Commuting and product-zero probability in finite rings

TL;DR

This paper investigates the probabilities of commuting elements and zero products in finite rings, establishing structural properties related to these probabilities that mirror classical group theory results.

Contribution

It introduces bounds on Lie-ideals and ideals in finite rings based on commuting and zero-product probabilities, extending known group theory theorems to ring theory.

Findings

Existence of Lie-ideals with bounded index related to commuting probability.

Existence of ideals with bounded index related to zero-product probability.

Analogies to P. Neumann's theorem in the context of finite rings.

Abstract

Let cp(R) be the probability that two random elements of a finite ring R commute and zp(R) the probability that the product of two random elements in R is zero. We show that if cp(R)=e, then there exists a Lie-ideal D in the Lie-ring (R,[.,.]) with e-bounded index and with [D,D] of e-bounded order. If zp(R)=e, then there exists an ideal D in R with e-bounded index and D^2 of e-bounded order. These results are analogous to the well-known theorem of P. Neumann on the commuting probability in finite groups.