𝑀 PROPN iajs-1818 242 11 for for ADP iajs-1818 242 12 all all DET iajs-1818 242 13 𝑖 𝑖 ADP iajs-1818 242 14 1,2 1,2 NUM iajs-1818 242 15 , , PUNCT iajs-1818 242 16 … … PUNCT iajs-1818 242 17 , , PUNCT iajs-1818 242 18 𝑛. 𝑛. NOUN iajs-1818 242 19 proof proof NOUN iajs-1818 242 20 . . PUNCT iajs-1818 243 1 1 1 NUM iajs-1818 243 2   NOUN iajs-1818 243 3 2 2 NUM iajs-1818 243 4 via via ADP iajs-1818 243 5 theorem theorem ADJ iajs-1818 243 6 4.3 4.3 NUM iajs-1818 243 7 . . SYM iajs-1818 243 8 2 2 NUM iajs-1818 243 9   NOUN iajs-1818 243 10 1 1 NUM iajs-1818 243 11 when when SCONJ iajs-1818 243 12 𝑁 𝑁 PROPN iajs-1818 243 13 is be AUX iajs-1818 243 14 relatively relatively ADV iajs-1818 243 15 coclosed coclose VERB iajs-1818 243 16 rickart rickart NOUN iajs-1818 243 17 to to ADP iajs-1818 243 18 𝑀 𝑀 PROPN iajs-1818 243 19 for for ADP iajs-1818 243 20 all all DET iajs-1818 243 21 𝑖 𝑖 ADP iajs-1818 243 22 1,2 1,2 NUM iajs-1818 243 23 , , PUNCT iajs-1818 243 24 … … PUNCT iajs-1818 243 25 , , PUNCT iajs-1818 243 26 𝑛 𝑛 NOUN iajs-1818 243 27 and and CCONJ iajs-1818 243 28 𝑁 𝑁 PROPN iajs-1818 243 29 has have VERB iajs-1818 243 30 the the DET iajs-1818 243 31 ccip ccip NOUN iajs-1818 243 32 . . PUNCT iajs-1818 244 1 to to PART iajs-1818 244 2 show show VERB iajs-1818 244 3 that that SCONJ iajs-1818 244 4 𝑁 𝑁 PROPN iajs-1818 244 5 is be AUX iajs-1818 244 6 relatively relatively ADV iajs-1818 244 7 coclosed coclose VERB iajs-1818 244 8 rickart rickart NOUN iajs-1818 244 9 to to ADP iajs-1818 244 10   PROPN iajs-1818 244 11 𝑀 𝑀 PROPN iajs-1818 244 12 . . PUNCT iajs-1818 245 1 let let VERB iajs-1818 245 2 𝑓 𝑓 PRON iajs-1818 245 3 ∈ ∈ PROPN iajs-1818 245 4 hom hom X iajs-1818 245 5 𝑁 𝑁 PROPN iajs-1818 245 6 , , PUNCT iajs-1818 245 7   PROPN iajs-1818 245 8 𝑀 𝑀 PROPN iajs-1818 245 9 . . PUNCT iajs-1818 246 1 let let VERB iajs-1818 246 2 us we PRON iajs-1818 246 3 consider consider VERB iajs-1818 246 4 the the DET iajs-1818 246 5 following follow VERB iajs-1818 246 6 sequence sequence NOUN iajs-1818 246 7 𝑁 𝑁 PROPN iajs-1818 246 8 → → SYM iajs-1818 246 9   PROPN iajs-1818 246 10 𝑀 𝑀 PROPN iajs-1818 246 11 → → SYM iajs-1818 246 12 𝑀 𝑀 PROPN iajs-1818 246 13 , , PUNCT iajs-1818 246 14 it it PRON iajs-1818 246 15 is be AUX iajs-1818 246 16 evident evident ADJ iajs-1818 246 17 that that SCONJ iajs-1818 246 18 im𝑓 im𝑓 NOUN iajs-1818 246 19 ∑ ∑ PROPN iajs-1818 246 20 i i PRON iajs-1818 246 21 m m VERB iajs-1818 246 22 𝜌 𝜌 X iajs-1818 246 23 𝑓 𝑓 PRON iajs-1818 246 24 where where SCONJ iajs-1818 246 25 𝜌 𝜌 PRON iajs-1818 246 26 is be AUX iajs-1818 246 27 the the DET iajs-1818 246 28 natural natural ADJ iajs-1818 246 29 projection projection NOUN iajs-1818 246 30 map map NOUN iajs-1818 246 31 of of ADP iajs-1818 246 32   PROPN iajs-1818 246 33 𝑀 𝑀 PROPN iajs-1818 246 34 onto onto ADP iajs-1818 246 35 𝑀 𝑀 PROPN iajs-1818 246 36 for for ADP iajs-1818 246 37 𝑖 𝑖 SYM iajs-1818 246 38 1,2 1,2 NUM iajs-1818 246 39 , , PUNCT iajs-1818 246 40 … … PUNCT iajs-1818 246 41 , , PUNCT iajs-1818 246 42 𝑛. 𝑛. NOUN iajs-1818 246 43 yield yield VERB iajs-1818 246 44 𝑓 𝑓 PRON iajs-1818 246 45 𝜌 𝜌 X iajs-1818 246 46 𝑓 𝑓 PROPN iajs-1818 246 47 , , PUNCT iajs-1818 246 48 𝜌 𝜌 X iajs-1818 246 49 𝑓 𝑓 PROPN iajs-1818 246 50 , , PUNCT iajs-1818 246 51 … … PUNCT iajs-1818 246 52 , , PUNCT iajs-1818 246 53 𝜌 𝜌 X iajs-1818 246 54 𝑓 𝑓 PRON iajs-1818 246 55 and and CCONJ iajs-1818 246 56 ker ker X iajs-1818 246 57 𝑓 𝑓 DET iajs-1818 246 58 ∩ ∩ ADJ iajs-1818 246 59 ker ker NOUN iajs-1818 246 60 𝜌 𝜌 X iajs-1818 246 61 𝑓 𝑓 PROPN iajs-1818 246 62 . . PUNCT iajs-1818 247 1 but but CCONJ iajs-1818 247 2 𝜌 𝜌 X iajs-1818 247 3 𝑓 𝑓 PRON iajs-1818 247 4 hom hom X iajs-1818 247 5 𝑁 𝑁 PROPN iajs-1818 247 6 , , PUNCT iajs-1818 247 7 𝑀 𝑀 PROPN iajs-1818 247 8 and and CCONJ iajs-1818 247 9 𝑁 𝑁 PROPN iajs-1818 247 10 is be AUX iajs-1818 247 11 relatively relatively ADV iajs-1818 247 12 coclosed coclose VERB iajs-1818 247 13 rickart rickart NOUN iajs-1818 247 14 to to ADP iajs-1818 247 15 𝑀 𝑀 PROPN iajs-1818 247 16 implies imply VERB iajs-1818 247 17 that that SCONJ iajs-1818 247 18 ker ker NOUN iajs-1818 248 1 𝜌 𝜌 X iajs-1818 248 2 𝑓 𝑓 PROPN iajs-1818 248 3 is be AUX iajs-1818 248 4 a a DET iajs-1818 248 5 coclosed coclosed ADJ iajs-1818 248 6 submodule submodule NOUN iajs-1818 248 7 in in ADP iajs-1818 248 8 𝑁. 𝑁. PROPN iajs-1818 248 9 because because SCONJ iajs-1818 248 10 𝑁 𝑁 PROPN iajs-1818 248 11 has have VERB iajs-1818 248 12 the the DET iajs-1818 248 13 ccip ccip NOUN iajs-1818 248 14 , , PUNCT iajs-1818 248 15 therefore therefore ADV iajs-1818 248 16 ker ker VERB iajs-1818 248 17 𝑓 𝑓 DET iajs-1818 248 18 ∩ ∩ ADJ iajs-1818 248 19 ker ker NOUN iajs-1818 248 20 𝜌 𝜌 X iajs-1818 248 21 𝑓 𝑓 PROPN iajs-1818 248 22 is be AUX iajs-1818 248 23 a a DET iajs-1818 248 24 coclosed coclosed ADJ iajs-1818 248 25 submodule submodule NOUN iajs-1818 248 26 in in ADP iajs-1818 248 27 𝑁 𝑁 PROPN iajs-1818 248 28 , , PUNCT iajs-1818 248 29 and and CCONJ iajs-1818 248 30 hence hence ADV iajs-1818 248 31 𝑁 𝑁 PROPN iajs-1818 248 32 is be AUX iajs-1818 248 33 relatively relatively ADV iajs-1818 248 34 coclosed coclose VERB iajs-1818 248 35 rickart rickart NOUN iajs-1818 248 36 to to ADP iajs-1818 248 37   PROPN iajs-1818 248 38 𝑀 𝑀 PROPN iajs-1818 248 39 . . PUNCT iajs-1818 249 1 as as ADP iajs-1818 249 2 an an DET iajs-1818 249 3 immediate immediate ADJ iajs-1818 249 4 result result NOUN iajs-1818 249 5 corollary corollary NOUN iajs-1818 249 6 . . PUNCT iajs-1818 250 1 when when SCONJ iajs-1818 250 2 𝑀 𝑀 PROPN iajs-1818 250 3 ∈ ∈ PROPN iajs-1818 250 4 is be AUX iajs-1818 250 5 a a DET iajs-1818 250 6 family family NOUN iajs-1818 250 7 modules module NOUN iajs-1818 250 8 over over ADP iajs-1818 250 9 𝑅 𝑅 PROPN iajs-1818 250 10 where where SCONJ iajs-1818 250 11   NOUN iajs-1818 250 12 1,2 1,2 NUM iajs-1818 250 13 , , PUNCT iajs-1818 250 14 … … PUNCT iajs-1818 250 15 , , PUNCT iajs-1818 250 16 𝑛 𝑛 PROPN iajs-1818 250 17 . . PUNCT iajs-1818 251 1 the the DET iajs-1818 251 2 next next ADJ iajs-1818 251 3 are be AUX iajs-1818 251 4 equivalent equivalent ADJ iajs-1818 251 5 ( ( PUNCT iajs-1818 251 6 1 1 NUM iajs-1818 251 7 ) ) PUNCT iajs-1818 251 8 if if SCONJ iajs-1818 251 9 𝑀 𝑀 PROPN iajs-1818 251 10 has have VERB iajs-1818 251 11 the the DET iajs-1818 251 12 ccip ccip NOUN iajs-1818 251 13 for for ADP iajs-1818 251 14 all all DET iajs-1818 251 15 𝑗 𝑗 PRON iajs-1818 251 16 1,2 1,2 NUM iajs-1818 251 17 , , PUNCT iajs-1818 251 18 … … PUNCT iajs-1818 251 19 , , PUNCT iajs-1818 251 20 𝑛 𝑛 PROPN iajs-1818 251 21 , , PUNCT iajs-1818 251 22 then then ADV iajs-1818 251 23 𝑀 𝑀 PROPN iajs-1818 251 24 is be AUX iajs-1818 251 25 relatively relatively ADV iajs-1818 251 26 coclosed coclose VERB iajs-1818 251 27 rickart rickart NOUN iajs-1818 251 28 to to ADP iajs-1818 251 29   PROPN iajs-1818 251 30 𝑀 𝑀 PROPN iajs-1818 251 31 . . PUNCT iajs-1818 252 1 ( ( PUNCT iajs-1818 252 2 2 2 X iajs-1818 252 3 ) ) PUNCT iajs-1818 252 4 𝑀 𝑀 PROPN iajs-1818 252 5 is be AUX iajs-1818 252 6 relatively relatively ADV iajs-1818 252 7 coclosed coclose VERB iajs-1818 252 8 rickart rickart NOUN iajs-1818 252 9 to to ADP iajs-1818 252 10 𝑀 𝑀 PROPN iajs-1818 252 11 for for ADP iajs-1818 252 12 all all DET iajs-1818 252 13 𝑖 𝑖 ADP iajs-1818 252 14 1,2 1,2 NUM iajs-1818 252 15 , , PUNCT iajs-1818 252 16 … … PUNCT iajs-1818 252 17 , , PUNCT iajs-1818 252 18 𝑛. 𝑛. NOUN iajs-1818 252 19 a a DET iajs-1818 252 20 characterization characterization NOUN iajs-1818 252 21 of of ADP iajs-1818 252 22 cosemisimple cosemisimple NOUN iajs-1818 252 23 rings ring NOUN iajs-1818 252 24 via via ADP iajs-1818 252 25 relatively relatively ADV iajs-1818 252 26 coclosed coclose VERB iajs-1818 252 27 rickart rickart NOUN iajs-1818 252 28 modules module NOUN iajs-1818 252 29 is be AUX iajs-1818 252 30 given give VERB iajs-1818 252 31 as as ADP iajs-1818 252 32 follows follow NOUN iajs-1818 252 33 . . PUNCT iajs-1818 253 1 theorem theorem VERB iajs-1818 253 2 . . PUNCT iajs-1818 254 1 the the DET iajs-1818 254 2 next next ADJ iajs-1818 254 3 are be AUX iajs-1818 254 4 equivalent equivalent ADJ iajs-1818 254 5 ( ( PUNCT iajs-1818 254 6 1 1 NUM iajs-1818 254 7 ) ) PUNCT iajs-1818 254 8 𝑅 𝑅 NOUN iajs-1818 254 9 is be AUX iajs-1818 254 10 a a DET iajs-1818 254 11 cosemisimple cosemisimple NOUN iajs-1818 254 12 right right ADJ iajs-1818 254 13 𝑅-module 𝑅-module PROPN iajs-1818 254 14 . . PUNCT iajs-1818 255 1 ( ( PUNCT iajs-1818 255 2 2 2 X iajs-1818 255 3 ) ) PUNCT iajs-1818 255 4 all all DET iajs-1818 255 5 𝑅-modules 𝑅-modules PROPN iajs-1818 255 6 are be AUX iajs-1818 255 7 cosemisimple cosemisimple NOUN iajs-1818 255 8 . . PUNCT iajs-1818 256 1 ( ( PUNCT iajs-1818 256 2 3 3 X iajs-1818 256 3 ) ) PUNCT iajs-1818 256 4 all all DET iajs-1818 256 5 𝑅-modules 𝑅-modules PROPN iajs-1818 256 6 are be AUX iajs-1818 256 7 relatively relatively ADV iajs-1818 256 8 coclosed coclose VERB iajs-1818 256 9 rickart rickart NOUN iajs-1818 256 10 to to ADP iajs-1818 256 11 any any DET iajs-1818 256 12 𝑅-module 𝑅-module PROPN iajs-1818 256 13 . . PUNCT iajs-1818 257 1 ( ( PUNCT iajs-1818 257 2 4 4 X iajs-1818 257 3 ) ) PUNCT iajs-1818 257 4 all all DET iajs-1818 257 5 𝑅-modules 𝑅-modules PROPN iajs-1818 257 6 have have VERB iajs-1818 257 7 the the DET iajs-1818 257 8 ccip ccip NOUN iajs-1818 257 9 . . PUNCT iajs-1818 258 1 ( ( PUNCT iajs-1818 258 2 5 5 X iajs-1818 258 3 ) ) PUNCT iajs-1818 258 4 all all PRON iajs-1818 258 5 injective injective ADJ iajs-1818 258 6 over over ADP iajs-1818 258 7 𝑅 𝑅 PROPN iajs-1818 258 8 have have VERB iajs-1818 258 9 the the DET iajs-1818 258 10 ccip ccip NOUN iajs-1818 258 11 . . PUNCT iajs-1818 259 1 ( ( PUNCT iajs-1818 259 2 6 6 X iajs-1818 259 3 ) ) PUNCT iajs-1818 259 4 all all PRON iajs-1818 259 5 injective injective ADJ iajs-1818 259 6 over over ADP iajs-1818 259 7 𝑅are 𝑅are PROPN iajs-1818 259 8 cosemisimple cosemisimple NOUN iajs-1818 259 9 . . PUNCT iajs-1818 260 1 ( ( PUNCT iajs-1818 260 2 7 7 X iajs-1818 260 3 ) ) PUNCT iajs-1818 260 4 all all DET iajs-1818 260 5 quasi quasi ADJ iajs-1818 260 6 injective injective ADJ iajs-1818 260 7 over over ADP iajs-1818 260 8 𝑅 𝑅 PROPN iajs-1818 260 9 have have VERB iajs-1818 260 10 the the DET iajs-1818 260 11 ccip ccip NOUN iajs-1818 260 12 . . PUNCT iajs-1818 261 1 ( ( PUNCT iajs-1818 261 2 8) 8) NUM iajs-1818 261 3 all all DET iajs-1818 261 4 quasi quasi ADJ iajs-1818 261 5 injective injective ADJ iajs-1818 261 6 over over ADP iajs-1818 261 7 𝑅 𝑅 PROPN iajs-1818 261 8 are be AUX iajs-1818 261 9 cosemisimple cosemisimple NOUN iajs-1818 261 10 . . PUNCT iajs-1818 262 1 proof proof NOUN iajs-1818 262 2 . . PUNCT iajs-1818 263 1 1 1 NUM iajs-1818 263 2   ADP iajs-1818 263 3 2 2 NUM iajs-1818 263 4 follows follow VERB