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 $a\in 1+J(R), b\in J(R)$, $l_a-r_{\sigma(b)}: R\to R$ 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 $a\in U(R),b\in J(R)$. The necessary condition for $T_3(R,\sigma)$ to be strongly clean is also obtained.