$\ast$-Semiclean rings

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 ∗ -version of the semiclean ring, i.e. the ∗ -semiclean ring. A ∗ -ring is ∗ -semiclean if each element is a sum of a ∗ -periodic element and a unit. The term ∗ -semiclean is a stronger notion than semiclean. In this paper, many properties of ∗ -semiclean rings are discussed. It is proved that if p ∈ P ( R ) such that pRp and (1 − p ) R (1 − p ) are ∗ -semiclean rings, then R is also a ∗ -semiclean ring. As a result, the matrix ring M n ( R ) over a ∗ -semiclean ring is ∗ -semiclean. A characterization that when the group rings RC r and RG are ∗ -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 2 n are ∗ -semiclean, where R is a commutative local ring. We have also demonstrated that the group ring Z 2 D 6 is a ∗ -semiclean ring (which is not a ∗ -clean ring).