on the probability of zero divisor elements in group rings

Let r be a non trivial finite commutative ring with identity and g be a non trivial group.we denote by p(rg) the probability that the product of two randomly chosen elements of a finitegroup ring rg is zero. we show that p(rg) <1/4 if and only if rg  z2c2, z3c2, z2c3. furthermore,we give the upper bound and lower bound for p(rg). in particular, we present the general formulafor p(rg), where r is a finite field of characteristic p and |g| ≤ 4.