When Is the Complement of the Zero-Divisor Graph of a Commutative Ring Complemented?
نویسندگان
چکیده
منابع مشابه
When does the complement of the zero-divisor graph of a commutative ring
In this article, we determine up to isomorphism of rings, rings R such that R has the following properties: (i) R is a commutative ring with identity which admits at least two nonzero zero-divisors, (ii) the complement of the zero-divisor graph of R is connected and it admits a cut vertex. Indeed, it is proved that there are exactly two such rings up to isomorphism of rings.
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متن کاملWhen does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex?
The rings considered in this article are commutative with identity which admit at least two nonzero annihilating ideals. Let $R$ be a ring. Let $mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $mathbb{A}(R)^{*} = mathbb{A}(R)backslash {(0)}$. The annihilating-ideal graph of $R$, denoted by $mathbb{AG}(R)$ is an undirected simple graph whose vertex set is $mathbb{A}(R...
متن کاملThe Zero-Divisor Graph of a Commutative Ring
Ž . Ž . Let R be a commutative ring with 1 and let Z R be its set of Ž . Ž . zero-divisors. We associate a simple graph G R to R with vertices Ž . Ž . 4 Z R * s Z R y 0 , the set of nonzero zero-divisors of R, and for disŽ . tinct x, y g Z R *, the vertices x and y are adjacent if and only if xy s 0. Ž . Thus G R is the empty graph if and only if R is an integral domain. The main object of this...
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ژورنال
عنوان ژورنال: ISRN Algebra
سال: 2012
ISSN: 2090-6293
DOI: 10.5402/2012/282054