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| 摘要: |
| 称一个元素e为potent元，如果存在整数k使得$e^k=e$,本文称R中的元素e有potent指数k如果k是满足$e^k=e(k\geq 2)$的最小整数.Potent元素的基本性质被给出.本文定义环为伪clean环，如果它的每个元素都是potent元和Jacobson根中元素的和.给出了一些伪clean环的性质的例子.本文也证明了$\mathbb{Z}_m$( $2\leq m\in \mathbb{Z}$ )是伪clean的，并且伪clean环是clean环.此外，本文也证明了伪clean环是直有限的且有稳定秩一. |
| 关键词: clean环 强J-clean环 伪clean环 potent元. |
| DOI： |
| 分类号:16U99 16E50 |
| 基金项目:国家自然科学基金项目（面上项目，重点项目，重大项目） |
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| On Potent Elements and Pseudo Clean Rings |
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Yan Ding,Hanpeng Gao
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| Abstract: |
| An element $e$ is called $potent$ if there exists some integer $k\geq2$ such that $e^k=e$. In this paper, an element $e\in R$ in a ring $R$ is said have $potent$ $index$ $k$ if $k$ is the smallest positive integer such that $e^k=e (k\geq 2)$. The basic properties of potent elements are obtained. We also define a ring $R$ is $pseudo$ $clean$ provided that every element of $R$ can be written as the
sum of a potent element and an element in its Jacobson radical. Some properties and examples of $pseudo$ $clean$ are given. We also show that $\mathbb{Z}_m$ is $pseudo$ $clean$ for every $2\leq m\in \mathbb{Z}$ and $pseudo$ $clean$ rings are clean. Furthermore, we prove $pseudo$ $clean$ rings are directly finite and have stable range one. |
| Key words: clean rings, strongly $J$-clean rings, pseudo clean rings potent elements. |
