The Probability that the Product of three Elements in a Finite Ring is Zero

Let [Formula: see text] be a finite commutative ring. In this paper, we consider the probability that the product of randomly chosen three elements of [Formula: see text] is zero, which we denote by [Formula: see text]. First we obtain bounds for [Formula: see text] for various classes of rings and then show that for any ring [Formula: see text] with identity [Formula: see text]. Finally, we characterize all commutative rings [Formula: see text] with identity with certain range of [Formula: see text].