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

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 $z{p}_{{}_k}(R)$. We investigate bounds for $z{p}_{{}_k}(R)$ that turn out to be sharp bounds for certain classes of rings. Further, we determine the maximum value of $z{p}_{{}_k}(R)$ that can be obtained for any ring $R$, and classify all rings within some specific range of $z{p}_{{}_k}(R)$.