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arXiv · 2610.07892

Nilpotent Product Probability of Finite Rings

Dibyasman Sarma·Tikaram Subedi
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Abstract

For a finite ring $R$, we investigate the probability that the product of two randomly chosen elements in $R$ is nilpotent, which we call the nilpotent product probability (NPP) of $R$ and denote by $P_{nil}(R)$. We derive bounds for $P_{nil}(R)$ that depend on the structure of $R$, and prove that among all commutative rings with identity, the class of local rings with residue field $\mathbb{Z}_{2}$ attains the maximum value of NPP, which is equal to $3/4$. Further, we determine the set of all values of NPP for commutative rings with identity and classify all rings attaining those values. Finally, we investigate the relationship between NPP of $R$ and NPP of some subrings of the matrix ring over $R$.

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Dibyasman Sarma, Tikaram Subedi. 2026-10-06. Nilpotent Product Probability of Finite Rings. https://arxiv.org/abs/2610.07892

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