A ring π has unbounded generating number (UGN) if,for every positive integer π, there is no π -module epimorphism π π β π π+1. For a ring π = β¨gβπΊ π g gradedby a group πΊ such that the base ring π 1 has UGN, weidentify several sets of conditions under which π mustalso have UGN. The most important of these are: (1)πΊ is amenable, and there is a positive integer π suchthat, for every g β πΊ, π g β (π 1)π as π 1-modules for someπ = 1, β¦ , π; (2) πΊ is supramenable, and there is a positive integer π such that, for every g β πΊ, π g β (π 1)π asπ 1-modules for some π = 0, β¦ , π. 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 ringπ , the smallest positive integer π such that there is anπ -module epimorphism π π β π π+1 is called the generating number of π , denoted gn(π ). If π has UGN, then wedefine gn(π ) βΆ= β΅0. We describe several classes of examples of a ring π graded by an amenable group πΊ such thatgn(π ) β gn(π 1).
MSC 2020
16P99, 16S35, 16W50, 20F65, 43A07 (primary), 16D90 (secondary)