iajs-1818 263 5 by by ADP iajs-1818 263 6 [ [ X iajs-1818 263 7 1 1 NUM iajs-1818 263 8 , , PUNCT iajs-1818 263 9 theorem theorem VERB iajs-1818 263 10 1.12 1.12 NUM iajs-1818 263 11 ] ] PUNCT iajs-1818 263 12 . . PUNCT iajs-1818 264 1 1 1 NUM iajs-1818 264 2   NOUN iajs-1818 264 3 3 3 NUM iajs-1818 264 4 , , PUNCT iajs-1818 264 5 2 2 NUM iajs-1818 264 6   NOUN iajs-1818 264 7 4 4 NUM iajs-1818 264 8   NOUN iajs-1818 264 9 5 5 NUM iajs-1818 264 10 , , PUNCT iajs-1818 264 11 6 6 NUM iajs-1818 264 12   NOUN iajs-1818 264 13 5 5 NUM iajs-1818 264 14 and and CCONJ iajs-1818 264 15 8 8 NUM iajs-1818 264 16   NOUN iajs-1818 264 17 7 7 NUM iajs-1818 264 18 are be AUX iajs-1818 264 19 obvious obvious ADJ iajs-1818 264 20 .. .. PUNCT iajs-1818 264 21 3 3 NUM iajs-1818 264 22   NOUN iajs-1818 264 23 1 1 NUM iajs-1818 264 24 when when SCONJ iajs-1818 264 25 𝐼 𝐼 PROPN iajs-1818 264 26 is be AUX iajs-1818 264 27 an an DET iajs-1818 264 28 ideal ideal NOUN iajs-1818 264 29 of of ADP iajs-1818 264 30 𝑅. 𝑅. NOUN iajs-1818 264 31 because because SCONJ iajs-1818 264 32 all all DET iajs-1818 264 33 𝑅-modules 𝑅-modules PROPN iajs-1818 264 34 are be AUX iajs-1818 264 35 relatively relatively ADV iajs-1818 264 36 coclosed coclose VERB iajs-1818 264 37 rickart rickart NOUN iajs-1818 264 38 to to ADP iajs-1818 264 39 any any DET iajs-1818 264 40 𝑅-module 𝑅-module PROPN iajs-1818 264 41 . . PUNCT iajs-1818 265 1 then then ADV iajs-1818 265 2 the the DET iajs-1818 265 3 𝑅-module 𝑅-module PROPN iajs-1818 265 4 𝑅 𝑅 PROPN iajs-1818 265 5 is be AUX iajs-1818 265 6 relatively relatively ADV iajs-1818 265 7 coclosed coclose VERB iajs-1818 265 8 rickart rickart NOUN iajs-1818 265 9 to to ADP iajs-1818 265 10 the the DET iajs-1818 265 11 module module NOUN iajs-1818 265 12 as as ADP iajs-1818 265 13 𝑅-module 𝑅-module PROPN iajs-1818 265 14 . . PUNCT iajs-1818 266 1 because because SCONJ iajs-1818 266 2 there there PRON iajs-1818 266 3 exists exist VERB iajs-1818 266 4 the the DET iajs-1818 266 5 natural natural ADJ iajs-1818 266 6 epimorphism epimorphism NOUN iajs-1818 266 7 𝜋 𝜋 PROPN iajs-1818 266 8 ∶ ∶ PROPN iajs-1818 266 9 𝑅 𝑅 PROPN iajs-1818 266 10 → → PROPN iajs-1818 266 11 , , PUNCT iajs-1818 266 12 hece hece PROPN iajs-1818 266 13 ker ker PROPN iajs-1818 266 14 𝜋 𝜋 PRON iajs-1818 266 15 𝐼 𝐼 PROPN iajs-1818 266 16 is be AUX iajs-1818 266 17 a a DET iajs-1818 266 18 coclosed coclosed ADJ iajs-1818 266 19 right right ADJ iajs-1818 266 20 ideal ideal NOUN iajs-1818 266 21 of of ADP iajs-1818 266 22 𝑅. 𝑅. NOUN iajs-1818 266 23 this this PRON iajs-1818 266 24 implies imply VERB iajs-1818 266 25 that that SCONJ iajs-1818 266 26 𝑅 𝑅 PROPN iajs-1818 266 27 is be AUX iajs-1818 266 28 cosemisimple cosemisimple NOUN iajs-1818 266 29 right right ADJ iajs-1818 266 30 𝑅-module 𝑅-module NOUN iajs-1818 266 31 . . PUNCT iajs-1818 267 1 4 4 NUM iajs-1818 267 2   NOUN iajs-1818 267 3 2 2 NUM iajs-1818 267 4 when when SCONJ iajs-1818 267 5 𝑀 𝑀 PROPN iajs-1818 267 6 is be AUX iajs-1818 267 7 an an DET iajs-1818 267 8 𝑅-module 𝑅-module PROPN iajs-1818 267 9 and and CCONJ iajs-1818 267 10 𝑁 𝑁 PROPN iajs-1818 267 11 a a DET iajs-1818 267 12 submodule submodule NOUN iajs-1818 267 13 of of ADP iajs-1818 267 14 𝑀 𝑀 PROPN iajs-1818 267 15 , , PUNCT iajs-1818 267 16 implies imply VERB iajs-1818 267 17 𝑀 𝑀 PROPN iajs-1818 267 18   PROPN iajs-1818 267 19 has have VERB iajs-1818 267 20 ccip ccip PROPN iajs-1818 267 21 . . PUNCT iajs-1818 268 1 let let VERB iajs-1818 268 2 𝑓 𝑓 DET iajs-1818 268 3 ∶ ∶ NOUN iajs-1818 268 4 𝑀 𝑀 PROPN iajs-1818 268 5 → → PUNCT iajs-1818 268 6 be be AUX iajs-1818 268 7 the the DET iajs-1818 268 8 natural natural ADJ iajs-1818 268 9 epimorphisim epimorphisim NOUN iajs-1818 268 10 . . PUNCT iajs-1818 269 1 by by ADP iajs-1818 269 2 lemma lemma PROPN iajs-1818 269 3 4.5 4.5 NUM iajs-1818 269 4 , , PUNCT iajs-1818 269 5 ker ker NOUN iajs-1818 269 6 𝑓 𝑓 PRON iajs-1818 269 7 is be AUX iajs-1818 269 8 a a DET iajs-1818 269 9 coclosed coclose VERB iajs-1818 269 10 submodule submodule NOUN iajs-1818 269 11 in in ADP iajs-1818 269 12 𝑀. 𝑀. PROPN iajs-1818 269 13 hence hence ADV iajs-1818 269 14 𝑀 𝑀 PROPN iajs-1818 269 15 is be AUX iajs-1818 269 16 coclosed coclose VERB iajs-1818 269 17 rickart rickart NOUN iajs-1818 269 18 . . PUNCT iajs-1818 270 1 5 5 NUM iajs-1818 270 2   NOUN iajs-1818 270 3 4 4 NUM iajs-1818 270 4 when when SCONJ iajs-1818 270 5 𝑀 𝑀 PROPN iajs-1818 270 6 is be AUX iajs-1818 270 7 a a DET iajs-1818 270 8 module module NOUN iajs-1818 270 9 over over ADP iajs-1818 270 10 an an DET iajs-1818 270 11 𝑅 𝑅 PROPN iajs-1818 270 12 , , PUNCT iajs-1818 270 13 then then ADV iajs-1818 270 14 there there PRON iajs-1818 270 15 is be VERB iajs-1818 270 16 an an DET iajs-1818 270 17 injective injective ADJ iajs-1818 270 18 𝑅-module 𝑅-module PROPN iajs-1818 270 19 𝑁 𝑁 PROPN iajs-1818 270 20 and and CCONJ iajs-1818 270 21 a a DET iajs-1818 270 22 monomorphisim monomorphisim NOUN iajs-1818 270 23 𝑓 𝑓 PROPN iajs-1818 270 24 : : PUNCT iajs-1818 270 25 𝑀 𝑀 PROPN iajs-1818 270 26 → → SYM iajs-1818 270 27 𝑁 𝑁 PROPN iajs-1818 270 28 implies imply VERB iajs-1818 270 29 that that SCONJ iajs-1818 270 30 𝑀 𝑀 PROPN iajs-1818 270 31 is be AUX iajs-1818 270 32 isomorphic isomorphic ADJ iajs-1818 270 33 to to ADP iajs-1818 270 34 im𝑓. im𝑓. NOUN iajs-1818 270 35 when when SCONJ iajs-1818 270 36 𝑓 𝑓 PRON iajs-1818 270 