崔建,殷晓斌.具有对合运算的$n$-clean环[J].数学研究及应用,2016,36(2):194~200
具有对合运算的$n$-clean环
$n$-Clean Rings with Involutions
投稿时间:2015-04-12  最后修改时间:2015-09-18
DOI:10.3770/j.issn:2095-2651.2016.02.008
中文关键词:  $*$-clean环  $n$-$*$-clean环  clean环  $n$-clean环
英文关键词:$*$-clean ring  $n$-$*$-clean ring  clean ring  $n$-clean ring
基金项目:国家自然科学基金(Grant No.11401009),安徽省自然科学基金(Grant No.1408085QA01),安徽高校省级自然科学研究重点项目(Grant No.KJ2014A082),安徽师范大学教育基金会(Grant No.2013qnzx35).
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 
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中文摘要:
      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.
英文摘要:
      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.
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