Generating numbers of rings graded by amenable and supramenable groups

TL;DR

This paper investigates the unbounded generating number property in rings graded by amenable and supramenable groups, identifying conditions under which the property is preserved and exploring implications for group amenability.

Contribution

It establishes new conditions linking group amenability to the unbounded generating number property in graded rings, and characterizes group amenability through ring-theoretic properties.

Findings

Conditions under which graded rings inherit UGN from the base ring.

Characterizations of amenability via ring properties.

Examples of rings with differing generating numbers from their base rings.

Abstract

A ring $R$ has {\it unbounded generating number} (UGN) if, for every positive integer $n$, there is no $R$-module epimorphism $R^n\to R^{n+1}$. For a ring $R=\bigoplus_{g\in G} R_g$ graded by a group $G$ such that the base ring $R_1$ 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 $g\in G$, $R_g\cong (R_1)^i$ as $R_1$-modules for some $i=1,\dots,r$; (2) $G$ is supramenable, and there is a positive integer $r$ such that, for every $g\in G$, $R_g\cong (R_1)^i$ as $R_1$-modules for some $i=0,\dots,r$. 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 $R^n\to R^{n+1}$ is called the {\it generating number} of $R$, denoted ${\rm gn}(R)$. If $R$ has UGN, then we define ${\rm gn}(R):=\aleph_0$. We describe several classes of examples of a ring $R$ graded by an amenable group $G$ such that ${\rm gn}(R)\neq {\rm gn}(R_1)$.