Idempotent elements and uniquely clean property of skew monoid rings

Let R be a ring with an endomorphism σ and F ∪ {0} the free monoid generated by U = {u1, ..., ut} with 0 added, and M = F ∪ {0}/(I) where I is the set of certain monomial in U such that Mn = 0, for some n. Then we can form the non-semiprime skew monoid ring R[M; σ]. An element a ∈ R is uniquely strongly clean if a has a unique expression as a = e + u, where e is an idempotent and u is a unit with ea = ae. We show that a σ-compatible ring R is uniquely clean if and only if R[M; σ] is a uniquely clean ring. If R is strongly π-regular and uniquely strongly clean, then R[M; σ] is uniquely strongly clean. It is also shown that idempotents of R[M; σ] (and hence the ring R[x; σ]=(xn)) are conjugate to idempotents of R and we apply this to show that R[M; σ] over a projective-free ring R is projective-free. It is also proved that if R is semi-abelian and σ(e) = e for each idempotent e ∈ R, then R[M; σ] is a semi-abelian ring.