UNIT CLEANNESS PROPERTY OF SOME RINGS

By Nicholson, a ring is called clean if every element of R can be written as the sum of a unit and an idempotent. As defined in [3], a ring R is unit-clean (uniquely unit-clean) if for every element x ∈ R, there exists v ∈ U(R) such that xv is clean (uniquely clean). In this article, we show that every unit-clean ring is pm-ring, and we investigate the conditions under which quasilocal rings, pm-rings and unit-clean rings are equivalent. Also, we prove that every 0-dimensional commutative ring is a unit-clean ring. Finally, by an example, we show that a localization of a unit-clean ring need not be unit-clean.