8386

7767

كارشناسي ارشد

رياضي محض ﴿جبر﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1392

يازده،118ص.

ص.ع.به فارسي و انگليسي

عاطفه قرباني

محمود بهبودي

15/10/92

محمدرضا ودادي، حسين خبازيان

رياضي

ID7767

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

Direct Sums of Rickart Modules Raheleh Dastanpour raheleh dastanpour@math iut ac ir 2013 Depatrment of Mathematical Sciences Isfahan University of Tecnology Isfahan 84156 83111 IranSupervisor Dr Atefeh Ghorbani a ghorbani@cc iut ac irAdvisor Dr Mahmood Behboodi mbehbood@cc iut ac ir2010 MSC Primary 16D10 Secondary 16D40 16D80Keywords Rickart rings and modules Baer rings and modules idempotents annihilators endomor phisms direct sum of modules right semi hereditary free and projective modules Abstract This thesis is based on the article Direct Sums of Rickart Modules written by G Lee S T Rizviand C S Roman Already the concept of Rickart ring was defined by Kaplansky Suppose R is aunitary ring R is called right Rickart if the right annihilator of every element of R as a right ideal generates by an idempotent element of R Lee Rizvi and Roman introduced the notion of Rickartmodules motivated by a need to put the notion of right Rickart rings in a general module theoreticsetting and by the question If R is a right Rickart ring and e2 e R what kind of Rickart propertywill the right R module eR have Assume M is a right R module M is called Rickart if for everyendomorphism of M ker M It has been shown that every direct summand of a Rickart moduleis a Rickart module but examples show that a direct sum of Rickart modules is not always Rickart Inthis thesis we consider this question When are the direct sums of Rickart modules also Rickart nWe show that if Mi is Mj injective for all i j I 1 2 n then Mi is a Rickart module i 1if and only if Mi is Mj Rickart for all i j I As a consequence we obtain that for a nonsingularCS module M E M M is always a Rickart module It is proved that if Mi is Mj C2 for all ni j I 1 2 n then i 1 Mi is a Rickart module if and only if Mi is Mj Rickart for all i j I From this we immediately conclude that for a Rickart module M with C2 condition every finite direct sum of copies of M is Rickart Also we show that if Mi is a fully invariant submodule of Mj j I for all i I then Mj is a Rickart module if and only if every Mi is Rickart Then we consider j Iindecomposable Rickart modules We show that M is an indecomposable Rickart module with C2

عاطفه قرباني

محمود بهبودي

محمدرضا ودادي، حسين خبازيان