Uniquely strongly clean triangular matrices

Authors

HUANYIN CHEN
ORHAN GÜRGÜN
HANDAN KOSE

Abstract

A ring $R$ is uniquely (strongly) clean provided that for any $a\in R$ there exists a unique idempotent $e\in R$ \big($e\in comm(a)$\big) such that $a-e\in U(R)$. We prove, in this note, that a ring $R$ is uniquely clean and uniquely bleached if and only if $R$ is abelian, ${\mathbb{T}}_{n}(R)$ 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 ${\mathbb{T}}_{n}(R)$ is uniquely strongly clean for some $n\geq 1$. In the commutative case, more explicit results are obtained.

DOI

10.3906/mat-1408-13

Keywords

Uniquely strongly clean ring, uniquely bleached ring, triangular matrix ring

First Page

645

Last Page

649

Recommended Citation

CHEN, HUANYIN; GÜRGÜN, ORHAN; and KOSE, HANDAN (2015) "Uniquely strongly clean triangular matrices," Turkish Journal of Mathematics: Vol. 39: No. 5, Article 4. https://doi.org/10.3906/mat-1408-13
Available at: https://journals.tubitak.gov.tr/math/vol39/iss5/4