{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:NP2LPVQNIS6F4DZSQNFDQIT73G","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"bec399837d36d2f8271874c0df18e33ad1500b5d38fcd18f49acfdb061cbef64","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2009-06-06T07:39:28Z","title_canon_sha256":"ad93184c1ad1940607b1b96f5dafc21a8b1d5b41e1ae59d8b04347f37eb5f1b7"},"schema_version":"1.0","source":{"id":"0906.1253","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.1253","created_at":"2026-05-18T04:31:41Z"},{"alias_kind":"arxiv_version","alias_value":"0906.1253v2","created_at":"2026-05-18T04:31:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.1253","created_at":"2026-05-18T04:31:41Z"},{"alias_kind":"pith_short_12","alias_value":"NP2LPVQNIS6F","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"NP2LPVQNIS6F4DZS","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"NP2LPVQN","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:86ca7722d23ebfcfe97bd83c1f34eaadc2e15bae5b3db3a4829c341b2f8115ca","target":"graph","created_at":"2026-05-18T04:31:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n\\geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and only if every finitely generated left $R$-module and every finitely generated right $R$-module have torsionfree dimension at most $n$, if and only if every finitely generated left (or right) $R$-module has Gorenstein dimension at most $n$. For any $n \\geq 1$, we study the properties of the finitely generated $R$-modules $M$ with $\\Ext_R^i(M, R)=0$ for any $1","authors_text":"Chonghui Huang, Zhaoyong Huang","cross_cats":["math.KT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2009-06-06T07:39:28Z","title":"Torsionfree Dimension of Modules and Self-Injective Dimension of Rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.1253","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8d683bacb1c41efe896c3b68af2c2ffabf9cae1e7816869a897fe15dcbd8f6b0","target":"record","created_at":"2026-05-18T04:31:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"bec399837d36d2f8271874c0df18e33ad1500b5d38fcd18f49acfdb061cbef64","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2009-06-06T07:39:28Z","title_canon_sha256":"ad93184c1ad1940607b1b96f5dafc21a8b1d5b41e1ae59d8b04347f37eb5f1b7"},"schema_version":"1.0","source":{"id":"0906.1253","kind":"arxiv","version":2}},"canonical_sha256":"6bf4b7d60d44bc5e0f32834a38227fd9a976b8ae03e23e1147023d717af6738e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6bf4b7d60d44bc5e0f32834a38227fd9a976b8ae03e23e1147023d717af6738e","first_computed_at":"2026-05-18T04:31:41.551948Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:41.551948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7gaOc65PIEARcWcbuaiogke6k2C1plhi6PREg81AV7ru0FYGUyC8W0Ukkyb/NuIASbhH/Z9GXjJFLTZqAyH+DA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:41.552477Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.1253","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8d683bacb1c41efe896c3b68af2c2ffabf9cae1e7816869a897fe15dcbd8f6b0","sha256:86ca7722d23ebfcfe97bd83c1f34eaadc2e15bae5b3db3a4829c341b2f8115ca"],"state_sha256":"bceac2e6acbb4b7bee458058b9dc8d928bad4a1bae2b92d767fb95d607073f12"}