关于Pseudo Weakly J-clean环（英文）

On Pseudo WeaklyJ-clean Rings

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.

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 .