Conference abstracts

Session S05 - Rings and Algebras

No date set

## $*$ - Clean Group Algebras

### Gianfranco Osmar Manrique Portuguez

### Universidade Federal Fluminense, Brasil - gian1427@gmail.com

An element of a(n associative) ring (with $ 1 $) is clean if it is the sum of a unit and an idempotent. A ring is clean if every element in it is clean. The property of cleanness was formulated by Nicholson [4] in the course of his study of exchange rings. From then on, several related concepts were proposed: uniquely clean rings, strongly clean rings, weakly clean rings, $ * $ - clean rings, r - clean rings, nil - clean rings, to cite a few. In the realm of group rings, these properties have been studied from 2001 [2] on with the aim of characterizing the rings $R $ and groups $G $ such that the group ring $RG$ is clean.

In 2010 Vas proposed the definition of a $\ast$ - clean ring (``star''- clean) [5]: a $\ast$ - ring (i.e., rings with an involution) in which every element may be written as a sum of a unit and a projection. Clearly, every $\ast$-clean ring is a $star$ - ring and is a clean ring. In [5], Vas asked: when is a $\ast$ - ring clean, but not $\ast$-clean?

Every group $G $ having an element g $\neq$ 1 , with $| \langle g \rangle | \neq 2$, is endowed with the classical involution $g \mapsto g^{-1}$. Because of that, group rings $RG$ are almost always $\ast$ - rings: if $R $ is a commutative rings, for instance, an involution in $RG $ is obtained from the $R $ - linear extension of the classical involution in $G $ (and is also called the classical involution in $RG $ ). The $\ast$-cleanness of group rings was first approached in 2011 [3]. Even though some instances of group rings are answers to Vas's question [1], very little is still known about conditions under which a group ring with the classical involution is $\ast$-clean (not even the case of the group ring $RG $, where $R $ is a commutative ring and $G$ is a cyclic group, is fully stablished!).

In this talk, I present some recent results [1]. Let $R $ be a commutative local ring. I will present $ R S_3 $ as an answer to Vas 's question, and I will provide necessary and sufficient conditions for the group ring $ R Q_8 $ to ber $\ast$ - clean, where $ Q_8 $ is the quaternion group of $ 8 $ elements.

REFERENCES :

[1] Y. Gao, J. Chen, Y. Li, em Some $\ast$-clean Group Rings, Algebra Colloquium 22 (2015) 169--180.

[2] J. Han, W. K. Nicholson, em Extensions of clean rings, Communications in Algebra 29 (2001), 2589--2595.

[3] C. Li, Y. Zhou, em On strongly $\ast$-clean rings, Journal of Algebra and Its Applications 6 (2011), 1363--1370.

[4] W. K. Nicholson, em Lifting idempotents and exchange rings, Transactions of the AMS 229 (1977), 269 -- 278.

[5] L. Vas, em $\ast$-Clean rings; some clean and almost clean Baer $\ast$-rings and von Neumann algebras, J. Algebra 324 (2010), 3388--3400.