A ring $R$ has unbounded generating number (UGN) if, for every positive integer $n$, there is no $R$-module epimorphism $R^n \to R^{n+1}$. Every nonzero commutative ring has UGN, but not every noncommutative ring does.

For a ring $R=\oplus_{g\in G} R_g$ graded by a group $G$, one can ask when UGN for the canonical subring $R_1$ lifts to UGN for the whole ring $R$, under suitable hypotheses on the grading. As it turns out, amenability of $G$ plays a key role in this question.

In this talk, I will explain the mechanisms behind this lifting procedure, the role played by the grading group, and present ring-theoretic characterizations of amenability for groups via the UGN property.

The talk is based on joint work with K. Lorensen.