حلقه ها و مدول هاي تحويل يافته، ZI و آرمنداريز

چكيده انگليسي :

On Reduced ZI and Armendariz Modules and Rings Salahedin Mohamadi s mohamadi@math iut ac ir 6 April 2011 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Mohamadreza Vedadi mvedadi@cc iut ac irAdvisor Dr atefeh ghorbani aghorbani@sci ui ac ir2000 MSC Primary 16S36 Secondary 16D80 Key words Reduced modules Semiprime modules Semiprimitive modules ZI modules Armendariz mod ules ps armendariz modules Nonsingular module AbstractA ring is called a reduced ring if it has no nonzero nilpotent elements In this thesise we continue the study ofnonsingular armendariz ps Armendariz ZI and reduced modules and rings show that reduced modules aresymmetric and symmetric modules are ZI and show that ats modules over reduced rings are reduced Also at modules over symmetric rings are symmetric Similary show that reduced modules are ps Armendarizand ps Armendariz modules are ZI Next characterize rings over which all modules are reduced symmetric Also introduce the concepts of regular and strongly regular rings V rings and p V rings and study its someproperties and extend these results to modules and show that regular ZI rings are strongly regular andtherefore reduced left and right V ring We study the relationships of reduced modules with semiprime semiprimitive modules and other related classes of modules and rings and extend several results known forreduced rings and quo rings to reduced modules and quo modules We prove that for a semiprime or a modulewith zero jacobson radical the concepts of reduced symmetric ps armendariz and ZI modules coincide Newexample of reduced modules are furnished at modules over reduced rings and modules with zero jacobsonradical over left quo rings are reduced We introduce the concepts of Baer quasi Baer principally quasi Baerand p p and rst observe the Baer modules are quasi Baer and quasi Baer modules are principaly quasi Baerbut quasi Baer modules need not be Baer and also principally quasi Baer need not be quasi Baer We nextprove that for a ZI module the concepts of Baer and quasi Baer modules and also the concepts of principallyquasi Baer and p p modules coincide Similary show that for a principally quasi Baer module and also for ap p module the concepts of reduced and ZI modules coincide Finally show that reduced rings are nonsingulr Zbut reduced modules need not be nonsingular For example for each prime integer p the Z module pZ isreduced but not nonsingular Also if R is an in nite product of elds and B is the direct sum of the same R elds then B is a reduced and singular R module Also prove that nonsingular rings modules need notbe reduced but left duo left nonsingular rings are reduced In view of the fact that all left or right duorings and all reversible rings are ZI it is natural to ask whether all left nonsingular ZI rings nonsingular ZImodules are reduced 1

محمدي، صلاح الدين

حلقه ها و مدول هاي تحويل يافته، ZI و آرمنداريز

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