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# $2$-Clean Rings

Published:2009-03-01
Printed: Mar 2009
• Z. Wang
• J. L. Chen
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## Abstract

A ring $R$ is said to be $n$-clean if every element can be written as a sum of an idempotent and $n$ units. The class of these rings contains clean rings and $n$-good rings in which each element is a sum of $n$ units. In this paper, we show that for any ring $R$, the endomorphism ring of a free $R$-module of rank at least 2 is $2$-clean and that the ring $B(R)$ of all $\omega\times \omega$ row and column-finite matrices over any ring $R$ is $2$-clean. Finally, the group ring $RC_{n}$ is considered where $R$ is a local ring.
 Keywords: $2$-clean rings, $2$-good rings, free modules, row and column-finite matrix rings, group rings
 MSC Classifications: 16D70 - Structure and classification (except as in 16Gxx), direct sum decomposition, cancellation 16D40 - Free, projective, and flat modules and ideals [See also 19A13] 16S50 - Endomorphism rings; matrix rings [See also 15-XX]