Strongly clean triangular matrix rings with endomorphisms

A ring R is strongly clean provided that every element in R is the sum of an idempotent and a unit that commutate. Let Tn(R; σ) be the skew triangular matrix ring over a local ring R where σ is an endomorphism of R. We show that T2(R; σ) is strongly clean if and only if for any aϵ 1+J(R); b ϵ J(R), la -rσ (b): R→ R is surjective. Further, T3(R; σ) is strongly clean if la-rσ (b); la-rσ2 (b) and lb-rσ (a)are surjective for any a ϵ U(R); b ϵ J(R). The necessary condition for T3(R; σ) to be strongly clean is also obtained. © 2015 Iranian Mathematical Society.