Uniquely Strongly Clean Triangular Matrix Rings

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.