On the probability of zero divisor elements in group rings

Abstract

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 $P(RG)<\frac{1}{4}$ if and only if $RG\ncong \mathbb{Z}_2C_2,\mathbb{Z}_3C_2‎, ‎\mathbb{Z}_2C_3$‎. ‎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 $|G|\leq 4$‎.

Keywords

• ‎group ring‎
• ‎probability‎
• ‎unit group‎
• ‎zero divisor