{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:IWVL5MN5NSLOVTY2IKGWXIX4UE","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":"6c1c859584e89535b426bd3a679820bcdbad2241ff8433702a43ecb709c037b2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-04T14:57:14Z","title_canon_sha256":"edbbe8e288eb58184a871ccd9b3828ec0748728a4d65b60349ba100ad2285b40"},"schema_version":"1.0","source":{"id":"1605.01305","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.01305","created_at":"2026-05-18T01:15:37Z"},{"alias_kind":"arxiv_version","alias_value":"1605.01305v1","created_at":"2026-05-18T01:15:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.01305","created_at":"2026-05-18T01:15:37Z"},{"alias_kind":"pith_short_12","alias_value":"IWVL5MN5NSLO","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IWVL5MN5NSLOVTY2","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IWVL5MN5","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:885ecd0b79e2139ecb5130f220da2aa3333e12560851ec5a12f0c1d693bd9237","target":"graph","created_at":"2026-05-18T01:15:37Z","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":"The rank of a ring $R$ is the supremum of minimal cardinalities of generating sets of $I$ as $I$ ranges over ideals of $R$. Matson showed that every positive integer occurs as the rank of some ring $R$. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension $0$ or $1$, we give four different constructions of rings of rank $n$ (for all positive integers n). Two constructions use one-dimensional domains, and the former of these directly generalizes Matson's construction. Our third construction uses Artinian rings (dimension zero), and our last construction use","authors_text":"Pete L. Clark","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-04T14:57:14Z","title":"A note on rings of finite rank"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01305","kind":"arxiv","version":1},"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":"sha256:44e57f17e111bab6bdacb41d3ceafbe8d561e8e9fbc9ffb1a180815abf7ceded","target":"record","created_at":"2026-05-18T01:15:37Z","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":"6c1c859584e89535b426bd3a679820bcdbad2241ff8433702a43ecb709c037b2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-05-04T14:57:14Z","title_canon_sha256":"edbbe8e288eb58184a871ccd9b3828ec0748728a4d65b60349ba100ad2285b40"},"schema_version":"1.0","source":{"id":"1605.01305","kind":"arxiv","version":1}},"canonical_sha256":"45aabeb1bd6c96eacf1a428d6ba2fca118a49aa2d01ad1220fa339b3653b8c41","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45aabeb1bd6c96eacf1a428d6ba2fca118a49aa2d01ad1220fa339b3653b8c41","first_computed_at":"2026-05-18T01:15:37.195822Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:37.195822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D0wTqNSPYKrKFejeB0PzEVKSu3fc+zp/IJkJMSuvytAHIt+JDWiTVlC0IGZbcPlahviqXTHQ9VZoUY9k48eEAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:37.196529Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.01305","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:44e57f17e111bab6bdacb41d3ceafbe8d561e8e9fbc9ffb1a180815abf7ceded","sha256:885ecd0b79e2139ecb5130f220da2aa3333e12560851ec5a12f0c1d693bd9237"],"state_sha256":"fda3fd43cca112cc6bdfa0a66984149423f019de318abbd133b08c71a65dc024"}