37 is be AUX iajs-1818 270 38 an an DET iajs-1818 270 39 epimorphism epimorphism NOUN iajs-1818 270 40 , , PUNCT iajs-1818 270 41 then then ADV iajs-1818 270 42 the the DET iajs-1818 270 43 result result NOUN iajs-1818 270 44 follows follow VERB iajs-1818 270 45 . . PUNCT iajs-1818 271 1 suppose suppose VERB iajs-1818 271 2 not not PART iajs-1818 271 3 , , PUNCT iajs-1818 271 4 so so CCONJ iajs-1818 271 5 there there PRON iajs-1818 271 6 is be VERB iajs-1818 271 7 an an DET iajs-1818 271 8 injective injective ADJ iajs-1818 271 9 𝑅-module 𝑅-module PROPN iajs-1818 271 10 𝐾 𝐾 PROPN iajs-1818 271 11 and and CCONJ iajs-1818 271 12 ihsciconf ihsciconf ADJ iajs-1818 271 13 2017 2017 NUM iajs-1818 271 14 special special ADJ iajs-1818 271 15 issue issue NOUN iajs-1818 271 16 ibn ibn PROPN iajs-1818 271 17 al al PROPN iajs-1818 271 18 - - PUNCT iajs-1818 271 19 haitham haitham PROPN iajs-1818 271 20 journal journal PROPN iajs-1818 271 21 for for ADP iajs-1818 271 22 pure pure ADJ iajs-1818 271 23 and and CCONJ iajs-1818 271 24 applied apply VERB iajs-1818 271 25 science science NOUN iajs-1818 271 26 https://doi.org/ https://doi.org/ NOUN iajs-1818 271 27 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 271 28 for for ADP iajs-1818 271 29 more more ADJ iajs-1818 271 30 information information NOUN iajs-1818 271 31 about about ADP iajs-1818 271 32 the the DET iajs-1818 271 33 conference conference NOUN iajs-1818 271 34 please please INTJ iajs-1818 271 35 visit visit VERB iajs-1818 271 36 the the DET iajs-1818 271 37 websites website NOUN iajs-1818 271 38 : : PUNCT iajs-1818 271 39 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 271 40     SPACE iajs-1818 271 41 www.ihsciconf.org www.ihsciconf.org NOUN iajs-1818 271 42       SPACE iajs-1818 271 43 mathematics|461 mathematics|461 NOUN iajs-1818 271 44         SPACE iajs-1818 271 45 and and CCONJ iajs-1818 271 46 a a DET iajs-1818 271 47 monomorphisim monomorphisim ADJ iajs-1818 271 48 𝑔 𝑔 NOUN iajs-1818 271 49 : : PUNCT iajs-1818 271 50 → → SYM iajs-1818 271 51 𝐾. 𝐾. PROPN iajs-1818 271 52 let let VERB iajs-1818 271 53 ℎ ℎ NOUN iajs-1818 271 54 : : PUNCT iajs-1818 271 55 𝑁 𝑁 PROPN iajs-1818 271 56 → → SYM iajs-1818 271 57 be be AUX iajs-1818 271 58 the the DET iajs-1818 271 59 natural natural ADJ iajs-1818 271 60 epimorphism epimorphism NOUN iajs-1818 271 61 . . PUNCT iajs-1818 272 1 let let VERB iajs-1818 272 2 us we PRON iajs-1818 272 3 consider consider VERB iajs-1818 272 4 𝑔ℎ 𝑔ℎ NOUN iajs-1818 272 5 : : PUNCT iajs-1818 272 6 𝑁 𝑁 PROPN iajs-1818 272 7 → → SYM iajs-1818 272 8 𝐾. 𝐾. PROPN iajs-1818 272 9 because because SCONJ iajs-1818 272 10 𝑁 𝑁 PROPN iajs-1818 272 11   ADJ iajs-1818 272 12 𝐾 𝐾 PROPN iajs-1818 272 13 is be AUX iajs-1818 272 14 injective injective ADJ iajs-1818 272 15 then then ADV iajs-1818 272 16 𝑁 𝑁 PROPN iajs-1818 272 17   ADJ iajs-1818 272 18 𝐾 𝐾 PROPN iajs-1818 272 19 has have VERB iajs-1818 272 20 the the DET iajs-1818 272 21 ccip ccip PROPN iajs-1818 272 22 . . PUNCT iajs-1818 273 1 by by ADP iajs-1818 273 2 lemma lemma PROPN iajs-1818 273 3 4.4 4.4 NUM iajs-1818 273 4 , , PUNCT iajs-1818 273 5 ker ker PROPN iajs-1818 273 6 𝑔ℎ 𝑔ℎ PROPN iajs-1818 273 7 is be AUX iajs-1818 273 8 a a DET iajs-1818 273 9 coclosed coclosed ADJ iajs-1818 273 10 submodule submodule NOUN iajs-1818 273 11 of of ADP iajs-1818 273 12 𝑁 𝑁 PROPN iajs-1818 273 13     SPACE iajs-1818 273 14 and and CCONJ iajs-1818 273 15 due due ADP iajs-1818 273 16 to to ADP iajs-1818 273 17 𝑔 𝑔 PROPN iajs-1818 273 18 is be AUX iajs-1818 273 19 a a DET iajs-1818 273 20 monomorphisim monomorphisim NOUN iajs-1818 273 21 , , PUNCT iajs-1818 273 22 implies imply VERB iajs-1818 273 23 ker ker PROPN iajs-1818 273 24 𝑔ℎ 𝑔ℎ PROPN iajs-1818 273 25 ker ker PROPN iajs-1818 273 26 ℎ ℎ PROPN iajs-1818 273 27 im𝑓 im𝑓 PROPN iajs-1818 273 28 is be AUX iajs-1818 273 29 coclosed coclose VERB iajs-1818 273 30 of of ADP iajs-1818 273 31 𝑁. 𝑁. PROPN iajs-1818 273 32 because because SCONJ iajs-1818 273 33 𝑁 𝑁 PROPN iajs-1818 273 34 has have VERB iajs-1818 273 35 the the DET iajs-1818 273 36 ccip ccip NOUN iajs-1818 273 37 , , PUNCT iajs-1818 273 38 then then ADV iajs-1818 273 39 by by ADP iajs-1818 273 40 lemma lemma PROPN iajs-1818 273 41 4.4 4.4 NUM iajs-1818 273 42 , , PUNCT iajs-1818 273 43 im𝑓 im𝑓 NOUN iajs-1818 273 44 has have VERB iajs-1818 273 45 the the DET iajs-1818 273 46 ccip ccip PROPN iajs-1818 273 47 . . PUNCT iajs-1818 274 1 but but CCONJ iajs-1818 274 2 𝑀 𝑀 PROPN iajs-1818 274 3 ≅ ≅ PROPN iajs-1818 274 4 im𝑓 im𝑓 PROPN iajs-1818 274 5 and and CCONJ iajs-1818 274 6 due due ADJ iajs-1818 274 7 to to PART iajs-1818 274 8     SPACE iajs-1818 274 9 proposition proposition NOUN iajs-1818 274 10 4.6 4.6 NUM iajs-1818 274 11 , , PUNCT iajs-1818 274 12 therefore therefore ADV iajs-1818 274 13 𝑀 𝑀 PROPN iajs-1818 274 14 has have VERB iajs-1818 274 15 the the DET iajs-1818 274 16 ccip ccip NOUN iajs-1818 274 17 . . PUNCT iajs-1818 275 1 5 5 NUM iajs-1818 275 2   NOUN iajs-1818 275 3 6 6 NUM iajs-1818 275 4 when when SCONJ iajs-1818 275 5 𝑀 𝑀 PROPN iajs-1818 275 6 is be AUX iajs-1818 275 7 an an DET iajs-1818 275 8 injective injective NOUN iajs-1818 275 9 with with ADP iajs-1818 275 10 𝑁 𝑁 PROPN iajs-1818 275 11 a a DET iajs-1818 275 12 submodule submodule NOUN iajs-1818 275 13 in in ADP iajs-1818 275 14 𝑀 𝑀 PROPN iajs-1818 275 15 with with ADP iajs-1818 