 |
|
$2$-Clean Rings |
Canad. Math. Bull. 52(2009), 145-153
Published:2009-03-01
Printed: Mar 2009
Printed: Mar 2009
- Z. Wang
- J. L. Chen
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] |