Strongly Clean Matrix Rings over a Skew Monoid Ring

Let [Formula: see text] be a ring with an endomorphism [Formula: see text], [Formula: see text] the free monoid generated by [Formula: see text] with 0 added, and [Formula: see text] a factor of [Formula: see text] obtained by setting certain monomials in [Formula: see text] to 0 such that [Formula: see text] for some [Formula: see text]. Then we can form the non-semiprime skew monoid ring [Formula: see text]. A local ring [Formula: see text] is called bleached if for any [Formula: see text] and any [Formula: see text], the abelian group endomorphisms [Formula: see text] and [Formula: see text] of [Formula: see text] are surjective. Using [Formula: see text], 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 [Formula: see text] for which the matrix ring [Formula: see text] is strongly clean. We investigate the strong cleanness of the full matrix rings over the skew monoid ring [Formula: see text].