275 16 𝑓 𝑓 PRON iajs-1818 275 17 : : PUNCT iajs-1818 275 18 𝑀 𝑀 PROPN iajs-1818 275 19 → → PUNCT iajs-1818 275 20 is be AUX iajs-1818 275 21 the the DET iajs-1818 275 22 natural natural ADJ iajs-1818 275 23 epimorphism epimorphism NOUN iajs-1818 275 24 . . PUNCT iajs-1818 276 1 when when SCONJ iajs-1818 276 2 is be AUX iajs-1818 276 3 injective injective ADJ iajs-1818 276 4 , , PUNCT iajs-1818 276 5 then then ADV iajs-1818 276 6 𝑀 𝑀 PROPN iajs-1818 276 7   PROPN iajs-1818 276 8 is be AUX iajs-1818 276 9 injective injective ADJ iajs-1818 276 10 with with ADP iajs-1818 276 11 the the DET iajs-1818 276 12 ccip ccip NOUN iajs-1818 276 13 . . PUNCT iajs-1818 277 1 by by ADP iajs-1818 277 2 lemma lemma PROPN iajs-1818 277 3 4.5 4.5 NUM iajs-1818 277 4 , , PUNCT iajs-1818 277 5 ker ker NOUN iajs-1818 277 6 𝑓 𝑓 DET iajs-1818 277 7 𝑁 𝑁 PROPN iajs-1818 277 8 is be AUX iajs-1818 277 9 a a DET iajs-1818 277 10 coclosed coclosed NOUN iajs-1818 277 11 of of ADP iajs-1818 277 12 𝑀. 𝑀. PROPN iajs-1818 277 13 suppose suppose VERB iajs-1818 277 14 not not PART iajs-1818 277 15 , , PUNCT iajs-1818 277 16 when when SCONJ iajs-1818 277 17 𝐸 𝐸 PROPN iajs-1818 277 18 is be AUX iajs-1818 277 19 the the DET iajs-1818 277 20 injective injective ADJ iajs-1818 277 21 hull hull NOUN iajs-1818 277 22 of of ADP iajs-1818 277 23 with with ADP iajs-1818 277 24 𝑖 𝑖 SYM iajs-1818 277 25 : : PUNCT iajs-1818 277 26 → → SYM iajs-1818 277 27 𝐸 𝐸 PROPN iajs-1818 277 28 is be AUX iajs-1818 277 29 the the DET iajs-1818 277 30 inclusion inclusion NOUN iajs-1818 277 31 function function NOUN iajs-1818 277 32 . . PUNCT iajs-1818 278 1 consider consider VERB iajs-1818 278 2 𝑖𝑓 𝑖𝑓 ADP iajs-1818 278 3 : : PUNCT iajs-1818 278 4 𝑀 𝑀 PROPN iajs-1818 278 5 → → SYM iajs-1818 278 6 𝐸 𝐸 PROPN iajs-1818 278 7 . . PUNCT iajs-1818 279 1 since since SCONJ iajs-1818 279 2 𝑀𝐸 𝑀𝐸 PROPN iajs-1818 279 3 is be AUX iajs-1818 279 4 injective injective ADJ iajs-1818 279 5 , , PUNCT iajs-1818 279 6 implies imply VERB iajs-1818 279 7 𝑀𝐸 𝑀𝐸 PROPN iajs-1818 279 8 has have VERB iajs-1818 279 9 the the DET iajs-1818 279 10 ccip ccip NOUN iajs-1818 279 11 . . PUNCT iajs-1818 280 1 but but CCONJ iajs-1818 280 2 𝑖 𝑖 PRON iajs-1818 280 3 is be AUX iajs-1818 280 4 a a DET iajs-1818 280 5 monomorphisim monomorphisim NOUN iajs-1818 280 6 , , PUNCT iajs-1818 280 7 then then ADV iajs-1818 280 8 ker ker PROPN iajs-1818 281 1 𝑖𝑓 𝑖𝑓 X iajs-1818 281 2 ker ker NOUN iajs-1818 282 1 𝑓 𝑓 DET iajs-1818 282 2 𝑁 𝑁 PROPN iajs-1818 282 3 is be AUX iajs-1818 282 4 a a DET iajs-1818 282 5 coclosed coclose VERB iajs-1818 282 6 in in ADP iajs-1818 282 7 𝑀. 𝑀. PROPN iajs-1818 282 8 therefore therefore ADV iajs-1818 282 9 𝑀 𝑀 PROPN iajs-1818 282 10 is be AUX iajs-1818 282 11 cosemisimple cosemisimple NOUN iajs-1818 282 12 . . PUNCT iajs-1818 283 1 7 7 NUM iajs-1818 283 2   NOUN iajs-1818 283 3 8 8 NUM iajs-1818 283 4 by by ADP iajs-1818 283 5 the the DET iajs-1818 283 6 same same ADJ iajs-1818 283 7 way way NOUN iajs-1818 283 8 of of ADP iajs-1818 283 9 the the DET iajs-1818 283 10 statement statement NOUN iajs-1818 283 11 5 5 NUM iajs-1818 283 12   NOUN iajs-1818 283 13 6 6 NUM iajs-1818 283 14 . . PUNCT iajs-1818 284 1 theorem theorem NOUN iajs-1818 284 2 let let VERB iajs-1818 284 3 𝑀 𝑀 PRON iajs-1818 284 4 be be AUX iajs-1818 284 5 a a DET iajs-1818 284 6 module module NOUN iajs-1818 284 7 over over ADP iajs-1818 284 8 𝑅. 𝑅. NOUN iajs-1818 284 9 the the DET iajs-1818 284 10 next next ADJ iajs-1818 284 11 are be AUX iajs-1818 284 12 equivalent equivalent ADJ iajs-1818 284 13 ( ( PUNCT iajs-1818 284 14 1 1 NUM iajs-1818 284 15 ) ) PUNCT iajs-1818 284 16 𝑀 𝑀 PROPN iajs-1818 284 17 is be AUX iajs-1818 284 18 coclosed coclose VERB iajs-1818 284 19 rickart rickart NOUN iajs-1818 284 20 with with ADP iajs-1818 284 21 ccip ccip PROPN iajs-1818 284 22 . . PUNCT iajs-1818 285 1 ( ( PUNCT iajs-1818 285 2 2 2 X iajs-1818 285 3 ) ) PUNCT iajs-1818 285 4 the the DET iajs-1818 285 5 right right ADJ iajs-1818 285 6 annihilator annihilator NOUN iajs-1818 285 7 in in ADP iajs-1818 285 8 𝑀 𝑀 PROPN iajs-1818 285 9 of of ADP iajs-1818 285 10 each each DET iajs-1818 285 11 finitely finitely ADV iajs-1818 285 12 generated generate VERB iajs-1818 285 13 left leave VERB iajs-1818 285 14 ideal ideal NOUN iajs-1818 285 15 𝐼 𝐼 ADP iajs-1818 285 16 𝑓 𝑓 PROPN iajs-1818 285 17 , , PUNCT iajs-1818 285 18 𝑓 𝑓 X iajs-1818 285 19 , , PUNCT iajs-1818 285 20 … … PUNCT iajs-1818 285 21 , , PUNCT iajs-1818 285 22 𝑓 𝑓 PRON iajs-1818 285 23 of of ADP iajs-1818 285 24 end end NOUN iajs-1818 285 25 𝑀 𝑀 PROPN iajs-1818 285 26 is be AUX iajs-1818 285 27 a a DET iajs-1818 285 28 coclosed coclose VERB iajs-1818 285 29 . . PUNCT iajs-1818 286 1 submodule submodule NOUN iajs-1818 286 2 of of ADP iajs-1818 286 3 𝑀. 