Quasi-clean rings and strongly quasi-clean rings

An element a of a ring R is called a quasi-idempotent if a(2 )= ka for some central unit k of R, or equivalently, a = ke, where k is a central unit and e is an idempotent of R. A ring R is called a quasi-Boolean ring if every element of R is quasi-idempotent. A ring R is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or R has no image isomorphic to Z(2); For an indecomposable commutative semilocal ring R with at least two maximal ideals, M-n(R)(n >= 2) is strongly quasi-clean if and only if M-n(R) is quasi-clean if and only if min{|R/m|, m is a maximal ideal of R} > n+ 1. For a prime p and a positive integer n >= 2, M-n(Z((p))) is strongly quasi-clean if and only if p > n. Some open questions are also posed.