{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:DFR4IQKCZZ2DMFMYLGSVKWPVZG","short_pith_number":"pith:DFR4IQKC","canonical_record":{"source":{"id":"1811.00242","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-11-01T05:33:23Z","cross_cats_sorted":[],"title_canon_sha256":"9425304915edc7a37bb22e97b1b7af0c9cba886ae32db413cb094757dcad0a13","abstract_canon_sha256":"202141fba2a7e40383f7183ecaff80dcbf5f4d36b2072df31e98950a702f0006"},"schema_version":"1.0"},"canonical_sha256":"1963c44142ce7436159859a55559f5c9ad144b3c06c7396a579d2e56508f9086","source":{"kind":"arxiv","id":"1811.00242","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.00242","created_at":"2026-05-17T23:42:42Z"},{"alias_kind":"arxiv_version","alias_value":"1811.00242v2","created_at":"2026-05-17T23:42:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.00242","created_at":"2026-05-17T23:42:42Z"},{"alias_kind":"pith_short_12","alias_value":"DFR4IQKCZZ2D","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DFR4IQKCZZ2DMFMY","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DFR4IQKC","created_at":"2026-05-18T12:32:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:DFR4IQKCZZ2DMFMYLGSVKWPVZG","target":"record","payload":{"canonical_record":{"source":{"id":"1811.00242","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-11-01T05:33:23Z","cross_cats_sorted":[],"title_canon_sha256":"9425304915edc7a37bb22e97b1b7af0c9cba886ae32db413cb094757dcad0a13","abstract_canon_sha256":"202141fba2a7e40383f7183ecaff80dcbf5f4d36b2072df31e98950a702f0006"},"schema_version":"1.0"},"canonical_sha256":"1963c44142ce7436159859a55559f5c9ad144b3c06c7396a579d2e56508f9086","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:42.843069Z","signature_b64":"Xrr5kaUVAvN1r+RhM2As42Ld+yREwKv/jZ1XU4QRuCU9/fJVaLly8Y0NS1hK7z3qF4WtANu0cUB9psAAM0IMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1963c44142ce7436159859a55559f5c9ad144b3c06c7396a579d2e56508f9086","last_reissued_at":"2026-05-17T23:42:42.842500Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:42.842500Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.00242","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3lBlCdz5heoZ6osTlUKDkOo5E9a/erghhrH9auJL35ENIPTU2nWqI1k+Y7X8nGiu2PiLvL+nETU0FefHzhTcBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:35:39.297649Z"},"content_sha256":"e8514feaee255591417d338a3324d3c219f717b223c6aca229c5e8a1b9aab502","schema_version":"1.0","event_id":"sha256:e8514feaee255591417d338a3324d3c219f717b223c6aca229c5e8a1b9aab502"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:DFR4IQKCZZ2DMFMYLGSVKWPVZG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Radical factorization in commutative rings, monoids and multiplicative lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Andreas Reinhart, Bruce Olberding","submitted_at":"2018-11-01T05:33:23Z","abstract_excerpt":"In this paper we study the concept of radical factorization in the context of abstract ideal theory in order to obtain a unified approach to the theory of factorization into radical ideals and elements in the literature of commutative rings, monoids and ideal systems. Using this approach we derive new characterizations of classes of rings whose ideals are a product of radical ideals, and we obtain also similar characterizations for classes of ideal systems in monoids and star ideals in integral domains."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00242","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:42:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wIzxtWcPLhiXvtCj5ayiqB51jiGyfoVsKYrm4FyGKMD6m7+pCYx3vjzDZRdhOycbYtw+4AmT4VzHBZHCVJg9Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T02:35:39.298025Z"},"content_sha256":"e52ef477a136816e89214a5c3888136f3b25ba2f1fb062b72e517137b35f0366","schema_version":"1.0","event_id":"sha256:e52ef477a136816e89214a5c3888136f3b25ba2f1fb062b72e517137b35f0366"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DFR4IQKCZZ2DMFMYLGSVKWPVZG/bundle.json","state_url":"https://pith.science/pith/DFR4IQKCZZ2DMFMYLGSVKWPVZG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