Commuting and product-zero probability in finite rings

Abstract

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