{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/decomposing-finite-mathbb-z-algebras","title":"Decomposing Finite $\\mathbb{Z}$-Algebras","arxiv_id":"2308.01735","date":"2023-08-03","proceeding":null,"authors":["Martin Kreuzer","Alexei Miasnikov","Florian Walsh"],"abstract":"For a finite $\\mathbb{Z}$-algebra $R$, i.e., for a ring which is not necessarily associative or unitary, but whose additive group is finitely generated, we construct a decomposition of $R/{\\rm Ann}(R)$ into directly indecomposable factors under weak hypotheses. The method is based on constructing and decomposing a ring of scalars $S$, and then lifting the decomposition of $S$ to the bilinear map given by the multiplication of $R$, and finally to $R/{\\rm Ann}(R)$. All steps of the construction are given as explicit algorithms and it is shown that the entire procedure has a probabilistic polynomial time complexity in the bit size of the input, except for the possible need to calculate the prime factorization of one integer. In particular, in the case when ${\\rm Ann}(R) = 0$, these algorithms compute a direct decomposition of $R$ into directly indecomposable factors.","url_abs":"https://arxiv.org/abs/2308.01735v1","url_pdf":"https://arxiv.org/pdf/2308.01735v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"decomposing-finite-mathbb-z-algebras","repo_url":"https://github.com/abacus42/finite_z_algebras","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}