Strongly Clean Matrix Rings over a Skew Monoid Ring

Let R be a ring with an endomorphism sigma, F boolean OR{0} the free monoid generated by U = {u(1),(center dot center dot center dot),u(t)} with 0 added, and M a factor of F obtained by setting certain monomials in F to 0 such that M-n = 0 for some n. Then we can form the non-semiprime skew monoid ring R[M; sigma]. A local ring R is called bleached if for any j is an element of J(R) and any u is an element of U (R), the abelian group endomorphisms l(u)- r(j) and l(j) - r(j) of R are surjective. Using R[M; sigma], we provide various classes of both bleached and non-bleached local rings. One of the main problems concerning strongly clean rings is to characterize the rings R for which the matrix ring M-n(R) is strongly clean. We investigate the strong cleanness of the full matrix rings over the skew monoid ring R[M; sigma].