A ring R satisfies the strong rank condition (SRC) if, for every natural number n, the free R-submodules of Rn all have rank ≤n. Let G be a group and R a ring strongly graded by G such that the base ring R1 is a domain. Using an argument originated by Laurent Bartholdi for studying cellular automata, we prove that R satisfies SRC if and only if R1 satisfies SRC and G is amenable. The special case of this result for group rings allows us to prove a characterization of amenability involving the group von Neumann algebra that was conjectured by Wolfgang Lück. In addition, we include two applications to the study of group rings and their modules.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory