Full Descripion of ring varieties whose finite rings are uniquely determined by their zero-divisor graphs
Abstract
The zero-divisor graph
Γ(R)
of an associative ring
R
is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of
R
, and two distinct vertices
x
and
y
are joined by an edge iff either
xy=0
or
yx=0
.
In the present paper, we give a full description of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.