On zero-divisor graphs of finite rings

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 ) .