Unit exchange elements in rings

Abstract

Replacing left principal ideals by cosets in the monoid (R, ·) of a unital ring R, we say that an element a 2 R is left unit exchange (or suitable) if there is an idempotent e 2 R such that e − a 2 U(R)(a − a2) where U(R) denotes the set of units. Unit-regular and clean elements are left (and right) unit suitable, and left (or right) unit suitable elements are exchange (suitable). The paper studies the multiple facets of this new notion.