Commuting and product-zero probability in finite rings

2023/07/24 by Shumyatsky, Pavel, Vannacci, Matteo · 1 citation
#FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2307.12633

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 D² of e-bounded order. These results are analogous to the well-known theorem of P. Neumann on the commuting probability in finite groups.

Cited by

• Commuting degree for BCK-algebras
• The multiplication probability of a finite ring

Related

• The probability that the product of k elements in a finite ring is zero
• Some bounds for commuting probability of finite rings
• On commuting probability of finite rings
• On generalized commuting probability of finite rings
• On relative commuting probability of finite rings