ALMOST UNIT-CLEAN RINGS

Abstract

A ring R is almost unit-clean provided that every element in R is equivalent to the sum of an idempotent and a regular element. We prove that every ring in which every zero-divisor is strongly pi-regular is almost unit-clean and every matrix ring of elementary divisor domains is almost unit-clean. Furthermore, it is shown that the trivial extension R(M) of a commutative ring R and an R-module M is almost unit-clean if and only if each x is an element of R can be written in the form ux = r + e where u is an element of U(R),r is an element of R - (Z(R) boolean OR Z(M)) and e is an element of Id(R). We thereby construct many examples of such rings.