STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING

Abstract

Let R be an associative ring with an endomorphism σ and F ∪ {0} the free monoid generated by U = {u1, . . . , ut} with 0 added, and M a factor of F setting certain monomial in U to 0, enough so that, for some n, Mn = 0. Then we can form the skew monoid ring R[M; σ]. An element of a ring R is strongly clean if it is the sum of an idempotent and a unit that commute. In this paper, we prove that P g∈M rgg ∈ R[M; σ] is a strongly clean element, if re or 1 − re is strongly π-regular in R. As a corollary, we deduce that if R is a strongly π-regular ring, then the skew monoid ring R[M; σ] is strongly clean. These rings is a new family of non-semiprime strongly clean skew monoid rings.