{"paper":{"title":"Generating numbers of rings graded by amenable and supramenable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RA","authors_text":"Johan \\\"Oinert, Karl Lorensen","submitted_at":"2022-01-11T17:23:48Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.04087","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.04087/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}