16278

14528

كارشناسي ارشد

جبر

اصفهان : دانشگاه صنعتي اصفهان

1399

ده، 89 ص.: مصور، جدول، نمودار.

1399/11/11

كتابنامه

رياضي محض

رياضي

1399/11/15

2669402

Endoprime modules and their direct sums Zinat Hajihashemi z hajihashemi@math iut ac ir 2020 Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mahmood Behboodi mbehbood@iut ac irAdvisor Dr Bijan Taeri btaeri@iut ac ir2000 MSC 16D70 16S50 16N60 Keywords Prime ring endoprime module FI indecomposable semicentral reduced ring column row finite matrixring Abstract This thesis is written according to a paper titelled Endoprime modules and their direct sums by G Lee and S T Rizvi 2018 The purpose of this thesis is to further study the endoprime modules as one of the special classes ofquasi Baer modules As a module theoretic analogue of a prime ring we characterize an endoprime module via its endomorphism ringand a weak retractability condition The ring R is prime if every nonzero fully invariant submodule of RR is faithfulwhere S EndR R i e for every 0 I R lR I 0 2 see definition 3 11 Generalizing of prime ring to the module theory is not equivalent to prime module where a module M is said tobe Dauns prime if rR M rR N for each non zero submodule N of M 1 see definition 4 1 But Vedadiand Haghany did this work and by generalizing the concept of prime ring to module theory define endoprime modulethat is Right R module M is endoprime module if every nonzero fully invarient submodule of M is faithful as aleft S module where S EndR M i e for every 0 N M lS N 0 2 see definition 3 15 Forexample i RR is an endoprime module if and only if R is a prime ring ii Any vector space is an endoprime module iii Every simple module is an endoprime module iv Any module which has a simple endomorphism ring is an endoprime module v Every Rickart module is an endoprime module where M is called a Rickart module for each S EndR M rM Ker eM for some e2 e S There are some sufficient and necessary conditions for endoprime module M is an endoprime module if and only iffor any m M and S Sm 0 implies that either 0 or m 0 Corollary 1 4 see theorem 3 17 The following conditions are equivalent for a module M and S EndR M i M is an endoprime module ii lS Sm 0 for all 0 m M iii rM S 0 for all 0 S iv rM J 0 for all 0 J S Corollary 2 4 see corollary 3 18 The following conditions are equivalent for a module M and S EndR M i M is an endoprime module ii for any two sided J of S and any fully invariant submodule Nof M JN 0 implies that either J 0 or N 0 iii for any left ideal I of S and any fully invarient submoduleN of M IN 0 implies that either I 0 or N 0 iv for any right ideal I of S and any submodule N of M IN 0 implies that either I 0 or N 0

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