$n$-Clean Rings with Involutions

DOI：10.3770/j.issn:2095-2651.2016.02.008

 作者 单位 崔建 安徽师范大学数学系, 安徽 芜湖 241003 殷晓斌 安徽师范大学数学系, 安徽 芜湖 241003

称$*$-环是$*$-clean的,如果该环中每个元素都能写成一个投影元与一个单位之和. 给定整数$n$,我们称$*$-环$R$是$n$-$*$-clean的,如果对于任意的$a\in R$都有$a=p+u_1+\cdots+u_n$, 其中$p$是投影元且$u_1,\ldots,u_n$是单位.给出了$n$-$*$-clean环的基本性质以及许多$2$-$*$-clean环但不是$*$-clean环的例子.此外,讨论了$n$-$*$-clean环的扩张性质.

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.