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
.
Disclosure statement
No potential conflict of interest was reported by the authors.