Generating numbers of rings graded by amenable and supramenable groups

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