{"paper":{"title":"Subgroups of an abelian group, related ideals of the group ring, and quotients by those ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.AC","authors_text":"Hideyasu Kawai","submitted_at":"2019-08-08T08:21:17Z","abstract_excerpt":"Let $RG$ be the group ring of an abelian group $G$ over a commutative ring $R$ with identity. An injection $\\Phi$ from the subgroups of $G$ to the non-unit ideals of $RG$ is well-known. It is defined by $\\Phi(N)=I(R,N)RG$ where $I(R,N)$ is the augmentation ideal of $RN$, and each ideal $\\Phi(N)$ has a property : $RG/\\Phi(N)$ is $R$-algebra isomorphic to $R(G/N)$. Let $T$ be the set of non-unit ideals of $RG$. While the image of $\\Phi$ is rather a small subset of $T$, we give conditions on $R$ and $G$ for the image of $\\Phi$ to have some distribution in $T$. In the last section, we give criteri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02968","kind":"arxiv","version":1},"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/1908.02968/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"}