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 ?-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 ? R can be written in the form ux = r + e where u ? U(R), r ? R - (Z(R) ? Z(M)) and e ? Id(R). We thereby construct many examples of such rings. © 2019 Editura Academiei Romane. All rights reserved.