A generalization of semiclean rings

Abstract

Ye defined a ring to be semiclean if every element of it can be written as a sum of a unit element and a periodic element. In this paper we generalize the notion of a semiclean ring to an almost semiclean ring. A ring R is said to be almost semiclean if each element is a sum of a regular element and a periodic element. We discuss some basic properties of almost semiclean rings. For example, R is almost semiclean if and only if the polynomial ring over R is almost semiclean. We also discuss when the idealization is almost semiclean. Finally, we give examples which distinguish almost semiclean rings from other classes of rings. Communicated by Silvana Bazzoni. © 2020, © 2020 Taylor & Francis Group, LLC.