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

Ashrafi, N.; Nasibi, E.

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

Document Type : Research Paper

Authors

N. Ashrafi

E. Nasibi

Semnan University

Abstract

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

Keywords

clean ring

exchange ring

r-clean ring

von Neumann regular ring

Subjects