Uniquely Strongly Clean Triangular Matrix Rings

TL;DR

This paper characterizes when triangular matrix rings over a ring are uniquely strongly clean, showing that this property is equivalent to the ring being abelian, with specific results for commutative rings and generalizations of prior theorems.

Contribution

It establishes necessary and sufficient conditions for triangular matrix rings to be uniquely strongly clean, extending previous results and identifying new classes of such rings.

Findings

R is uniquely clean iff it is abelian.

T_n(R) is uniquely strongly clean for all n iff R is abelian.

Explicit results obtained in the commutative case.

Abstract

A ring $R$ is uniquely (strongly) clean provided that for any $a\in R$ there exists a unique idempotent $e\in R \big(\in comm(a)\big)$ such that $a-e\in U(R)$. Let $R$ be a uniquely bleached ring. We prove, in this note, that $R$ is uniquely clean if and only if $R$ is abelian, and $T_n(R)$ is uniquely strongly clean for all $n\geq 1$, if and only if $R$ is abelian, $T_n(R)$ is uniquely strongly clean for some $n\geq 1$. In the commutative case, the more explicit results are obtained. These also generalize the main theorems in [6] and [7], and provide many new class of such rings.