On zero-divisor graphs of finite rings

By Saieed Akbari, Ali Mohammadian. 2007

93 citations recorded by the source corpus.

Abstract The zero-divisor graph of a ring R is defined as the directed graph Γ ( R ) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x → y is an edge if and only if x y = 0 . Recently, it has been shown that for any finite ring R, Γ ( R ) has an even number of edges. Here we give a simple proof for this result. In this paper we investigate some properties of zero-divisor graphs of matrix rings and group rings. Among other results, we prove that for any two finite commutative rings R , S with identity and n , m ⩾ 2 , if Γ ( M n ( R ) ) ≃ Γ ( M m ( S ) ) , then n = m , | R | = | S | , and Γ ( R ) ≃ Γ ( S ) .

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cdf7d698545ed90f0cfbabb71f3b8ebd87d5f7b6

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In Algebra and Number Theory

1. M. Behboodi
2. Saieed Akbari

In Geometry and Topology

1. M. Behboodi
2. Saieed Akbari
3. Mohammad Javad Nikmehr

Citations in the corpus

• The Classification of the Annihilating-Ideal Graphs of Commutative Rings (2014, background): “There are many papers on assigning a graph to a ring (for instance, see [1, 2, 4, 12, 13, 16]).”