Compressed zero-divisor graphs of matrix rings over finite fields

Abstract

ABSTRACT We extend the notion of the compressed zero-divisor graph to noncommutative rings in a way that still induces a product preserving functor from the category of finite unital rings to the category of directed graphs. For a finite field F, we investigate the properties of , the graph of the matrix ring over F, and give a purely graph-theoretic characterization of this graph when . For we prove that every graph automorphism of is induced by a ring automorphism of . 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 , then . In particular, this holds if with .