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arXiv · 1307.7339

Uniquely Strongly Clean Triangular Matrix Rings

H. Chen·O. Gurgun·H. Kose
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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.

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H. Chen, O. Gurgun, H. Kose. 2013-08-29. Uniquely Strongly Clean Triangular Matrix Rings. https://arxiv.org/abs/1307.7339

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