Quasi-semi clean rings

Abstract

Recently, Tang et al. [1] studied a broader concept than the concept of clean rings called quasi-clean rings. In this paper, we use the method of Tang et al. [1] in generalizing clean to construct and study a generalization of semiclean rings of Ye [3] to a broader class of rings called quasi-semiclean rings. Let  be a ring with identity. Then,  is said to be a quasi-semicelan ring if every element  can be expressed as  where  is a unit of  and  is a quasi-periodic element of : that is  satisfies the equation  where  and  is a central unit, . We prove several properties of the class of quasi-semiclean rings similar to those of semiclean rings and also settle new properties and results. In particular, we show that clean rings and certain types of quasi-semiclean rings (quasi-clean rings to be precise) are the same concept and prove this equivalence. Furthermore, we investigate the quasi-semicleaness of various types of rings such as Morita context rings, trivial extensions of modules , and the Nagata ring .