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Generating numbers of rings graded by amenable and supramenable groups
Pennsylvania State University, United States.
Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.ORCID iD: 0000-0001-8095-0820
2024 (English)In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 109, no 1, article id e12826Article in journal (Refereed) Published
Abstract [en]

A ring 𝑅 has unbounded generating number (UGN) if,for every positive integer 𝑛, there is no 𝑅-module epimorphism 𝑅𝑛 β†’ 𝑅𝑛+1. For a ring 𝑅 = ⨁g∈𝐺 𝑅g gradedby a group 𝐺 such that the base ring 𝑅1 has UGN, weidentify several sets of conditions under which 𝑅 mustalso have UGN. The most important of these are: (1)𝐺 is amenable, and there is a positive integer π‘Ÿ suchthat, for every g ∈ 𝐺, 𝑅g β‰… (𝑅1)𝑖 as 𝑅1-modules for some𝑖 = 1, … , π‘Ÿ; (2) 𝐺 is supramenable, and there is a positive integer π‘Ÿ such that, for every g ∈ 𝐺, 𝑅g β‰… (𝑅1)𝑖 as𝑅1-modules for some 𝑖 = 0, … , π‘Ÿ. The pair of conditions(1) leads to three different ring-theoretic characterizations of the property of amenability for groups. We alsoconsider rings that do not have UGN; for such a ring𝑅, the smallest positive integer 𝑛 such that there is an𝑅-module epimorphism 𝑅𝑛 β†’ 𝑅𝑛+1 is called the generating number of 𝑅, denoted gn(𝑅). If 𝑅 has UGN, then wedefine gn(𝑅) ∢= β„΅0. We describe several classes of examples of a ring 𝑅 graded by an amenable group 𝐺 such thatgn(𝑅) β‰  gn(𝑅1).

MSC 2020

16P99, 16S35, 16W50, 20F65, 43A07 (primary), 16D90 (secondary)

Place, publisher, year, edition, pages
John Wiley & Sons, 2024. Vol. 109, no 1, article id e12826
National Category
Algebra and Logic
Identifiers
URN: urn:nbn:se:bth-25553DOI: 10.1112/jlms.12826ISI: 001119522700001Scopus ID: 2-s2.0-85174638607OAI: oai:DiVA.org:bth-25553DiVA, id: diva2:1810138
Available from: 2023-11-07 Created: 2023-11-07 Last updated: 2023-12-31Bibliographically approved

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Γ–inert, Johan

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