The multiplication probability of a finite ring

TL;DR

This work studies the probability that the product of two random elements in a finite ring equals a fixed element, a generalized version of the nullity and commutativity probabilities. The authors develop a general framework using left annihilators and a solvability criterion to compute Prob x (R) and show how these probabilities behave under direct products and quotienting by ideals. They obtain an explicit formula for simple rings, particularly matrix rings over finite fields, that depends on the rank of the fixed element, and they derive detailed results for finite local rings, including exact values and bounds based on the position of x in the Jacobson radical and its powers. Collectively, these results extend the understanding of product distributions in finite rings and provide actionable formulas for semisimple and local ring families such as Z n and rings with nilpotent Jacobson radical.

Abstract

We study the probability that the product of two randomly chosen elements in a finite ring $R$ is equal to some fixed element $x \in R$. We calculate this probability for semisimple rings and some special classes of local rings, and find the bounds for this probability for an arbitrary finite ring.

Theorems & Definitions (16)

• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1
• Theorem 3.2
• Lemma 4.1
• Theorem 4.2