Commuting probability for subrings and quotient rings

Abstract

We prove that the commuting probability of a finite ring is no larger than the commuting probabilities of its subrings and quotients, and characterize when equality occurs in such a comparison.2010 MSC: 05E15

The Probability of Zero Multiplication in Finite Rings

“…and thus the desired equality follows. [3] Since for any ring R, the factor ring R/J(R) is a semisimple ring, we can now calculate its nullity degree. We thus have the following upper bound for zp(R).…”

“…Group theoretical problems related to discrete probabilities have been extensively studied (see, for example, [2, 8, 9, 12, 16]). Recently, there has been interest in using a similar approach with rings, mainly concerning the commuting probability (also called the commutativity degree) [3–5, 13] and the probability of zero multiplication (also called the nullity degree) [6, 7] of finite rings. However, all recent results regarding the nullity degree have been obtained in the setting of finite commutative rings.…”

On commuting probabilities in finite groups and rings

“…Publications [28], [7], [15], [6] and [18] deal with commuting probability in groups. In papers [3] and [1]…”

On commuting probabilities in finite groups and rings

“…Buckley et. al., [3] and [1], classified all finite rings with commuting probability ≥ 11 32 and anticommuting probability ≥ 15 32 , respectively. Throughout this paper, |A| denotes cardinality of the set A.…”

“…In this paper, we positively resolve the first conjecture and partially resolve the second. 3 2 Main results…”