Strongly clean triangular matrix rings with endomorphisms

TL;DR

This paper characterizes when skew triangular matrix rings over a local ring are strongly clean, linking this property to specific surjectivity conditions involving ring endomorphisms.

Contribution

It provides necessary and sufficient conditions for the strong cleanness of skew triangular matrix rings over local rings, extending the understanding of ring decompositions.

Findings

$T_2(R,\sigma)$ is strongly clean iff certain maps are surjective.

$T_3(R,\sigma)$ is strongly clean under specific surjectivity conditions.

Necessary conditions for $T_3(R,\sigma)$ to be strongly clean are established.

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.