$n$-Clean Rings with Involutions
Received:April 12, 2015  Revised:September 18, 2015
Key Word: $*$-clean ring   $n$-$*$-clean ring   clean ring   $n$-clean ring
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11401009), the Natural Science Foundation of Anhui Province (Grant No.1408085QA01), the Key Natural Science Foundation of Anhui Educational Committee (Grant No.KJ2014A082) and the Educational Foundation of Anhui Normal University (Grant No.2013qnzx35).
 Author Name Affiliation Jian CUI Department of Mathematics, Anhui Normal University, Anhui 241003, P. R. China Xiaobin YIN Department of Mathematics, Anhui Normal University, Anhui 241003, P. R. China
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A $*$-ring is called $*$-clean if every element of the ring can be written as the sum of a projection and a unit. For an integer $n\geq 1,$ we call a $*$-ring $R$ $n$-$*$-clean if for any $a\in R,$ $a=p+u_1+\cdots+u_n$ where $p$ is a projection and $u_i$ are units for all $i$. Basic properties of $n$-$*$-clean rings are considered, and a number of illustrative examples of $2$-$*$-clean rings which are not $*$-clean are provided. In addition, extension properties of $n$-$*$-clean rings are discussed.