On Weakly $r$-Clean Rings

DOI：10.3770/j.issn:2095-2651.2017.05.005

 作者 单位 高汉鹏 安徽师范大学数学计算机科学学院, 安徽 芜湖 241000 殷晓斌 安徽师范大学数学计算机科学学院, 安徽 芜湖 241000

作为$r$-clean环与弱clean环的推广,本文介绍了弱$r$-clean环的概念. 环$R$称为弱$r$-clean环,若任意$a\in R$, 都存在幂等元$e$以及正则元$r$使得$a=r+e$或$a=r-e$. 一些弱$r$-clean环的性质和例子被给出. 此外,我们还证明了弱clean环与弱$r$-clean环在abelian环条件下是等价的.

As generalization of $r$-clean rings and weakly clean rings, we define a ring $R$ is weakly $r$-clean if for any $a\in R$ there exist an idempotent $e$ and a regular element $r$ such that $a=r+e$ or $a=r-e$. Some properties and examples of weakly $r$-clean rings are given. Furthermore, we prove the weakly clean rings and weakly $r$-clean rings are equivalent for abelian rings.