Generating numbers of rings graded by amenable and supramenable groups

Abstract

AbstractA ring has unbounded generating number (UGN) if, for every positive integer , there is no ‐module epimorphism . For a ring graded by a group such that the base ring has UGN, we identify several sets of conditions under which must also have UGN. The most important of these are: (1) is amenable, and there is a positive integer such that, for every , as ‐modules for some ; (2) is supramenable, and there is a positive integer such that, for every , as ‐modules for some . 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 , the smallest positive integer such that there is an ‐module epimorphism is called the generating number of , denoted . If has UGN, then we define . We describe several classes of examples of a ring graded by an amenable group such that .