Abstract. Let R be a ring with identity, X be the set of all nonzero, nonunits of R and G be the group of all units of R. A ring R is called[x]r = {xu | u ∈ G}) which are equivalence classes on X. It is shown that for a semisimple unit-duo ring R (for example, a strongly regular ring), there exist a finite number of equivalence classes on X if and only if R is artinian. By considering the zero divisor graph (denoted Γ(R)) determined by equivalence classes of zero divisors of a unit-duo ring R, it is shown that for a unit-duo ring R such that Γ(R) is a finite graph, R is local if and only if diam( Γ(R)) = 2.