关于Pseudo Weakly J-clean环

摘要

A ring R is called a pseudo weakly J‐clean ring if every element a∈ R can be written in the form of a= e+ w+ (1-e)Ra or a= -e+ w+ (1-e)Ra where e is an idempotent and w belongs to the Jacobson radical . This paper explores various properties of pseudo weakly J‐clean rings .A ring R is pseudo weakly J‐clean if and only if R[[x]] ,Hurwitz series ring H(R) ,trivial extension T(R ,M) and S(R ,σ) are pseudo weakly J‐clean , respectively .Furthermore ,it proves that the following are equivalent ,for any n∈ N ,Sn(R) is pseudo J‐clean , for any n∈N ,R[x]/(xn ) is pseudo weakly J‐clean ,where (xn ) is the ideal generated by xn .In particular ,it characterize S= R[D ,C] is pseudo weakly J‐clean under certain conditions .Also it shows that 2 is a unit in R , then R is pseudo J‐clean if and only if RC2 is pseudo J‐clean .%一个环R叫做 pseudo weakly J‐clean环，如果 R中的每一个元素都可以写成 a＝ e＋ w＋（1－e）Ra或a＝－e＋w＋（1－e）Ra的形式，其中 e是幂等元，w属于 Jacobson根。文章探究了 pseudo weakly J‐clean环的各种性质。环 R是pseudo weakly J‐clean环当且仅当幂级数环 R[[x]]，Hurwitz级数环 H（R），平凡扩张 T（R ，M）和 S（R ，σ）分别是 pseudo weakly J‐clean环。更进一步证明以下几点是等价的：任意的 n∈ N ，Sn （R）是 pseudo J‐clean；任意的 n∈ N ，R[x]／（xn ）是pseudo J‐clean ，（xn ）是由 xn生成的理想。特别的，阐述了在某种条件下 S＝ R[D ，C]是pseudo weakly J‐clean ；并且得出结论：当2是 R中的可逆元时，R是pseudo J‐clean当且仅当群环 RC2是pseudo J‐clean 。