{"paper":{"title":"Artinian and noetherian partial skew groupoid rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.RA","authors_text":"H\\'ector Pinedo, Johan \\\"Oinert, Patrik Nystedt","submitted_at":"2016-03-07T20:07:10Z","abstract_excerpt":"Let $\\alpha = \\{ \\alpha_g : R_{g^{-1}} \\rightarrow R_g \\}_{g \\in \\textrm{mor}(G)}$ be a partial action of a groupoid $G$ on a non-associative ring $R$ and let $S = R \\star_{\\alpha} G$ be the associated partial skew groupoid ring. We show that if $\\alpha$ is global and unital, then $S$ is left (right) artinian if and only if $R$ is left (right) artinian and $R_g = \\{ 0 \\},$ for all but finitely many $g \\in \\textrm{mor}(G)$. We use this result to prove that if $\\alpha$ is unital and $R$ is alternative, then $S$ is left (right) artinian if and only if $R$ is left (right) artinian and $R_g = \\{ 0 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02237","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}