Let R be a ring with an endomorphism σ , F ∪ { 0 } the free monoid generated by U = { u 1 , ... , 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 ; σ ]. A local ring R is called bleached if for any j ∈ J (R) and any u ∈ U (R) , the abelian group endomorphisms l u − r j and l j − r u of R are surjective. Using R [ M ; σ ] , 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 ; σ ]. [ABSTRACT FROM AUTHOR]