Characterizing categoricity in the class $Add(M)$

[2502.19641] Characterizing categoricity in the class $Add(M)$ Skip to main content Search Submit Donate Log in Mathematics > Rings and Algebras arXiv:2502.19641 (math) [Submitted on 27 Feb 2025 ( v1 ), last revised 27 Aug 2026 (this version, v7)] Title: Characterizing categoricity in the class $Add(M)$ Authors: Xiaolei Zhang View PDF HTML (experimental) Abstract: We study categoricity of the additive closure $\operatorname{Add}(M)$, consisting of all direct summands of arbitrary direct sums of copies of a fixed module $M$. For an $\eta$-generated module $M$, we prove that categoricity in a single cardinal $\lambda \ge \theta = \max\{|R|,\eta\}$ is equivalent to the stabilization condition $P^{(\eta)} \cong M^{(\eta)}$ for every nonzero $\eta$-generated $P \in \operatorname{Add}(M)$, or equivalently $\operatorname{Add}(P)=\operatorname{Add}(M)$. This condition implies that every $X \in \operatorname{Add}(M)$ of cardinality $\lambda > \nu_M = \max\{\theta, |\mathcal{S}_\eta(M)|\}$ is isomorphic to $M^{(\lambda)}$, where $\mathcal{S}_\eta(M)$ is a representative set of the nonzero $\eta$-generated modules in $\operatorname{Add}(M)$. Since $|\mathcal{S}_\eta(M)| \le 2^\theta$, we obtain a uniform ZFC tail above $2^\theta$. For projective modules, the criterion reduces to Bass's right $p$-connectedness. For pure-projective modules, it forces von Neumann regularity and simplicity: the pure-projective modules are eventually categorical if and only if $R$ is a simple von Neumann regular ring. In the commutative case, this recovers precisely the fields. Our approach combines Walker's decomposition theorem, support reduction, and Eilenberg absorption, and we highlight several subtle pitfalls in naive downward transfer arguments. Subjects: Rings and Algebras (math.RA) Cite as: arXiv:2502.19641 [math.RA] (or arXiv:2502.19641v7 [math.RA] for this version) https://doi.org/10.48550/arXiv.2502.19641 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Xiaolei Zhang [ view email ] [v1] Thu, 27 Feb 2025 00:22:19 UTC (7 KB) [v2] Sun, 2 Mar 2025 06:05:16 UTC (7 KB) [v3] Fri, 7 Mar 2025 12:19:18 UTC (7 KB) [v4] Tue, 22 Jul 2025 01:27:58 UTC (7 KB) [v5] Sun, 27 Jul 2025 02:20:28 UTC (7 KB) [v6] Thu, 26 Mar 2026 00:38:43 UTC (7 KB) [v7] Thu, 27 Aug 2026 22:22:37 UTC (12 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.RA < prev | next > new | recent | 2025-02 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation 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)

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