2022/01/11 by Karl Lorensen, Lorensen, Karl, Johan Öinert +1 · 1 citation
Computer Science · Mathematics · #16D90 #16P99 #16S35 #16W50 #20F65 #43A07 #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2201.04087
openalex publication_date 2022/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A ring R has \it unbounded generating number (UGN) if, for every positive integer n, there is no R-module epimorphism Rn→ Rn+1. For a ring R=\bigoplusg∈ G Rg graded by a group G such that the base ring R1 has UGN, we identify several sets of conditions under which R must also have UGN. The most important of these are: (1) G is amenable, and there is a positive integer r such that, for every g∈ G, Rg≅ (R1)i as R1-modules for some i=1,…,r; (2) G is supramenable, and there is a positive integer r such that, for every g∈ G, Rg≅ (R1)i as R1-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 also consider rings that do not have UGN; for such a ring R, the smallest positive integer n such that there is an R-module epimorphism Rn→ Rn+1 is called the \it generating number of R, denoted \rm gn(R). If R has UGN, then we define \rm gn(R):=ℵ0. We describe several classes of examples of a ring R graded by an amenable group G such that \rm gn(R)≠ \rm gn(R1).