Zero-divisor graphs of non-commutative rings

By Saieed Akbari, Ali Mohammadian. 2006

130 citations recorded by the source corpus.

Abstract In a manner analogous to the commutative case, the zero-divisor graph of a non-commutative ring R can be 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 . We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ ( R ) . In this paper it is shown that, with finitely many exceptions, if R is a ring and S is a finite semisimple ring which is not a field and Γ ( R ) ≃ Γ ( S ) , then R ≃ S . For any finite field F and each integer n ⩾ 2 , we prove that if R is a ring and Γ ( R ) ≃ Γ ( M n ( F ) ) , then R ≃ M n ( F ) . Redmond defined the simple undirected graph Γ ¯ ( R ) obtaining by deleting all directions on the edges in Γ ( R ) . We classify all ring R whose Γ ¯ ( R ) is a complete graph, a bipartite graph or a tree.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a343245f2688ee2bbda25c3bca9384f6a3774230

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Ali Mohammadian
2. Saieed Akbari
3. M. Behboodi
4. Z. Rakeei

In Geometry and Topology

1. Saieed Akbari
2. Ali Mohammadian
3. M. Behboodi
4. Z. Rakeei

Citations in the corpus

• The Annihilating-Ideal Graph of Commutative Rings I (2008, background): “In the literature, there are many papers on assigning a graph to a ring, a group, semigroup or a module (see, for example, [1–11, 14–18]).”
• On zero-divisor graphs of finite rings (2007, background): “For other properties of zerodivisor graphs, see [2,3].”