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1. CMB Online first
Simplicity of Partial Skew Group Rings of Abelian Groups Let $A$ be a ring with local units, $E$ a set of local units for $A$,
$G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of
$A$ that contain local units.
We show that $A\star_{\alpha} G$ is simple if and only if $A$ is
$G$-simple and the center of the corner $e\delta_0 (A\star_{\alpha} G)
e \delta_0$ is a field for all $e\in E$. We apply the result to
characterize simplicity of partial skew group rings in two cases,
namely for partial skew group rings arising from partial actions by
clopen subsets of a compact set and partial actions on the set level.
Keywords:partial skew group rings, simple rings, partial actions, abelian groups Categories:16S35, 37B05 |
2. CMB 2012 (vol 57 pp. 326)
On Zero-divisors in Group Rings of Groups with Torsion Nontrivial pairs of zero-divisors in group rings are
introduced and discussed. A problem on the existence of nontrivial
pairs of zero-divisors in group rings of free Burnside groups of odd
exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of
zero-divisors are also found in group rings of free products of groups
with torsion.
Keywords:Burnside groups, free products of groups, group rings, zero-divisors Categories:20C07, 20E06, 20F05, , 20F50 |
3. CMB 2009 (vol 52 pp. 145)
$2$-Clean Rings 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 Categories:16D70, 16D40, 16S50 |