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Neat-flat right $R$-modules are projective if and only if $R$ is a right $\\sum$-$CS$ ring. Every finitely generated neat-flat right $R$-module is projective if and only if $R$ is a right $C$-ring and every finitely generated free right $R$-module is extending. Every cyclic neat-flat ri"},"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":"1306.2860","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.RA","submitted_at":"2013-06-12T15:14:17Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"da830129c6630e40878a7f6a5175a61bb2c3f2010283383ee2a338b6fe179b14","abstract_canon_sha256":"dd813fb9b05007e08da1b83fbbfbe9c0d1def71bd250c33632c7096499ce3195"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:09.201342Z","signature_b64":"7xQVZbQl6hhGDdMGcaCCmLkrMzHYc3zlyr1UFaNtC9SO+l7quMmzO4YCpOHieBPJZRJzkUyKWiP1w9djARsFBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8bc8e98f0fe03dd7d032ce730a677cd840dafcc240a94d7fab38038da94607dc","last_reissued_at":"2026-05-18T03:21:09.200802Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:09.200802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Neat-Flat Modules","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.RA","authors_text":"Engin B\\\"uy\\\"uka\\c{s}{\\i}k, Y{\\i}lmaz Dur\\u{g}un","submitted_at":"2013-06-12T15:14:17Z","abstract_excerpt":"Let $R$ be a ring and $M$ be a right $R$-module. $M$ is called neat-flat if any short exact sequence of the form $0\\to K\\to N\\to M\\to 0$ is neat-exact i.e. any homomorphism from a simple right $R$-module $S$ to $M$ can be lifted to $N$. We prove that, a module is neat-flat if and only if it is simple-projective. Neat-flat right $R$-modules are projective if and only if $R$ is a right $\\sum$-$CS$ ring. Every finitely generated neat-flat right $R$-module is projective if and only if $R$ is a right $C$-ring and every finitely generated free right $R$-module is extending. 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