Uniquely strongly clean triangular matrices

Authors : HUANYIN CHEN, ORHAN GÜRGÜN, HANDAN KOSE

Pages : 645-649

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :A ring $R$ is uniquely insert ignore into journalissuearticles values(strongly); clean provided that for any $a\in R$ there exists a unique idempotent $e\in R$ \biginsert ignore into journalissuearticles values($e\in comminsert ignore into journalissuearticles values(a);$\big); such that $a-e\in Uinsert ignore into journalissuearticles values(R);$. We prove, in this note, that a ring $R$ is uniquely clean and uniquely bleached if and only if $R$ is abelian, ${\mathbb{T}}_{n}insert ignore into journalissuearticles values(R);$ is uniquely strongly clean for all $n\geq 1$, i.e. every $n\times n$ triangular matrix over $R$ is uniquely strongly clean, if and only if $R$ is abelian, and ${\mathbb{T}}_{n}insert ignore into journalissuearticles values(R);$ is uniquely strongly clean for some $n\geq 1$. In the commutative case, more explicit results are obtained.
Keywords : Uniquely strongly clean ring, uniquely bleached ring, triangular matrix ring