On the probability of zero divisor elements in group rings

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Science, Soran University , Kawa St, Soran, Erbil, Iraq

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 $|G|\leq 4$‎.

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Volume 11, Issue 4 - Serial Number 4
December 2022
Pages 253-257
  • Receive Date: 28 December 2020
  • Revise Date: 15 April 2021
  • Accept Date: 25 October 2021
  • Published Online: 01 December 2022