dc.contributor.author Kropholler, Peter H. dc.contributor.author Lorensen, Karl dc.date.accessioned 2019-08-16T06:31:21Z dc.date.available 2019-08-16T06:31:21Z dc.date.issued 2019-08-16 dc.identifier.uri http://publications.mfo.de/handle/mfo/2513 dc.description.abstract A ring $R$ satisfies the $\textit{strong rank condition}$ (SRC) if, for every natural number $n$, the free $R$-submodules of $R^n$ all have rank $\leq n$. Let $G$ be a group and $R$ a ring strongly graded by $G$ such that the base ring $R_1$ is a domain. Using an argument originated by Laurent Bartholdi for studying cellular automata, we prove that $R$ satisfies SRC if and only if en_US $R_1$ 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. dc.language.iso en_US en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2019,22 dc.subject Group-graded ring en_US dc.subject Strong rank en_US dc.subject Amenable group en_US dc.title Group-Graded Rings Satisfying the Strong Rank Condition en_US dc.type Preprint en_US dc.identifier.doi 10.14760/OWP-2019-22 local.scientificprogram Research in Pairs 2015 en_US local.series.id OWP-2019-22 en_US local.subject.msc 16 en_US local.subject.msc 20 en_US local.subject.msc 43 en_US
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