Ideal Zero-Product Probability in Finite Commutative Rings

[2608.14618] Ideal Zero-Product Probability in Finite Commutative Rings Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Commutative Algebra arXiv:2608.14618 (math) [Submitted on 11 Jul 2026] Title: Ideal Zero-Product Probability in Finite Commutative Rings Authors: Sukrit Chakraborty , Sourav Kanti Patra View a PDF of the paper titled Ideal Zero-Product Probability in Finite Commutative Rings, by Sukrit Chakraborty and Sourav Kanti Patra View PDF HTML (experimental) Abstract: We introduce the \emph{ideal zero-product probability}, a new probabilistic invariant associated with a finite commutative ring $R$, defined by \[ \zeta_k(R) = \frac{ |\{(I_1,\ldots,I_k)\in\mathcal I(R)^k : I_1\cdots I_k=(0)\}| } {|\mathcal I(R)|^k}, \] 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 $\zeta_k(R)$ 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. Comments: 33 pages, comments are welcome Subjects: Commutative Algebra (math.AC) ; Combinatorics (math.CO) Cite as: arXiv:2608.14618 [math.AC] (or arXiv:2608.14618v1 [math.AC] for this version) https://doi.org/10.48550/arXiv.2608.14618 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Sukrit Chakraborty [ view email ] [v1] Sat, 11 Jul 2026 13:00:33 UTC (19 KB) Full-text links: Access Paper: View a PDF of the paper titled Ideal Zero-Product Probability in Finite Commutative Rings, by Sukrit Chakraborty and Sourav Kanti Patra View PDF HTML (experimental) TeX Source view license Current browse context: math.AC < prev | next > new | recent | 2026-08 Change to browse by: math math.CO References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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