Open this publication in new window or tab >>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 → 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 𝑟 suchthat, for every g ∈ 𝐺, 𝑅g ≅ (𝑅1)𝑖 as 𝑅1-modules for some𝑖 = 1, … , 𝑟; (2) 𝐺 is supramenable, and there is a positive integer 𝑟 such that, for every g ∈ 𝐺, 𝑅g ≅ (𝑅1)𝑖 as𝑅1-modules for some 𝑖 = 0, … , 𝑟. 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 → 𝑅𝑛+1 is called the generating number of 𝑅, denoted gn(𝑅). If 𝑅 has UGN, then wedefine gn(𝑅) ∶= ℵ0. We describe several classes of examples of a ring 𝑅 graded by an amenable group 𝐺 such thatgn(𝑅) ≠ gn(𝑅1).
MSC 2020
16P99, 16S35, 16W50, 20F65, 43A07 (primary), 16D90 (secondary)
Place, publisher, year, edition, pages
John Wiley & Sons, 2024
National Category
Algebra and Logic
Identifiers
urn:nbn:se:bth-25553 (URN)10.1112/jlms.12826 (DOI)001119522700001 ()2-s2.0-85174638607 (Scopus ID)
2023-11-072023-11-072023-12-31Bibliographically approved