| As a branch of algebra,the importance of ring theory is self-evident.Clean ring is an important branch in ring theory.Since it was proposed by W.K.Nicholson in 1977,clean rings have been paid more and more attention because of its simple structure and many connections with other rings.In addition,it has attracted a lot of algebra scholars to do further researches and expanded it.Therefore,the research of clean rings is a very meaningful job.In this paper,some equivalent conditions of strongJn-clean ring are given,strong Jn-cleaness of single matrix and properties of 2-clean rings over a non commutative conditions are also given.Specific activities include the following:Firstly,some properties and equivalent conditions of strong Jn-clean rings are given when studying strong Jn-clean rings.The strong n-regular rings are introduced for leading strong Jn-clean rings.Then,some equivalent conditions between strong n-regular rings and strong Jn-clean rings are given.Particularly,it is proved that if R is a ring with all idempotents in the center,then R is strongJn-clean rings if and only if R/ J?R?is strongly p-regular and idempotents lift modulo J?R?.Some examples are also given to illustrate the connections between strongly Jn-clean rings and other rings.Secondly,the strong Jn-cleaness of single matrix is investigated.Under the communicative condition,some necessary conditions for strong cleaness of 2 ?2 matrix rings are given.Moreover,the strong Jn-cleaness of 2 ?2 matrix over a commutative local ring is studied,and a method to judge strong J2-cleaness of 2 ?2 matrix is also given by characteristic equation.Finally,non communicative 2-clean rings are studied.Some new equivalent conditions of 2-clean rings are given.Under the non commutative condition,some equivalence conditions between 2-clean ring and clean ring are proved.The 2-cleaness of the direct sum and homomorphism rings are also considered.In addition,some 2-cleaness of matrix rings are investigated and extended. |
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