{"paper":{"title":"Cartan maps and projective modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.GR","authors_text":"Guangjun Zhu, Ming-chang Kang","submitted_at":"2015-08-01T07:38:55Z","abstract_excerpt":"Let $R$ be a commutative ring, $\\pi$ be a finite group, $R\\pi$ be the group ring of $\\pi$ over $R$. Theorem 1. If $R$ is a commutative artinian ring and $\\pi$ is a finite group. Then the Cartan map $c:K_0(R\\pi)\\to G_0(R\\pi)$ is injective. Theorem 2. Suppose that $R$ is a Dedekind domain with $\\fn{char}R=p>0$ and $\\pi$ is a $p$-group. Then every finitely generated projective $R\\pi$-module is isomorphic to $F \\oplus \\c{A}$ where $F$ is a free module and $\\c{A}$ is a projective ideal of $R\\pi$. Moreover, $R$ is a principal ideal domain if and only if every finitely generated projective $R\\pi$-mod"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00095","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":""},"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"}