On the arithmetic of the join rings over finite fields

[2308.13428] On the arithmetic of the join rings over finite fields Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Rings and Algebras arXiv:2308.13428 (math) [Submitted on 25 Aug 2023 ( v1 ), last revised 8 Apr 2026 (this version, v3)] Title: On the arithmetic of the join rings over finite fields Authors: Sunil K. Chebolu , Jonathan Merzel , Ján Mináč , Tung T. Nguyen , Federico Pasini , Nguyên Duy Tân View a PDF of the paper titled On the arithmetic of the join rings over finite fields, by Sunil K. Chebolu and 5 other authors View PDF HTML (experimental) Abstract: Given a collection $\{ G_i\}_{i=1}^d$ of finite groups and a ring $R$, we have previously introduced and studied certain foundational properties of the join ring $\mathcal{J}_{G_1, G_2, \ldots, G_d}(R)$. This ring bridges two extreme worlds: matrix rings $M_n(R)$ on one end, and group rings $R[G]$ on the other. The construction of this ring was motivated by various problems in graph theory, network theory, nonlinear dynamics, and neuroscience. In this paper, we continue our investigations of this ring, focusing more on its arithmetic properties. We begin by constructing a generalized augmentation map that gives a structural decomposition of this ring. This decomposition allows us to compute the zeta function of the join of group rings. We show that the join of group rings is a natural home for studying the concept of simultaneous primitive roots for a given set of primes. This concept is related to the order of the unit group of the join of group rings. Finally, we characterize the join of group rings over finite fields with the property that the order of every unit divides a fixed number. Remarkably, Mersenne and Fermat primes unexpectedly emerge within the context of this exploration. Comments: 23 pages, accepted for publication in The Journal of Algebra Subjects: Rings and Algebras (math.RA) ; Number Theory (math.NT) Cite as: arXiv:2308.13428 [math.RA] (or arXiv:2308.13428v3 [math.RA] for this version) https://doi.org/10.48550/arXiv.2308.13428 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Sunil Chebolu [ view email ] [v1] Fri, 25 Aug 2023 15:13:29 UTC (22 KB) [v2] Sat, 13 Apr 2024 20:41:43 UTC (24 KB) [v3] Wed, 8 Apr 2026 21:51:24 UTC (25 KB) Full-text links: Access Paper: View a PDF of the paper titled On the arithmetic of the join rings over finite fields, by Sunil K. Chebolu and 5 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.RA < prev | next > new | recent | 2023-08 Change to browse by: math math.NT 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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