{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:7WSHWK74C4XO42S5APVAQHWZN2","short_pith_number":"pith:7WSHWK74","schema_version":"1.0","canonical_sha256":"fda47b2bfc172eee6a5d03ea081ed96eb0b2053bbbdd3b1549491f881a161589","source":{"kind":"arxiv","id":"1109.1391","version":1},"attestation_state":"computed","paper":{"title":"The Transcendence Degree over a Ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Gregor Kemper","submitted_at":"2011-09-07T08:48:57Z","abstract_excerpt":"For a finitely generated algebra over a field, the transcendence degree is known to be equal to the Krull dimension. The aim of this paper is to generalize this result to algebras over rings. A new definition of the transcendence degree of an algebra A over a ring R is given by calling elements of A algebraically dependent if they satisfy an algebraic equation over R whose trailing coefficient, with respect to some monomial ordering, is 1. The main result is that for a finitely generated algebra over a Noetherian Jacobson ring, the transcendence degree is equal to the Krull dimension."},"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":"1109.1391","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-09-07T08:48:57Z","cross_cats_sorted":[],"title_canon_sha256":"a8c731cf414fdbb7655cd49b71e6d1ba50182683c04259b816573d8a1f091544","abstract_canon_sha256":"ea31715cc3561076e356e873fb0ef3672e2149f0359016ef13b6102999648c1d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:55.805793Z","signature_b64":"SxvlLHJ6yWOHP8Q1epNMeks5uyF7FY6RnJBSgBKXOvlxjLio01rRA3PzhJ+ZxFKH3K+c6TU/nHD1K4+TRsfaAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fda47b2bfc172eee6a5d03ea081ed96eb0b2053bbbdd3b1549491f881a161589","last_reissued_at":"2026-05-18T04:13:55.805162Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:55.805162Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Transcendence Degree over a Ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Gregor Kemper","submitted_at":"2011-09-07T08:48:57Z","abstract_excerpt":"For a finitely generated algebra over a field, the transcendence degree is known to be equal to the Krull dimension. The aim of this paper is to generalize this result to algebras over rings. A new definition of the transcendence degree of an algebra A over a ring R is given by calling elements of A algebraically dependent if they satisfy an algebraic equation over R whose trailing coefficient, with respect to some monomial ordering, is 1. The main result is that for a finitely generated algebra over a Noetherian Jacobson ring, the transcendence degree is equal to the Krull dimension."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.1391","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1109.1391","created_at":"2026-05-18T04:13:55.805251+00:00"},{"alias_kind":"arxiv_version","alias_value":"1109.1391v1","created_at":"2026-05-18T04:13:55.805251+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.1391","created_at":"2026-05-18T04:13:55.805251+00:00"},{"alias_kind":"pith_short_12","alias_value":"7WSHWK74C4XO","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"7WSHWK74C4XO42S5","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"7WSHWK74","created_at":"2026-05-18T12:26:22.705136+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7WSHWK74C4XO42S5APVAQHWZN2","json":"https://pith.science/pith/7WSHWK74C4XO42S5APVAQHWZN2.json","graph_json":"https://pith.science/api/pith-number/7WSHWK74C4XO42S5APVAQHWZN2/graph.json","events_json":"https://pith.science/api/pith-number/7WSHWK74C4XO42S5APVAQHWZN2/events.json","paper":"https://pith.science/paper/7WSHWK74"},"agent_actions":{"view_html":"https://pith.science/pith/7WSHWK74C4XO42S5APVAQHWZN2","download_json":"https://pith.science/pith/7WSHWK74C4XO42S5APVAQHWZN2.json","view_paper":"https://pith.science/paper/7WSHWK74","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1109.1391&json=true","fetch_graph":"https://pith.science/api/pith-number/7WSHWK74C4XO42S5APVAQHWZN2/graph.json","fetch_events":"https://pith.science/api/pith-number/7WSHWK74C4XO42S5APVAQHWZN2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7WSHWK74C4XO42S5APVAQHWZN2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7WSHWK74C4XO42S5APVAQHWZN2/action/storage_attestation","attest_author":"https://pith.science/pith/7WSHWK74C4XO42S5APVAQHWZN2/action/author_attestation","sign_citation":"https://pith.science/pith/7WSHWK74C4XO42S5APVAQHWZN2/action/citation_signature","submit_replication":"https://pith.science/pith/7WSHWK74C4XO42S5APVAQHWZN2/action/replication_record"}},"created_at":"2026-05-18T04:13:55.805251+00:00","updated_at":"2026-05-18T04:13:55.805251+00:00"}