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Commutative weakly nil clean unital ringss Hamed Saremi h saremi@math iut ac ir june 20 2016 for the degree of Master of Science in farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mahmood behboodi mbehbood@cc iut ac irAdvisor Dr Atefeh Ghorbani a ghorbani@cc iut ac irMSC 16S34 16U60 16U99 16S70Keywords weakly nil clean ring weakly nil clean group ring nagata ringAbstract This M Sc thesis is based on the following paperPeter V Dancheva W Wm McGovernb Commutative weakly nil clean unital rings J Algebra 425 2015 410 422 A ring R is said to be clean ring if every element of R can be written as the sum of a unit and an idempotent Cleanrings were first introduced in a paper by Nicholson 25 in 1977 as a class of exchange rings A ring R is said to be nil clean ring if every element of R can be written as the sum of a nilpotent and an idempotent This class of rings were first introduced by Diesel In the past ten years there have been many investigations concerning variants of the clean and strongly clean prop erties Additionally several authors have studied versions of such properties in the case of non unital rings We define the concept of a weakly nil clean commutative ring which generalizes Diesel s 11 notion of a nil cleancommutative ring and investigate this class of rings A ring R is said weakly nil clean ring if every element of R can be written as the sum of or difference a nilpotent andan idempotent We obtain some fundamental properties of this rings In particular it is proved that these rings are clean We alsoconsider the questions of when the Nagata ring as well as the group ring is weakly nil clean Throughout the present paper all rings considered unless otherwise noted shall be assumed to be commutative andpossess an identity Our notation and terminology shall follow 10 and 15 For instance for such a ring R U R denotes the group of all units of R N R is the nilradical of R J R is the Jacobson radical of R and Id R is theset of all idempotents of R It is a known fact that U R N R U R If 1 x U R then x is called quasi regular We denote the set of quasi regular elements of R by L R Wedenote the set of all maximal ideals of R by Max R Recently A Diesl 11 modified the definition of a clean ring and obtained an interesting new concept he called nilclean In his article he proved many fundamental properties as well as developed a general theory of nil clean rings his interest as was Nicholson s was in the context of non commutative rings The notion of uniquely clean rings was firstly defined in 4 in the commutative case as those rings in which everyelement is uniquely the sum of a unit and an idempotent Later on in 24 the authors study such arbitrary rings againcalling them uniquely clean notice that this uniqueness is tantamount to the existence of a unique idempotent with thegiven sum property In a subsequent paper 25 the same authors coined the term semi boolean ringas a general ring I satisfying the condition that for all r I there is an e Id R and j J R such that r j e this wasdone in the context of rings not necessarily commutative nor possessing an identity they show that a uniquely clean

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