13956

12683

كارشناسي ارشد

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

۱۳۹۷

نه، ۷۱ص.: مصور

فارسي به انگليسي; انگليسي به فارسي

1397/07/17

كتابنامه

علوم رياضي

رياضي

ID12683

Nil clean and Strongly Nil clean Rings Seyedeh Elham Ahmadpoori e ahmadpoori94@yahoo com 2018 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr MohammadReza Vedadi mrvedadi@cc iut ac ir Advisor Dr Atefeh Ghorbani a ghorbani@cc iut ac ir 2010 MSC 16U99 16E90 16S34 16S50 16S70 Keywords Clean Nil clean Strongly clean Strongly nil clean AbstractThis M Sc Thesis based on The following paper Ko an Tamer Wang Zhou and Zhou Yiqiang Nil clean and strongly nil clean rings Journal ofPure and Applied Algebra 220 2 633 646 2016 Throughout R is an associative ring with identity and C R J R U R and Nil R denote Thecenter The Jacobson radical The unit group and The set of all nilpotent elements of R respectively An element a of a ring R is called nil clean if a e b where e2 e R and b is a nilpotent element ofR if furTher eb be Then a is called strongly nil clean The ring R is called nil clean resp stronglynil clean if each of its elements is nil clean resp strongly nil clean several equivalent conditions ofa strongly nil clean element is given an element a R is strongly nil clean if and only if a is strongly regular and a a2 is nilpotent if and only if a is uniquely strongly clean and a a2 is nilpotent This follows that a ring R is strongly nil clean if and only if R J R is a boolean ring and J R isa nil ideal It is proved that a similar result to The Jacobson Lemma holds for strongly nil cleanelements Let R be a ring and a b R If 1 ab is strongly nil clean Then so is 1 ba Then it isinvestigated when a Morita context is a strongly nil clean ring and determine when a formal matrixring is strongly nil clean The group ring of a locally nite 2 group over a strongly nil clean ring isstrongly nil clean In The sequel A non unital ring is considered as an associative ring with or without identity It isshown that a non unital ring R is strongly nil clean if and only if R J R is a boolean ring and J R is a nil ideal a ring R is strongly nil clean if and only if R I is strongly nil clean where I is a nil

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