On Weakly $r$-Clean Rings
Received:October 13, 2016  Revised:May 17, 2017
Key Word: $r$-clean rings   weakly clean rings   weakly $r$-clean rings   regular rings  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11401009; 11326062).
Author NameAffiliation
Hanpeng GAO Department of Mathematics, Anhui Normal University, Anhui 241000, P. R. China 
Xiaobin YIN Department of Mathematics, Anhui Normal University, Anhui 241000, P. R. China 
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Abstract:
      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.
Citation:
DOI:10.3770/j.issn:2095-2651.2017.05.005
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