𝑀. PROPN iajs-1818 286 4 proof proof NOUN iajs-1818 286 5 . . PUNCT iajs-1818 287 1 1 1 NUM iajs-1818 287 2   NOUN iajs-1818 287 3 2 2 NUM iajs-1818 287 4 let let VERB iajs-1818 287 5 𝐼 𝐼 PRON iajs-1818 287 6 be be AUX iajs-1818 287 7 a a DET iajs-1818 287 8 nonzero nonzero NOUN iajs-1818 287 9 left leave VERB iajs-1818 287 10 ideal ideal NOUN iajs-1818 287 11 of of ADP iajs-1818 287 12 end end NOUN iajs-1818 287 13 𝑀 𝑀 PROPN iajs-1818 287 14 where where SCONJ iajs-1818 287 15 𝑀 𝑀 PROPN iajs-1818 287 16 a a DET iajs-1818 287 17 coclosed coclose VERB iajs-1818 287 18 rickart rickart NOUN iajs-1818 287 19 module module NOUN iajs-1818 287 20 over over ADP iajs-1818 287 21 𝑅 𝑅 PROPN iajs-1818 287 22 with with ADP iajs-1818 287 23 a a DET iajs-1818 287 24 finite finite ADJ iajs-1818 287 25 number number NOUN iajs-1818 287 26 of of ADP iajs-1818 287 27 generators generator NOUN iajs-1818 287 28 { { PUNCT iajs-1818 287 29 𝑓 𝑓 PROPN iajs-1818 287 30 , , PUNCT iajs-1818 287 31 𝑓 𝑓 X iajs-1818 287 32 , , PUNCT iajs-1818 287 33 … … PUNCT iajs-1818 287 34 , , PUNCT iajs-1818 287 35 𝑓 𝑓 X iajs-1818 287 36 } } PUNCT iajs-1818 287 37 . . PUNCT iajs-1818 288 1 because because SCONJ iajs-1818 288 2 𝑀 𝑀 PROPN iajs-1818 288 3 is be AUX iajs-1818 288 4 coclosed coclose VERB iajs-1818 288 5 rickart rickart NOUN iajs-1818 288 6 , , PUNCT iajs-1818 288 7 then then ADV iajs-1818 288 8 𝑟 𝑟 X iajs-1818 288 9 𝑓 𝑓 PRON iajs-1818 288 10 is be AUX iajs-1818 288 11 a a DET iajs-1818 288 12 coclosed coclosed NOUN iajs-1818 288 13 of of ADP iajs-1818 288 14 𝑀 𝑀 PROPN iajs-1818 288 15 for for ADP iajs-1818 288 16 every every DET iajs-1818 288 17 𝑖 𝑖 SYM iajs-1818 288 18 1,2 1,2 NUM iajs-1818 288 19 , , PUNCT iajs-1818 288 20 … … PUNCT iajs-1818 288 21 , , PUNCT iajs-1818 288 22 𝑛. 𝑛. NOUN iajs-1818 288 23 but but CCONJ iajs-1818 288 24 𝑀 𝑀 PROPN iajs-1818 288 25 with with ADP iajs-1818 288 26 ccip ccip PROPN iajs-1818 288 27 , , PUNCT iajs-1818 288 28 it it PRON iajs-1818 288 29 follows follow VERB iajs-1818 288 30 that that SCONJ iajs-1818 288 31 ⋂ ⋂ PROPN iajs-1818 288 32 𝑟 𝑟 NOUN iajs-1818 288 33 𝑓 𝑓 PRON iajs-1818 288 34 𝑟 𝑟 PUNCT iajs-1818 288 35 𝐼 𝐼 PROPN iajs-1818 288 36 is be AUX iajs-1818 288 37 a a DET iajs-1818 288 38 coclosed coclosed NOUN iajs-1818 288 39 of of ADP iajs-1818 288 40 𝑀. 𝑀. PROPN iajs-1818 288 41 ( ( PUNCT iajs-1818 288 42 3 3 NUM iajs-1818 288 43 ) ) PUNCT iajs-1818 288 44   NOUN iajs-1818 289 1 1 1 NUM iajs-1818 289 2 when when SCONJ iajs-1818 289 3 𝑓 𝑓 DET iajs-1818 289 4 ∈ ∈ PROPN iajs-1818 289 5 end end NOUN iajs-1818 289 6 𝑀 𝑀 PROPN iajs-1818 289 7 , , PUNCT iajs-1818 289 8 then then ADV iajs-1818 289 9 𝑓 𝑓 PRON iajs-1818 289 10 is be AUX iajs-1818 289 11 a a DET iajs-1818 289 12 left left ADJ iajs-1818 289 13 ideal ideal NOUN iajs-1818 289 14 of of ADP iajs-1818 289 15 end end NOUN iajs-1818 289 16 𝑀 𝑀 PROPN iajs-1818 289 17 with with ADP iajs-1818 289 18 one one NUM iajs-1818 289 19 generator generator NOUN iajs-1818 289 20 . . PUNCT iajs-1818 290 1 by by ADP iajs-1818 290 2 hypothesis hypothesis NOUN iajs-1818 290 3 , , PUNCT iajs-1818 290 4 𝑟 𝑟 PRON iajs-1818 290 5 𝑓 𝑓 PRON iajs-1818 290 6 is be AUX iajs-1818 290 7 coclosed coclose VERB iajs-1818 290 8 of of ADP iajs-1818 290 9 𝑀 𝑀 PROPN iajs-1818 290 10 , , PUNCT iajs-1818 290 11 yeild yeild PROPN iajs-1818 290 12 𝑀 𝑀 PROPN iajs-1818 290 13 is be AUX iajs-1818 290 14 coclosed coclose VERB iajs-1818 290 15 rickart rickart NOUN iajs-1818 290 16 . . PUNCT iajs-1818 291 1 moreover moreover ADV iajs-1818 291 2 , , PUNCT iajs-1818 291 3 ⋂ ⋂ PROPN iajs-1818 291 4 𝑟 𝑟 X iajs-1818 291 5 𝑓 𝑓 DET iajs-1818 291 6 𝑟 𝑟 X iajs-1818 291 7 𝐼 𝐼 PROPN iajs-1818 291 8 for for ADP iajs-1818 291 9 any any DET iajs-1818 291 10 finitely finitely ADV iajs-1818 291 11 generated generate VERB iajs-1818 291 12 left leave VERB iajs-1818 291 13 ideal ideal NOUN iajs-1818 291 14 𝐼 𝐼 ADP iajs-1818 291 15 𝑓 𝑓 PROPN iajs-1818 291 16 , , PUNCT iajs-1818 291 17 𝑓 𝑓 X iajs-1818 291 18 , , PUNCT iajs-1818 291 19 … … PUNCT iajs-1818 291 20 , , PUNCT iajs-1818 291 21 𝑓 𝑓 PRON iajs-1818 291 22 of of ADP iajs-1818 291 23 end end NOUN iajs-1818 291 24 𝑀 𝑀 PROPN iajs-1818 291 25 . . PUNCT iajs-1818 292 1 hence hence ADV iajs-1818 292 2 ⋂ ⋂ PROPN iajs-1818 292 3 𝑟 𝑟 NOUN iajs-1818 292 4 𝑓 𝑓 PRON iajs-1818 292 5 is be AUX iajs-1818 292 6 a a DET iajs-1818 292 7 coclosed coclosed NOUN iajs-1818 292 8 of of ADP iajs-1818 292 9 𝑀 𝑀 PROPN iajs-1818 292 10 , , PUNCT iajs-1818 292 11 thus thus ADV iajs-1818 292 12 𝑀 𝑀 PROPN iajs-1818 292 13 has have VERB iajs-1818 292 14 the the DET iajs-1818 292 15 ccip ccip PROPN iajs-1818 292 16 . . PUNCT iajs-1818 293 1 references reference NOUN iajs-1818 293 2 [ [ X iajs-1818 293 3 1 1 X iajs-1818 293 4 ] ] PUNCT iajs-1818 293 5 j. j. PROPN iajs-1818 293 6 clark clark PROPN iajs-1818 293 7 ; ; PUNCT iajs-1818 293 8 ch ch PROPN iajs-1818 293 9 . . PROPN iajs-1818 293 10 lomp lomp PROPN iajs-1818 293 11 ; ; PUNCT iajs-1818 293 12 n. n. PROPN iajs-1818 293 13 vanaja vanaja PROPN iajs-1818 293 14 ; ; PUNCT iajs-1818 293 15 r. r. PROPN iajs-1818 293 16 wisbauer wisbauer NOUN iajs-1818 293 17 , , PUNCT iajs-1818 293 18 lifting lift VERB iajs-1818 293 19 modules module NOUN iajs-1818 293 20 supplements supplement NOUN iajs-1818 293 21 and and CCONJ iajs-1818 293 22 projectivity projectivity NOUN iajs-1818 293 23 in in ADP iajs-1818 293 24 module module NOUN iajs-1818 293 25 theory theory NOUN iajs-1818 293 26 . . PUNCT iajs-1818 294 1 birkhauser birkhauser PROPN iajs-1818 294 2 verlag verlag PROPN iajs-1818 294 3 , , PUNCT iajs-1818 294 4 basel basel PROPN iajs-1818 294 5 , , PUNCT iajs-1818 294 6 boston boston PROPN iajs-1818 294 7 , , PUNCT iajs-1818 294 8 berlin berlin PROPN iajs-1818 294 9 , , PUNCT iajs-1818 294 10 2006 2006 NUM iajs-1818 294 11 . . PUNCT iajs-1818 295 1 [ [ X iajs-1818 295 2 2 2 NUM iajs-1818 295 3 ] ] PUNCT iajs-1818 295 4 ghaleb ghaleb PROPN iajs-1818 295 5 ahmed ahmed PROPN iajs-1818 