Conference abstracts
Session S05 - Rings and Algebras
No date set
A study on clean rings
Laiz Valim da Rocha
Universidade Federal Fluminense, Brasil - laizvalim@gmail.com
A ring is said to be clean if every element can be written as sum of a unit and an idempotent. These rings were defined by Nicholson [5], while studying exchange rings. The class of clean rings is located among other well known classes of rings [3]. 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.
The study of $\ast$-clean rings was motivated by a question made by T. Y. Lam at the Conference on Algebra and Its Applications, in March 2005, at the Ohio University: which von Neumann algebras are clean as rings? Since von Neumann algebras are $\ast$-rings (i.e., rings with an involution), it is more natural to work with projections (idempotents that are symmetric under the ring involution) than with idempotents.
So, in 2010 Vaš defined $\ast$-clean rings [6]: a $\ast$-ring in which every element may be written as a sum of a unit and a projection. Clearly, every $\ast$-clean ring is a $\ast$-ring and is a clean ring.
Every group $G$ is endowed with the classical involution $g \mapsto g^{-1}$. If $R$ is a commutative ring, for instance, the $R$-linear extension of the classical involution in $G$ is the classical involution in $RG$. $\ast$-clean group rings were first studied in 2011 [4]. However very little is still known about when a group ring 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, we present clean rings, their relationship with other types of rings [3] and some recent results [1]. Let $R$ be a commutative local ring. I will provide necessary and sufficient conditions for the group rings $R C_3$ and $R C_4$ to be $\ast$-clean, where $C_n$ denote the cyclic group with $n$ elements.
REFERENCES
[1] Y. Gao, J. Chen, Y. Li. Some $\ast$-clean Group Rings, Algebra Colloquium 22 (2015) 169--180.
[2] J. Han, W. K. Nicholson. Extensions of clean rings, Communications in Algebra 29 (2001), 2589--2595.
[3] N. A. Immormino. Some Notes On Clean Rings, Bowling Green State University, 2012.
[4] C. Li, Y. Zhou. On strongly $\ast$-clean rings, Journal of Algebra and Its Applications 6 (2011), 1363--1370.
[5] W. K. Nicholson. Lifting idempotents and exchange rings}, Transactions of the AMS 229 (1977), 269 -- 278.
[6] L. Vaš. $\ast$-Clean rings; some clean and almost clean Baer $\ast$-rings and von Neumann algebras, J. Algebra 324 (2010), 3388--3400.