Almost unit-clean rings

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 investigate conditions under which a ring is almost unit-clean. 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 2 R can be written in the form ux = r+e where u 2 U(R); r 2 R 􀀀 (Z(R) [ Z(M)) and e 2 Id(R). We thereby construct many examples of such rings.