University of Isfahan International Journal of Group Theory 2251-7650 11 4 2022 12 01 On the probability of zero divisor elements in group rings 253 257 26054 10.22108/ijgt.2021.126694.1664 EN Haval M. Mohammed Salih Department of Mathematics, Faculty of Science, Soran University , Kawa St, Soran, Erbil, Iraq Journal Article 2020 12 28 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 $|G|\leq 4$‎. https://ijgt.ui.ac.ir/article_26054_2690f3deefb66baa0c7be9888228a25a.pdf