Handan KÖSE, Huanyin CHEN, Orhan GURGÜN

8593

Uniquely strongly clean triangular matrices

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.

Anahtar Kelimeler:

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

Uniquely strongly clean triangular matrices

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.

Keywords:

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

Turkish Journal of Mathematics-Cover

ISSN: 1300-0098
Yayın Aralığı: 6
Yayıncı: TÜBİTAK

Arşiv

Sayıdaki Diğer Makaleler

On pseudohyperbolic space motions

Tunahan TURHAN, Nural YÜKSEL, Nihat AYYILDIZ

Split extension classifiers in the category of precrossed modules of commutative algebras

Yaşar BOYACI, Tufan Sait KUZPINARI, Enver Önder USLU

Classification of metallic shaped hypersurfaces in real space forms

Cihan ÖZGÜR, Nihal ÖZGÜR YILMAZ

Super d-anti-magic labeling of subdivided kC5

Muhammad HUSSAIN, Ali TABRAIZ

Super d-anti-magic labeling of subdivided $kC_{5}$

MUHAMMAD HUSSAIN, ALI TABRAIZ

Uniquely strongly clean triangular matrices

Handan KÖSE, Huanyin CHEN, Orhan GURGÜN

Quadratic recursive towers of function fields over $\mathbb{F}_2$

HENNING STICHTENOTH, SEHER TUTDERE

On the second homology of the Sch\"{u}tzenberger product of monoids

MELEK YAĞCI, LEYLA BUGAY, HAYRULLAH AYIK

On isophote curves and their characterizations

Yusuf YAYLI, Fatih DOĞAN

Fiber product preserving bundle functors on fibered-fibered manifolds

W lodzimierz M. MIKULSKI