Group-graded rings satisfying the strong rank condition
| Autor: | Peter H. Kropholler, Karl Lorensen |
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| Rok vydání: | 2019 |
| Předmět: |
16W50
16P99 20F65 43A07 Ring (mathematics) Algebra and Number Theory Mathematics::Commutative Algebra Group (mathematics) 010102 general mathematics Amenable group Mathematics - Rings and Algebras 01 natural sciences Combinatorics Base (group theory) symbols.namesake Von Neumann algebra Rank condition Rings and Algebras (math.RA) 0103 physical sciences symbols FOS: Mathematics Rank (graph theory) 010307 mathematical physics 0101 mathematics Mathematics Group ring |
| DOI: | 10.48550/arxiv.1901.10001 |
| Popis: | 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 $R_1$ 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. Oberwolfach Preprints;2019,22 |
| Databáze: | OpenAIRE |
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