DFR4IQKCZZ2DMFMYLGSVKWPVZG/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T02:35:39Z","links":{"resolver":"https://pith.science/pith/DFR4IQKCZZ2DMFMYLGSVKWPVZG","bundle":"https://pith.science/pith/DFR4IQKCZZ2DMFMYLGSVKWPVZG/bundle.json","state":"https://pith.science/pith/DFR4IQKCZZ2DMFMYLGSVKWPVZG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DFR4IQKCZZ2DMFMYLGSVKWPVZG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:DFR4IQKCZZ2DMFMYLGSVKWPVZG","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":"202141fba2a7e40383f7183ecaff80dcbf5f4d36b2072df31e98950a702f0006","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-11-01T05:33:23Z","title_canon_sha256":"9425304915edc7a37bb22e97b1b7af0c9cba886ae32db413cb094757dcad0a13"},"schema_version":"1.0","source":{"id":"1811.00242","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.00242","created_at":"2026-05-17T23:42:42Z"},{"alias_kind":"arxiv_version","alias_value":"1811.00242v2","created_at":"2026-05-17T23:42:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.00242","created_at":"2026-05-17T23:42:42Z"},{"alias_kind":"pith_short_12","alias_value":"DFR4IQKCZZ2D","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_16","alias_value":"DFR4IQKCZZ2DMFMY","created_at":"2026-05-18T12:32:19Z"},{"alias_kind":"pith_short_8","alias_value":"DFR4IQKC","created_at":"2026-05-18T12:32:19Z"}],"graph_snapshots":[{"event_id":"sha256:e52ef477a136816e89214a5c3888136f3b25ba2f1fb062b72e517137b35f0366","target":"graph","created_at":"2026-05-17T23:42:42Z","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":"In this paper we study the concept of radical factorization in the context of abstract ideal theory in order to obtain a unified approach to the theory of factorization into radical ideals and elements in the literature of commutative rings, monoids and ideal systems. Using this approach we derive new characterizations of classes of rings whose ideals are a product of radical ideals, and we obtain also similar characterizations for classes of ideal systems in monoids and star ideals in integral domains.","authors_text":"Andreas Reinhart, Bruce Olberding","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-11-01T05:33:23Z","title":"Radical factorization in commutative rings, monoids and multiplicative lattices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00242","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:e8514feaee255591417d338a3324d3c219f717b223c6aca229c5e8a1b9aab502","target":"record","created_at":"2026-05-17T23:42:42Z","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":"202141fba2a7e40383f7183ecaff80dcbf5f4d36b2072df31e98950a702f0006","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-11-01T05:33:23Z","title_canon_sha256":"9425304915edc7a37bb22e97b1b7af0c9cba886ae32db413cb094757dcad0a13"},"schema_version":"1.0","source":{"id":"1811.00242","kind":"arxiv","version":2}},"canonical_sha256":"1963c44142ce7436159859a55559f5c9ad144b3c06c7396a579d2e56508f9086","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1963c44142ce7436159859a55559f5c9ad144b3c06c7396a579d2e56508f9086","first_computed_at":"2026-05-17T23:42:42.842500Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:42.842500Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xrr5kaUVAvN1r+RhM2As42Ld+yREwKv/jZ1XU4QRuCU9/fJVaLly8Y0NS1hK7z3qF4WtANu0cUB9psAAM0IMBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:42.843069Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.00242","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e8514feaee255591417d338a3324d3c219f717b223c6aca229c5e8a1b9aab502","sha256:e52ef477a136816e89214a5c3888136f3b25ba2f1fb062b72e517137b35f0366"],"state_sha256":"f05f625be5c1dde3be39dc3153df6bbc0f34be85a66daba638c44300cf8d1bf7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BR3y/y1BTbH8EBbh2FB5eNQlzA5xEXsPYpxwBmUOqNn0Q5rz0vn3/xY35XvdxoQJ/B5vv6/N1y3+EAzNzD34Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T02:35:39.302145Z","bundle_sha256":"2df1f9f9b743fefd8c2ff747674e38fbf529577c75c85db1e1ac732db7e6b1db"}}