{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4OHB2AEHLJKFE5HC3D3ULLSYXZ","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":"3c8536fe704885a15b7455da18acd90fb56a2b9a9f47b25203934d56dccb2754","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-11-13T19:13:38Z","title_canon_sha256":"0b85bc0685023a0e9213b3da3a0e15b81f3d1c1ee32d9baba45821cd3b5b5ffb"},"schema_version":"1.0","source":{"id":"1611.04173","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.04173","created_at":"2026-05-18T00:59:17Z"},{"alias_kind":"arxiv_version","alias_value":"1611.04173v1","created_at":"2026-05-18T00:59:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04173","created_at":"2026-05-18T00:59:17Z"},{"alias_kind":"pith_short_12","alias_value":"4OHB2AEHLJKF","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4OHB2AEHLJKFE5HC","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4OHB2AEH","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:051dcd315e5e4bfc1fcaf73614df22734ac04ed0dba0953fb0208ee8a7dec7c0","target":"graph","created_at":"2026-05-18T00:59:17Z","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 an integral domain. For elements $a,b \\in R$, let $[a,b]$ denote their greatest common divisor, if it exists. We say that $R$ has the Z-property if whenever $a,b,c,d$ and $e$ are nonzero nonunits of $R$ such that $abc=de$, then $[ab,d] \\neq 1$ or $[ab,e] \\neq 1$. The purpose of this paper is to study this property. The atomic integral domains that have this property constitute a class of half-factorial domains. Also, it is known that $R$ must have this property in order for the polynomial ring $R[x]$ to be half-factorial. We use it to give a characterization of half-factorial polyno","authors_text":"Mark Batell","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-11-13T19:13:38Z","title":"On a class of half-factorial domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04173","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":"sha256:07796a097b33131b0d51a5fc5249f27a7049ea172c8c937b35c32e054b7fe754","target":"record","created_at":"2026-05-18T00:59:17Z","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":"3c8536fe704885a15b7455da18acd90fb56a2b9a9f47b25203934d56dccb2754","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-11-13T19:13:38Z","title_canon_sha256":"0b85bc0685023a0e9213b3da3a0e15b81f3d1c1ee32d9baba45821cd3b5b5ffb"},"schema_version":"1.0","source":{"id":"1611.04173","kind":"arxiv","version":1}},"canonical_sha256":"e38e1d00875a545274e2d8f745ae58be402816f76d44f857bb93dd1e6b0172b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e38e1d00875a545274e2d8f745ae58be402816f76d44f857bb93dd1e6b0172b6","first_computed_at":"2026-05-18T00:59:17.296372Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:59:17.296372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RXLEFy+RwkEGb7CK/RYhueu19BAQWPnNN2F6TRtFispwE1dKKSxcmsJjFpJl+s0HM9ekgFXiSmSIMkY/3Z3gDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:59:17.296909Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.04173","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:07796a097b33131b0d51a5fc5249f27a7049ea172c8c937b35c32e054b7fe754","sha256:051dcd315e5e4bfc1fcaf73614df22734ac04ed0dba0953fb0208ee8a7dec7c0"],"state_sha256":"671447f1cc34d0b509df0cb380ff1d8d6f01ff77dc5af2fd2dc6f67f27fb096a"}