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 ‎$T_n(R,sigma)$ be the skew triangular matrix ring over a local‎ ‎ring $R$ where $sigma$ is an endomorphism of $R$‎. ‎We show that‎ ‎$T_2(R,sigma)$ is strongly clean if and only if for any $ain‎ ‎1+J(R)‎, ‎bin J(R)$‎, ‎$l_a-r_{sigma(b)}‎: ‎Rto R$ is surjective‎. ‎Further‎, ‎$T_3(R,sigma)$ is strongly clean if‎ ‎$l_{a}-r_{sigma(b)}‎, ‎l_{a}-r_{sigma^2(b)}$ and‎ ‎$l_{b}-r_{sigma(a)}$ are surjective for any $ain U(R),bin‎ ‎J(R)$‎. ‎The necessary condition for $T_3(R,sigma)$ to be strongly‎ ‎clean is also obtained‎. ‎