{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:DLVN3WC2PFCHOLEREZO74YWQ6R","short_pith_number":"pith:DLVN3WC2","canonical_record":{"source":{"id":"1308.5948","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-08-27T18:52:09Z","cross_cats_sorted":[],"title_canon_sha256":"56d520dfda485604938555c63b1977805797c15f106f2dd1e664078d91cc9e64","abstract_canon_sha256":"7015ea1650851874825a8912f17ea7300dd517d09ae6502b043880087210e155"},"schema_version":"1.0"},"canonical_sha256":"1aeaddd85a7944772c91265dfe62d0f475db80e005bdbd7faee798b57a9e8e74","source":{"kind":"arxiv","id":"1308.5948","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.5948","created_at":"2026-05-18T01:10:33Z"},{"alias_kind":"arxiv_version","alias_value":"1308.5948v2","created_at":"2026-05-18T01:10:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.5948","created_at":"2026-05-18T01:10:33Z"},{"alias_kind":"pith_short_12","alias_value":"DLVN3WC2PFCH","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DLVN3WC2PFCHOLER","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DLVN3WC2","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:DLVN3WC2PFCHOLEREZO74YWQ6R","target":"record","payload":{"canonical_record":{"source":{"id":"1308.5948","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-08-27T18:52:09Z","cross_cats_sorted":[],"title_canon_sha256":"56d520dfda485604938555c63b1977805797c15f106f2dd1e664078d91cc9e64","abstract_canon_sha256":"7015ea1650851874825a8912f17ea7300dd517d09ae6502b043880087210e155"},"schema_version":"1.0"},"canonical_sha256":"1aeaddd85a7944772c91265dfe62d0f475db80e005bdbd7faee798b57a9e8e74","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:33.776463Z","signature_b64":"HcNTKLdVSh3nhvPWZD7p/+fZ/5mrV1myTdRsy4OrP8+eWmbSQjIs/efsItUNymgDcOvFn+qcLv0qrXSTsmcwDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1aeaddd85a7944772c91265dfe62d0f475db80e005bdbd7faee798b57a9e8e74","last_reissued_at":"2026-05-18T01:10:33.775853Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:33.775853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1308.5948","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":"2026-05-18T01:10:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XWTxOHJqIruiXGpQLpYBQZuZfoqeJOp9+RmvWcfNzM9yEesUY+Uk3YbLQYcJjOQ2d98v/1JYpxCmgqgFsVDmCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:47:30.508272Z"},"content_sha256":"d256d5974b81f4a6eaab55391624b7423a9c96d6f4bd64b34b77cf2e09b5f27f","schema_version":"1.0","event_id":"sha256:d256d5974b81f4a6eaab55391624b7423a9c96d6f4bd64b34b77cf2e09b5f27f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:DLVN3WC2PFCHOLEREZO74YWQ6R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ratliff-Rush closures and linear growth of primary decompositions of ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Monireh Sedghi","submitted_at":"2013-08-27T18:52:09Z","abstract_excerpt":"Let $R$ be a commutative Noetherian ring, $E$ a non-zero finitely generated $R$-module and $I$ an ideal of $R$. One purpose of this paper is to show that the sequences $\\Ass_RE/ \\widetilde{I_E^n}$ and $\\Ass_R\\widetilde{I^n _E}/\\widetilde{I^{n+1}_E}$, $n = 1,2, \\dots$, of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff \\cite[Theorem 3.1]{MR}. In addition, a characterization concerning the set $\\widetilde{A^*}(I,E)$ is included. A second purpose of this paper is to prove that $I$ has linear growth primary decompositions for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5948","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":"2026-05-18T01:10:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U1Nc0C6xlDefimWcfZnN1hTKyv6hvzw7PEWO85KFyQSY8ERebNxvFw8SfS5yDn7x2CAvFQjYl1IwFsyANfegCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:47:30.508867Z"},"content_sha256":"35176b9c10100767d94470a493be57d316d7954f346d164d8899aac1bbb62ea1","schema_version":"1.0","event_id":"sha256:35176b9c10100767d94470a493be57d316d7954f346d164d8899aac1bbb62ea1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DLVN3WC2PFCHOLEREZO74YWQ6R/bundle.json","state_url":"https://pith.science