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 finite group ring RG is zero‎. ‎We show that P(RG)<\frac{۱}{۴} if and only if RG\ncong \mathbb{Z}_۲C_۲,\mathbb{Z}_۳C_۲‎, ‎\mathbb{Z}_۲C_۳‎. ‎Furthermore‎, ‎we give the upper bound and lower bound for P(RG)‎. ‎In particular‎, ‎we present the general formula for P(RG)‎, ‎where R is a finite field of characteristic p and |G|\leq ۴‎.

Keywords:

‎group ring‎ , ‎probability‎ , ‎unit group‎ , ‎zero divisor