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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1508.00095","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-08-01T07:38:55Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"06f152f632e947b8f4cd3411ea9f04b407ebf7b6d09a7d58aad13d3c64b9ea87","abstract_canon_sha256":"45330d926ff6b83ffbd697803480bab8b72122c8ead5002eb28e8a759365b001"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:36.832065Z","signature_b64":"BnlyYvoNvOq154YHPdDhaBgQa/akxzHCxTRY0VmMqVwVW3dPnyYud765jVz3KC6chm0xxHeJI0wop7CseLtUBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1003994d1701d1f1b2a034586df7b7b8a76dd8c9c40bfb9e2d4f9e810c993608","last_reissued_at":"2026-05-18T01:32:36.831500Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:36.831500Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $R\\pi$. 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