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Turkish Journal of Mathematics

DOI

10.55730/1300-0098.3437

Abstract

A ring $R$ is called semiclean if every element of $R$ can be expressed as sum of a periodic element and a unit. In this paper, we introduce a new class of ring, which is the $\ast$-version of the semiclean ring, i.e. the $\ast$-semiclean ring. A $\ast$-ring is $\ast$-semiclean if each element is a sum of a $\ast$-periodic element and a unit. The term $\ast$-semiclean is a stronger notion than semiclean. In this paper, many properties of $\ast$-semiclean rings are discussed. It is proved that if $p \in P(R)$ such that $pRp$ and $(1-p)R(1-p)$ are $\ast$-semiclean rings, then $R$ is also a $\ast$-semiclean ring. As a result, the matrix ring $M_{n}(R)$ over a $\ast$-semiclean ring is $\ast$-semiclean. A characterization that when the group rings $RC_{r}$ and $RG$ are $\ast$-semiclean is done, where $R$ is a finite commutative local ring, $C_{r}$ is a cyclic group of order $r$, and $G$ is a locally finite abelian group. We have also found sufficient conditions when the group rings $RC_{3}$, $RC_{4}$, $RQ_{8}$, and $RQ_{2n}$ are $\ast$-semiclean, where $R$ is a commutative local ring. We have also demonstrated that the group ring $\mathbb{Z}_{2}D_{6}$ is a $\ast$-semiclean ring (which is not a $\ast$-clean ring).

Keywords

Group rings, semiclean rings, $\ast$-periodic element

First Page

1406

Last Page

1422

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