Strongly Clean Power Series Rings

An element a in a ring R with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And a∈ R is called strongly π-regular if both chains aR⊇ a²R⊇⋯ and Ra⊇ Ra²⊇⋯ terminate. A ring R is called strongly clean (respectively, strongly π-regular) if every element of R is strongly clean (respectively, strongly π-regular). Strongly π-regular elements of a ring are all strongly clean. Let σ be an endomorphism of R. It is proved that for rᵢxⁱ∈ R[[x,σ]], if r₀ or 1-r₀ is strongly π-regular in R, then rᵢxⁱ is strongly clean in R[[x,σ]]. In particular, if R is strongly π-regular, then R[[x,σ]] is strongly clean. It is also proved that if R is a strongly π-regular ring, then R[x,σ]/(xⁿ) is strongly clean for all n≥1 and that the group ring of a locally finite group over a strongly regular or commutative strongly π-regular ring is strongly clean.