Strongly clean triangular matrix rings with endomorphisms

Abstract

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; ?) be the skew triangular matrix ring over a local ring R where ? is an endomorphism of R. We show that T2(R; ?) is strongly clean if and only if for any a? 1+J(R); b ? J(R), la -r?(b): R› R is surjective. Further, T3(R; ?) is strongly clean if la-r?(b); la-r?2(b) and lb-r?(a)are surjective for any a ? U(R); b ? J(R). The necessary condition for T3(R; ?) to be strongly clean is also obtained. © 2015 Iranian Mathematical Society.