Ideal Zero-Product Probability in Finite Commutative Rings

Abstract

We introduce the \emph{ideal zero-product probability}, a new probabilistic invariant associated with a finite commutative ring $R$, defined by \[ ζ_k(R) = \frac{ |\{(I_1,\ldots,I_k)\in\mathcal I(R)^k : I_1\cdots I_k=(0)\}| } {|\mathcal I(R)|^k}, \] where $\mathcal I(R)$ denotes the set of all ideals of $R$. This quantity gives the probability that the product of $k$ independently and uniformly chosen ideals is the zero ideal. We establish its basic properties, including invariance under ring isomorphisms, monotonicity in $k$, and multiplicativity with respect to finite direct products. Explicit formulas are obtained for finite fields, finite Boolean rings, finite chain rings, and finite principal ideal rings. For finite chain rings, we prove that the invariant depends only on the Loewy length. We also derive closed formulas using inclusion--exclusion and bounded compositions, together with generating functions. Finally, we study the asymptotic behaviour of $ζ_k(R)$ and prove that it converges to $1$ as $k\to\infty$. We also compare the ideal zero-product probability with the classical zero-product probability of ring elements on finite chain rings, obtaining explicit formulas and showing that the two invariants capture fundamentally different structural information.