295 6 , , PUNCT iajs-1818 295 7 pure pure ADJ iajs-1818 295 8 rickart rickart NOUN iajs-1818 295 9 modules module NOUN iajs-1818 295 10 and and CCONJ iajs-1818 295 11 their their PRON iajs-1818 295 12 generalization generalization NOUN iajs-1818 295 13 , , PUNCT iajs-1818 295 14     SPACE iajs-1818 295 15 international international ADJ iajs-1818 295 16 journal journal NOUN iajs-1818 295 17 of of ADP iajs-1818 295 18 mathematics mathematic NOUN iajs-1818 295 19 trend trend NOUN iajs-1818 295 20 and and CCONJ iajs-1818 295 21 technology technology NOUN iajs-1818 295 22 30 30 NUM iajs-1818 295 23 ( ( PUNCT iajs-1818 295 24 2 2 NUM iajs-1818 295 25 ) ) PUNCT iajs-1818 295 26 82 82 NUM iajs-1818 295 27 - - SYM iajs-1818 295 28 95 95 NUM iajs-1818 295 29 . . PUNCT iajs-1818 295 30 2016 2016 NUM iajs-1818 296 1 [ [ X iajs-1818 296 2 3 3 NUM iajs-1818 296 3 ] ] X iajs-1818 296 4 ghaleb ghaleb NOUN iajs-1818 296 5 ahmed ahmed PROPN iajs-1818 296 6 , , PUNCT iajs-1818 296 7     SPACE iajs-1818 296 8 dual dual ADJ iajs-1818 296 9 coclosed coclose VERB iajs-1818 296 10 rickart rickart NOUN iajs-1818 296 11 modules module NOUN iajs-1818 296 12 , , PUNCT iajs-1818 296 13 ibn ibn PROPN iajs-1818 296 14 al al PROPN iajs-1818 296 15 - - PUNCT iajs-1818 296 16 haitham haitham PROPN iajs-1818 296 17 journal journal PROPN iajs-1818 296 18 for for ADP iajs-1818 296 19 pure pure ADJ iajs-1818 296 20 and and CCONJ iajs-1818 296 21 applied applied ADJ iajs-1818 296 22 science science NOUN iajs-1818 296 23 , , PUNCT iajs-1818 296 24 to to PART iajs-1818 296 25 appear appear VERB iajs-1818 296 26 . . PUNCT iajs-1818 297 1 [ [ X iajs-1818 297 2 4 4 NUM iajs-1818 297 3 ] ] X iajs-1818 297 4 ghawi ghawi X iajs-1818 297 5 , , PUNCT iajs-1818 297 6 th.y th.y NOUN iajs-1818 297 7 . . PUNCT iajs-1818 298 1 modules module NOUN iajs-1818 298 2 with with ADP iajs-1818 298 3 closed closed ADJ iajs-1818 298 4 intersection intersection NOUN iajs-1818 298 5 ( ( PUNCT iajs-1818 298 6 sum sum NOUN iajs-1818 298 7 ) ) PUNCT iajs-1818 298 8 property property NOUN iajs-1818 298 9 , , PUNCT iajs-1818 298 10 ph.d ph.d PROPN iajs-1818 298 11 . . PUNCT iajs-1818 299 1 thesis thesis NOUN iajs-1818 299 2 , , PUNCT iajs-1818 299 3 university university PROPN iajs-1818 299 4 of of ADP iajs-1818 299 5 al al PROPN iajs-1818 299 6 - - PUNCT iajs-1818 299 7 mustansiriyah mustansiriyah PROPN iajs-1818 299 8 , , PUNCT iajs-1818 299 9 iraq iraq PROPN iajs-1818 299 10 , , PUNCT iajs-1818 299 11 2015 2015 NUM iajs-1818 299 12 . . PUNCT iajs-1818 300 1 [ [ X iajs-1818 300 2 5 5 NUM iajs-1818 300 3 ] ] X iajs-1818 300 4 i.m.a i.m.a NOUN iajs-1818 300 5 . . PUNCT iajs-1818 300 6 hadi hadi PROPN iajs-1818 300 7 , , PUNCT iajs-1818 300 8 ghaleb ghaleb PROPN iajs-1818 300 9 ahmed ahmed PROPN iajs-1818 300 10 , , PUNCT iajs-1818 300 11     SPACE iajs-1818 300 12 modules module NOUN iajs-1818 300 13 with with ADP iajs-1818 300 14 ⊛(⊛’or ⊛(⊛’or PROPN iajs-1818 300 15 ⊛ ⊛ NUM iajs-1818 300 16 ’’ ’' PUNCT iajs-1818 300 17 ) ) PUNCT iajs-1818 300 18 condition condition NOUN iajs-1818 300 19 , , PUNCT iajs-1818 300 20 the the DET iajs-1818 300 21 bulletin bulletin NOUN iajs-1818 300 22 of of ADP iajs-1818 300 23 society society NOUN iajs-1818 300 24 for for ADP iajs-1818 300 25 mathematical mathematical ADJ iajs-1818 300 26 services service NOUN iajs-1818 300 27 and and CCONJ iajs-1818 300 28 standards standard NOUN iajs-1818 300 29 4 4 NUM iajs-1818 300 30 , , PUNCT iajs-1818 300 31 12 12 NUM iajs-1818 300 32 - - SYM iajs-1818 300 33 21 21 NUM iajs-1818 300 34 . . PUNCT iajs-1818 300 35 2012 2012 NUM iajs-1818 300 36 [ [ X iajs-1818 300 37 6 6 NUM iajs-1818 300 38 ] ] X iajs-1818 300 39 i.m.a i.m.a NOUN iajs-1818 300 40 . . PUNCT iajs-1818 300 41 hadi hadi PROPN iajs-1818 300 42 , , PUNCT iajs-1818 300 43 ghaleb ghaleb PROPN iajs-1818 300 44 ahmed ahmed PROPN iajs-1818 300 45 , , PUNCT iajs-1818 300 46     SPACE iajs-1818 300 47 strongly strongly ADV iajs-1818 300 48 ( ( PUNCT iajs-1818 300 49 comletely comletely ADV iajs-1818 300 50 ) ) PUNCT iajs-1818 300 51 hollow hollow ADJ iajs-1818 300 52 submodules submodule NOUN iajs-1818 300 53 i i PRON iajs-1818 300 54 , , PUNCT iajs-1818 300 55     SPACE iajs-1818 300 56 ibn ibn PROPN iajs-1818 300 57 alhaitham alhaitham NOUN iajs-1818 300 58 journal journal NOUN iajs-1818 300 59 for for ADP iajs-1818 300 60 pure pure ADJ iajs-1818 300 61 and and CCONJ iajs-1818 300 62 applied apply VERB iajs-1818 300 63 science science NOUN iajs-1818 300 64 3 3 NUM iajs-1818 300 65 ( ( PUNCT iajs-1818 300 66 25 25 NUM iajs-1818 300 67 ) ) PUNCT iajs-1818 300 68 , , PUNCT iajs-1818 300 69 393 393 NUM iajs-1818 300 70 - - SYM iajs-1818 300 71 403 403 NUM iajs-1818 300 72 . . PUNCT iajs-1818 301 1 2012 2012 NUM iajs-1818 301 2 ihsciconf ihsciconf PROPN iajs-1818 301 3 2017 2017 NUM iajs-1818 301 4 special special ADJ iajs-1818 301 5 issue issue NOUN iajs-1818 301 6 ibn ibn PROPN iajs-1818 301 7 al al PROPN iajs-1818 301 8 - - PUNCT iajs-1818 301 9 haitham haitham PROPN iajs-1818 301 10 journal journal PROPN iajs-1818 301 11 for for ADP iajs-1818 301 12 pure pure ADJ iajs-1818 301 13 and and CCONJ iajs-1818 301 14 applied apply VERB iajs-1818 301 15 science science NOUN iajs-1818 301 16 https://doi.org/ https://doi.org/ NOUN iajs-1818 301 17 10.30526/2017.ihsciconf.1818 10.30526/2017.ihsciconf.1818 NUM iajs-1818 