Commuting and product-zero probability in finite rings
Rings and Algebras
2023-07-25 v1 Group Theory
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.
Cite
@article{arxiv.2307.12633,
title = {Commuting and product-zero probability in finite rings},
author = {Pavel Shumyatsky and Matteo Vannacci},
journal= {arXiv preprint arXiv:2307.12633},
year = {2023}
}