/pith/DLVN3WC2PFCHOLEREZO74YWQ6R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DLVN3WC2PFCHOLEREZO74YWQ6R/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":"2026-06-04T03:47:30Z","links":{"resolver":"https://pith.science/pith/DLVN3WC2PFCHOLEREZO74YWQ6R","bundle":"https://pith.science/pith/DLVN3WC2PFCHOLEREZO74YWQ6R/bundle.json","state":"https://pith.science/pith/DLVN3WC2PFCHOLEREZO74YWQ6R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DLVN3WC2PFCHOLEREZO74YWQ6R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:DLVN3WC2PFCHOLEREZO74YWQ6R","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":"7015ea1650851874825a8912f17ea7300dd517d09ae6502b043880087210e155","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-08-27T18:52:09Z","title_canon_sha256":"56d520dfda485604938555c63b1977805797c15f106f2dd1e664078d91cc9e64"},"schema_version":"1.0","source":{"id":"1308.5948","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1308.5948","created_at":"2026-05-18T01:10:33Z"},{"alias_kind":"arxiv_version","alias_value":"1308.5948v2","created_at":"2026-05-18T01:10:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.5948","created_at":"2026-05-18T01:10:33Z"},{"alias_kind":"pith_short_12","alias_value":"DLVN3WC2PFCH","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DLVN3WC2PFCHOLER","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DLVN3WC2","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:35176b9c10100767d94470a493be57d316d7954f346d164d8899aac1bbb62ea1","target":"graph","created_at":"2026-05-18T01:10:33Z","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 a commutative Noetherian ring, $E$ a non-zero finitely generated $R$-module and $I$ an ideal of $R$. One purpose of this paper is to show that the sequences $\\Ass_RE/ \\widetilde{I_E^n}$ and $\\Ass_R\\widetilde{I^n _E}/\\widetilde{I^{n+1}_E}$, $n = 1,2, \\dots$, of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff \\cite[Theorem 3.1]{MR}. In addition, a characterization concerning the set $\\widetilde{A^*}(I,E)$ is included. A second purpose of this paper is to prove that $I$ has linear growth primary decompositions for ","authors_text":"Monireh Sedghi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-08-27T18:52:09Z","title":"Ratliff-Rush closures and linear growth of primary decompositions of ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.5948","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:d256d5974b81f4a6eaab55391624b7423a9c96d6f4bd64b34b77cf2e09b5f27f","target":"record","created_at":"2026-05-18T01:10:33Z","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":"7015ea1650851874825a8912f17ea7300dd517d09ae6502b043880087210e155","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-08-27T18:52:09Z","title_canon_sha256":"56d520dfda485604938555c63b1977805797c15f106f2dd1e664078d91cc9e64"},"schema_version":"1.0","source":{"id":"1308.5948","kind":"arxiv","version":2}},"canonical_sha256":"1aeaddd85a7944772c91265dfe62d0f475db80e005bdbd7faee798b57a9e8e74","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1aeaddd85a7944772c91265dfe62d0f475db80e005bdbd7faee798b57a9e8e74","first_computed_at":"2026-05-18T01:10:33.775853Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:33.775853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HcNTKLdVSh3nhvPWZD7p/+fZ/5mrV1myTdRsy4OrP8+eWmbSQjIs/efsItUNymgDcOvFn+qcLv0qrXSTsmcwDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:33.776463Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.5948","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d256d5974b81f4a6eaab55391624b7423a9c96d6f4bd64b34b77cf2e09b5f27f","sha256:35176b9c10100767d94470a493be57d316d7954f346d164d8899aac1bbb62ea1"],"state_sha256":"ae86d8c112b6d14c380994d6976464985c869d1f0cb7737cb5e3189642ae469d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0fzCTKmKcTsmKLrLgTDn8XRyH83vxPy9MbonTLh4nBRJHwNAv933ll+9Fus8dFlgHKKvBI8k7g+BdNoaVngNBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T03:47:30.512644Z","bundle_sha256":"f9de054fdf5be81c174479f07b90a4815e496b8ac00134d73a9df187f4c6bea2"}}