On commuting probability of finite rings

TL;DR

This paper investigates the probability that two elements commute in finite rings, introduces a generalized version of this probability, and shows its invariance under a specific algebraic equivalence called ${ ext{Z}}$-isoclinism.

Contribution

It introduces a generalized commuting probability for finite rings and proves its invariance under ${ ext{Z}}$-isoclinism, extending understanding of ring commutativity properties.

Findings

Derived bounds for the commuting probability of finite rings.

Defined and analyzed the generalized commuting probability.

Proved invariance of the generalized probability under ${ ext{Z}}$-isoclinism.

Abstract

The commuting probability of a finite ring $R$, denoted by $\Pr(R)$, is the probability that any two randomly chosen elements of $R$ commute. In this paper, we obtain several bounds for $\Pr(R)$ through a generalization of $\Pr(R)$. Further, we define ${\Z}$-isoclinism between two pairs of rings and show that the generalized commuting probability, defined in this paper, is invariant under ${\Z}$-isoclinism between two pairs of finite rings.