Compressed zero-divisor graphs of noncommutative rings

Abstract

We extend the notion of the compressed zero-divisor graph Θ(R) to noncommutative rings in a way that still induces a product preserving functor Θ from the category of finite unital rings to the category of directed graphs. For a finite field F , we investigate the properties of Θ(Mn(F )), the graph of the matrix ring over F , and give a purely graph-theoretic characterization of this graph when n = 3. For n = 2 we prove that every graph automorphism of Θ(Mn(F )) is induced by a ring automorphism of Mn(F ). We also show that for finite unital rings R and S, where S is semisimple and has no homomorphic image isomorphic to a field, if Θ(R) ∼ = Θ(S), then R ∼ = S. In particular, this holds if S = Mn(F ) with n = 1.