The probability that the product of k elements in a finite ring is zero

TL;DR

This paper investigates the probability that the product of k randomly chosen elements in a finite commutative ring is zero, providing bounds, maximum values, and classifications for various rings.

Contribution

It establishes sharp bounds for the zero-product probability and classifies rings based on their probability values, advancing understanding of ring structure probabilities.

Findings

Derived sharp bounds for zero-product probability in finite rings

Identified the maximum zero-product probability across all rings

Classified rings within specific probability ranges

Abstract

In this paper, for a fixed integer $k\ge 2$, we study the probability that the product of $k$ randomly chosen elements in a finite commutative ring $R$ is zero, which we denote by $zp_{_k}(R)$. We investigate bounds for $zp_{_k}(R)$ that turn out to be sharp bounds for certain classes of rings. Further, we determine the maximum value of $zp_{_k}(R)$ that can be obtained for any ring $R$, and classify all rings within some specific range of $zp_{_k}(R)$.