{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:XHUV4V4SK2LLJT43PRR6T6I3V3","short_pith_number":"pith:XHUV4V4S","schema_version":"1.0","canonical_sha256":"b9e95e57925696b4cf9b7c63e9f91baefe9d766c1de03020542884899fe30711","source":{"kind":"arxiv","id":"1708.02546","version":1},"attestation_state":"computed","paper":{"title":"A principal ideal theorem for compact sets of rank one valuation rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Bruce Olberding","submitted_at":"2017-08-08T16:20:46Z","abstract_excerpt":"Let $F$ be a field, and let Zar$(F)$ be the space of valuation rings of $F$ with respect to the Zariski topology. We prove that if $X$ is a quasicompact set of rank one valuation rings in Zar$(F)$ whose maximal ideals do not intersect to $0$, then the intersection of the rings in $X$ is an integral domain with quotient field $F$ such that every finitely generated ideal is a principal ideal. To prove this result, we develop a duality between (a) quasicompact sets of rank one valuation rings whose maximal ideals do not intersect to $0$, and (b) one-dimensional Pr\\\"ufer domains with nonzero Jacob"},"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":"1708.02546","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-08-08T16:20:46Z","cross_cats_sorted":[],"title_canon_sha256":"bf334b1a384a0b16019f76e1e5318c3e27209f5af553220e60bd5466422c1bd3","abstract_canon_sha256":"908218044728df39f7baa2f96e418a605d0f862e97cfe6a162c710bda2ef84f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:23.954408Z","signature_b64":"7tyDRbWBidvhkWeAJYLVFNi7HLGGGDbX8URNi0r4LLTafZv4964Znpaj6z6w0vxHqGrCILHwA7C+3xCqrMZmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9e95e57925696b4cf9b7c63e9f91baefe9d766c1de03020542884899fe30711","last_reissued_at":"2026-05-18T00:38:23.953671Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:23.953671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A principal ideal theorem for compact sets of rank one valuation rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Bruce Olberding","submitted_at":"2017-08-08T16:20:46Z","abstract_excerpt":"Let $F$ be a field, and let Zar$(F)$ be the space of valuation rings of $F$ with respect to the Zariski topology. We prove that if $X$ is a quasicompact set of rank one valuation rings in Zar$(F)$ whose maximal ideals do not intersect to $0$, then the intersection of the rings in $X$ is an integral domain with quotient field $F$ such that every finitely generated ideal is a principal ideal. To prove this result, we develop a duality between (a) quasicompact sets of rank one valuation rings whose maximal ideals do not intersect to $0$, and (b) one-dimensional Pr\\\"ufer domains with nonzero Jacob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02546","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":"1708.02546","created_at":"2026-05-18T00:38:23.953776+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.02546v1","created_at":"2026-05-18T00:38:23.953776+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.02546","created_at":"2026-05-18T00:38:23.953776+00:00"},{"alias_kind":"pith_short_12","alias_value":"XHUV4V4SK2LL","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"XHUV4V4SK2LLJT43","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"XHUV4V4S","created_at":"2026-05-18T12:31:53.515858+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/XHUV4V4SK2LLJT43PRR6T6I3V3","json":"https://pith.science/pith/XHUV4V4SK2LLJT43PRR6T6I3V3.json","graph_json":"https://pith.science/api/pith-number/XHUV4V4SK2LLJT43PRR6T6I3V3/graph.json","events_json":"https://pith.science/api/pith-number/XHUV4V4SK2LLJT43PRR6T6I3V3/events.json","paper":"https://pith.science/paper/XHUV4V4S"},"agent_actions":{"view_html":"https://pith.science/pith/XHUV4V4SK2LLJT43PRR6T6I3V3","download_json":"https://pith.science/pith/XHUV4V4SK2LLJT43PRR6T6I3V3.json","view_paper":"https://pith.science/paper/XHUV4V4S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.02546&json=true","fetch_graph":"https://pith.science/api/pith-number/XHUV4V4SK2LLJT43PRR6T6I3V3/graph.json","fetch_events":"https://pith.science/api/pith-number/XHUV4V4SK2LLJT43PRR6T6I3V3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XHUV4V4SK2LLJT43PRR6T6I3V3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XHUV4V4SK2LLJT43PRR6T6I3V3/action/storage_attestation","attest_author":"https://pith.science/pith/XHUV4V4SK2LLJT43PRR6T6I3V3/action/author_attestation","sign_citation":"https://pith.science/pith/XHUV4V4SK2LLJT43PRR6T6I3V3/action/citation_signature","submit_replication":"https://pith.science/pith/XHUV4V4SK2LLJT43PRR6T6I3V3/action/replication_record"}},"created_at":"2026-05-18T00:38:23.953776+00:00","updated_at":"2026-05-18T00:38:23.953776+00:00"}