Singular ideals over arbitrary fields for the cyclic-headed snakes

[2506.19254] Singular ideals over arbitrary fields for the cyclic-headed snakes Skip to main content Search Submit Donate Log in Mathematics > Rings and Algebras arXiv:2506.19254 (math) [Submitted on 24 Jun 2025 ( v1 ), last revised 28 Aug 2026 (this version, v2)] Title: Singular ideals over arbitrary fields for the cyclic-headed snakes Authors: Ramla Abdellatif , Lisa Orloff Clark , Roy Jansen , Stephen Marsland View PDF HTML (experimental) Abstract: We study the Steinberg algebras with coefficients in an arbitrary field K for the cyclic-headed snake groupoids, which are basic examples of non-Hausdorff groupoids. We are particularly interested in elements of this algebra that are no longer continuous, known as singular functions. These functions form an ideal, which may contain proper subsets that are themselves ideals of the Steinberg algebra. We provide three conditions under which such 'proper subset' ideals exist: first, when the number of heads of the snake divides the characteristic of the base field; second, when the base field is of non-prime characteristic; and third, when certain cyclotomic polynomials split over the base field. We also show the existence of many further subset ideals not covered by these conditions. We fully explore the cases of the two- and three-headed snakes. In the three-headed snake, we prove that the ideal of singular functions properly contains non-zero ideals of the Steinberg algebra if, and only if, the base field K is a splitting field of x^2 + x + 1, the third cyclotomic polynomial. Consequently, there are always proper subset ideals when K has characteristic a prime not congruent to -1 mod 3. Comments: 20 pages, comments welcome Subjects: Rings and Algebras (math.RA) ; Number Theory (math.NT) MSC classes: 16S99, 22A22, 16D25, 18B40 Cite as: arXiv:2506.19254 [math.RA] (or arXiv:2506.19254v2 [math.RA] for this version) https://doi.org/10.48550/arXiv.2506.19254 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Roy Jansen [ view email ] [v1] Tue, 24 Jun 2025 02:35:55 UTC (20 KB) [v2] Fri, 28 Aug 2026 03:43:05 UTC (25 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.RA < prev | next > new | recent | 2025-06 Change to browse by: math math.NT 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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