On commuting probability of finite rings

Abstract

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