301 18 for for ADP iajs-1818 301 19 more more ADJ iajs-1818 301 20 information information NOUN iajs-1818 301 21 about about ADP iajs-1818 301 22 the the DET iajs-1818 301 23 conference conference NOUN iajs-1818 301 24 please please INTJ iajs-1818 301 25 visit visit VERB iajs-1818 301 26 the the DET iajs-1818 301 27 websites website NOUN iajs-1818 301 28 : : PUNCT iajs-1818 301 29 http://www.ihsciconf.org/conf/ http://www.ihsciconf.org/conf/ VERB iajs-1818 301 30     SPACE iajs-1818 301 31 www.ihsciconf.org www.ihsciconf.org NOUN iajs-1818 301 32       SPACE iajs-1818 301 33 mathematics|462 mathematics|462 NOUN iajs-1818 301 34         SPACE iajs-1818 302 1 [ [ X iajs-1818 302 2 7 7 NUM iajs-1818 302 3 ] ] X iajs-1818 302 4 i.m.a i.m.a NOUN iajs-1818 302 5 . . PUNCT iajs-1818 302 6 hadi hadi PROPN iajs-1818 302 7 , , PUNCT iajs-1818 302 8 ghaleb ghaleb PROPN iajs-1818 302 9 ahmed ahmed PROPN iajs-1818 302 10 , , PUNCT iajs-1818 302 11     SPACE iajs-1818 302 12 strongly strongly ADV iajs-1818 302 13 ( ( PUNCT iajs-1818 302 14 completely completely ADV iajs-1818 302 15 ) ) PUNCT iajs-1818 302 16 hollow hollow ADJ iajs-1818 302 17 submodules submodules PROPN iajs-1818 302 18 ii ii PROPN iajs-1818 302 19 , , PUNCT iajs-1818 302 20 ibn ibn PROPN iajs-1818 302 21 alhaitham alhaitham NOUN iajs-1818 302 22 journal journal NOUN iajs-1818 302 23 for for ADP iajs-1818 302 24 pure pure ADJ iajs-1818 302 25 and and CCONJ iajs-1818 302 26 applied apply VERB iajs-1818 302 27 science science NOUN iajs-1818 302 28 1 1 NUM iajs-1818 302 29 ( ( PUNCT iajs-1818 302 30 26 26 NUM iajs-1818 302 31 ) ) PUNCT iajs-1818 302 32 , , PUNCT iajs-1818 302 33 292302 292302 NUM iajs-1818 302 34 . . PUNCT iajs-1818 303 1 2013 2013 NUM iajs-1818 304 1 [ [ X iajs-1818 304 2 8 8 NUM iajs-1818 304 3 ] ] X iajs-1818 304 4 i.m.a i.m.a NOUN iajs-1818 304 5 . . PUNCT iajs-1818 304 6 hadi hadi PROPN iajs-1818 304 7 , , PUNCT iajs-1818 304 8 ghaleb ghaleb PROPN iajs-1818 304 9 ahmed ahmed PROPN iajs-1818 304 10 , , PUNCT iajs-1818 304 11 t t PROPN iajs-1818 304 12 - - PUNCT iajs-1818 304 13 regular regular ADJ iajs-1818 304 14 modules module NOUN iajs-1818 304 15 , , PUNCT iajs-1818 304 16     SPACE iajs-1818 304 17 international international ADJ iajs-1818 304 18 journal journal NOUN iajs-1818 304 19 of of ADP iajs-1818 304 20 advanced advanced ADJ iajs-1818 304 21 scientific scientific ADJ iajs-1818 304 22 and and CCONJ iajs-1818 304 23 technical technical ADJ iajs-1818 304 24 research research NOUN iajs-1818 304 25 6 6 NUM iajs-1818 304 26 ( ( PUNCT iajs-1818 304 27 1 1 NUM iajs-1818 304 28 ) ) PUNCT iajs-1818 304 29 128 128 NUM iajs-1818 304 30 - - SYM iajs-1818 304 31 137 137 NUM iajs-1818 304 32 . . PUNCT iajs-1818 304 33 2016 2016 NUM iajs-1818 305 1 [ [ X iajs-1818 305 2 9 9 NUM iajs-1818 305 3 ] ] X iajs-1818 305 4 g. g. PROPN iajs-1818 305 5 lee lee PROPN iajs-1818 305 6 , , PUNCT iajs-1818 305 7 s. s. PROPN iajs-1818 305 8 t. t. PROPN iajs-1818 305 9 rizvi rizvi PROPN iajs-1818 305 10 and and CCONJ iajs-1818 305 11 c. c. PROPN iajs-1818 305 12 s. s. PROPN iajs-1818 305 13 roman roman PROPN iajs-1818 305 14 , , PUNCT iajs-1818 305 15 rickart rickart NOUN iajs-1818 305 16 modules module NOUN iajs-1818 305 17 , , PUNCT iajs-1818 305 18 comm comm NOUN iajs-1818 305 19 . . PUNCT iajs-1818 306 1 algebra algebra NOUN iajs-1818 306 2 38 38 NUM iajs-1818 306 3 2010 2010 NUM iajs-1818 306 4 , , PUNCT iajs-1818 306 5 4005 4005 NUM iajs-1818 306 6 - - SYM iajs-1818 306 7 4027 4027 NUM iajs-1818 306 8 . . PUNCT iajs-1818 307 1 [ [ X iajs-1818 307 2 10 10 NUM iajs-1818 307 3 ] ] PUNCT iajs-1818 307 4 a.s.mijbass a.s.mijbass NOUN iajs-1818 307 5 , , PUNCT iajs-1818 307 6 quasi quasi ADJ iajs-1818 307 7 - - ADJ iajs-1818 307 8 dedekind dedekind ADJ iajs-1818 307 9 modules module NOUN iajs-1818 307 10 , , PUNCT iajs-1818 307 11 ph.d ph.d PROPN iajs-1818 307 12 . . PUNCT iajs-1818 308 1 thesis thesis NOUN iajs-1818 308 2 , , PUNCT iajs-1818 308 3 university university NOUN iajs-1818 308 4 of of ADP iajs-1818 308 5 baghdad baghdad PROPN iajs-1818 308 6 , , PUNCT iajs-1818 308 7 1997 1997 NUM iajs-1818 308 8 . . PUNCT iajs-1818 309 1 [ [ X iajs-1818 309 2 11 11 NUM iajs-1818 309 3 ] ] PUNCT iajs-1818 309 4 nuhad nuhad PROPN iajs-1818 309 5 s. s. PROPN iajs-1818 309 6 al al PROPN iajs-1818 309 7 - - PUNCT iajs-1818 309 8 mothafar mothafar PROPN iajs-1818 309 9 , , PUNCT iajs-1818 309 10 ghaleb ghaleb PROPN iajs-1818 309 11 ahmed ahmed PROPN iajs-1818 309 12 , , PUNCT iajs-1818 309 13 2 2 NUM iajs-1818 309 14 - - PUNCT iajs-1818 309 15 regular regular ADJ iajs-1818 309 16 modules module NOUN iajs-1818 309 17 , , PUNCT iajs-1818 309 18 ibn ibn PROPN iajs-1818 309 19 al al PROPN iajs-1818 309 20 - - PUNCT iajs-1818 309 21 haitham haitham PROPN iajs-1818 309 22 journal journal PROPN iajs-1818 309 23 for for ADP iajs-1818 309 24 pure pure ADJ iajs-1818 309 25 and and CCONJ iajs-1818 309 26 applied apply VERB iajs-1818 309 27 science science NOUN iajs-1818 309 28 28 28 NUM iajs-1818 309 29 ( ( PUNCT iajs-1818 309 30 2 2 NUM iajs-1818 309 31 ) ) PUNCT iajs-1818 309 32 ( ( PUNCT iajs-1818 309 33 2015 2015 NUM iajs-1818 309 34 ) ) PUNCT iajs-1818 309 35 , , PUNCT iajs-1818 309 36 184 184 NUM iajs-1818 309 37 - - SYM iajs-1818 309 38 192 192 NUM iajs-1818 309 39 . . 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