Rings in which elements are the sum of an‎ ‎idempotent and a regular element

Abstract

Let R be an associative ring with unity. An element <br />a in R is said to be r-clean if a = e+r, where e is an idempotent and <br />r is a regular (von Neumann) element in R. If every element of R is <br />r-clean, then R is called an r-clean ring. In this paper, we prove that <br />the concepts of clean ring and r-clean ring are equivalent for abelian <br />rings. Further we prove that if 0 and 1 are the only idempotents <br />in R, then an r-clean ring is an exchange ring. Also we show that <br />the center of an r-clean ring is not necessary r-clean, but if 0 and <br />1 are the only idempotents in R, then the center of an r-clean ring <br />is r-clean. Finally we give some properties and examples of r-clean <br />rings