{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:MTPXUWK3NRIE7IP6RW273OQU7L","short_pith_number":"pith:MTPXUWK3","canonical_record":{"source":{"id":"1508.07518","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-30T00:39:02Z","cross_cats_sorted":[],"title_canon_sha256":"21211945594121f65800f2814634d0c4374d63cf521f92c7e1f1fb473bf6a2f9","abstract_canon_sha256":"c43b479e575c223943abae695ae9a436d7bdcbf1b7e0e9f6ad598f6044e9ca50"},"schema_version":"1.0"},"canonical_sha256":"64df7a595b6c504fa1fe8db5fdba14fadef894715e2be296dea89612c36c2e86","source":{"kind":"arxiv","id":"1508.07518","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.07518","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"arxiv_version","alias_value":"1508.07518v2","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.07518","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"pith_short_12","alias_value":"MTPXUWK3NRIE","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MTPXUWK3NRIE7IP6","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MTPXUWK3","created_at":"2026-05-18T12:29:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:MTPXUWK3NRIE7IP6RW273OQU7L","target":"record","payload":{"canonical_record":{"source":{"id":"1508.07518","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-30T00:39:02Z","cross_cats_sorted":[],"title_canon_sha256":"21211945594121f65800f2814634d0c4374d63cf521f92c7e1f1fb473bf6a2f9","abstract_canon_sha256":"c43b479e575c223943abae695ae9a436d7bdcbf1b7e0e9f6ad598f6044e9ca50"},"schema_version":"1.0"},"canonical_sha256":"64df7a595b6c504fa1fe8db5fdba14fadef894715e2be296dea89612c36c2e86","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:36.849731Z","signature_b64":"996GR0FaTEont4yjfxJsbUdOk6C1/+DYE6P9QvigPiI3j+yVvypvrhZXPz5+/3XTNULAMPuosZJWbe/bY0kkCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64df7a595b6c504fa1fe8db5fdba14fadef894715e2be296dea89612c36c2e86","last_reissued_at":"2026-05-18T01:03:36.849177Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:36.849177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.07518","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":"2026-05-18T01:03:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hgRekS9kSJvI6LFAnsfXS5N0bcLdjSg4OQL9siJcC2peLSn/OLkTjiEivb/IRa8QGDqakA718F6ao8DPf92nBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T11:02:57.029958Z"},"content_sha256":"95ddaa64b0d21ec46750323f57e00e5d33c1b17ac0a99a40dff5f3bd31c9e094","schema_version":"1.0","event_id":"sha256:95ddaa64b0d21ec46750323f57e00e5d33c1b17ac0a99a40dff5f3bd31c9e094"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:MTPXUWK3NRIE7IP6RW273OQU7L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graded-irreducible modules are irreducible","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Justin Chen, Youngsu Kim","submitted_at":"2015-08-30T00:39:02Z","abstract_excerpt":"We show that if a graded submodule of a Noetherian module cannot be written as a proper intersection of graded submodules, then it cannot be written as a proper intersection of submodules at all. More generally, we show that a natural extension of the index of reducibility to the graded setting coincides with the ordinary index of reducibility. We also investigate the question of uniqueness of the components in a graded-irreducible decomposition, as well as the relation between the index of reducibility of a non-graded ideal and that of its largest graded subideal."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07518","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":"2026-05-18T01:03:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UHES2tBoVbdLGFFFZdYdlwenue0oY1Ssjg2ewfYN6GgjKgih9X8i3qsepDaTywmbRNadPW1FH1Mam6h21N2tAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T11:02:57.030637Z"},"content_sha256":"d3779d12231beec8831e7efc008518d8296e797b1a66afed08f228351e99bd2a","schema_version":"1.0","event_id":"sha256:d3779d12231beec8831e7efc008518d8296e797b1a66afed08f228351e99bd2a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MTPXUWK3NRIE7IP6RW273OQU7L/bundle.json","state_url":"https://pith.science/pith/MTPXUWK3NRIE7IP6RW273OQU7L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MTPXUWK3NRIE7IP6RW273OQU7L/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":"2026-05-31T11:02:57Z","links":{"resolver":"https://pith.science/pith/MTPXUWK3NRIE7IP6RW273OQU7L","bundle":"https://pith.science/pith/MTPXUWK3NRIE7IP6RW273OQU7L/bundle.json","state":"https://pith.science/pith/MTPXUWK3NRIE7IP6RW273OQU7L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MTPXUWK3NRIE7IP6RW273OQU7L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:MTPXUWK3NRIE7IP6RW273OQU7L","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":"c43b479e575c223943abae695ae9a436d7bdcbf1b7e0e9f6ad598f6044e9ca50","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-30T00:39:02Z","title_canon_sha256":"21211945594121f65800f2814634d0c4374d63cf521f92c7e1f1fb473bf6a2f9"},"schema_version":"1.0","source":{"id":"1508.07518","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.07518","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"arxiv_version","alias_value":"1508.07518v2","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.07518","created_at":"2026-05-18T01:03:36Z"},{"alias_kind":"pith_short_12","alias_value":"MTPXUWK3NRIE","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"MTPXUWK3NRIE7IP6","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"MTPXUWK3","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:d3779d12231beec8831e7efc008518d8296e797b1a66afed08f228351e99bd2a","target":"graph","created_at":"2026-05-18T01:03:36Z","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":"We show that if a graded submodule of a Noetherian module cannot be written as a proper intersection of graded submodules, then it cannot be written as a proper intersection of submodules at all. More generally, we show that a natural extension of the index of reducibility to the graded setting coincides with the ordinary index of reducibility. We also investigate the question of uniqueness of the components in a graded-irreducible decomposition, as well as the relation between the index of reducibility of a non-graded ideal and that of its largest graded subideal.","authors_text":"Justin Chen, Youngsu Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-30T00:39:02Z","title":"Graded-irreducible modules are irreducible"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07518","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":"sha256:95ddaa64b0d21ec46750323f57e00e5d33c1b17ac0a99a40dff5f3bd31c9e094","target":"record","created_at":"2026-05-18T01:03:36Z","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":"c43b479e575c223943abae695ae9a436d7bdcbf1b7e0e9f6ad598f6044e9ca50","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-08-30T00:39:02Z","title_canon_sha256":"21211945594121f65800f2814634d0c4374d63cf521f92c7e1f1fb473bf6a2f9"},"schema_version":"1.0","source":{"id":"1508.07518","kind":"arxiv","version":2}},"canonical_sha256":"64df7a595b6c504fa1fe8db5fdba14fadef894715e2be296dea89612c36c2e86","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64df7a595b6c504fa1fe8db5fdba14fadef894715e2be296dea89612c36c2e86","first_computed_at":"2026-05-18T01:03:36.849177Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:36.849177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"996GR0FaTEont4yjfxJsbUdOk6C1/+DYE6P9QvigPiI3j+yVvypvrhZXPz5+/3XTNULAMPuosZJWbe/bY0kkCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:36.849731Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.07518","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95ddaa64b0d21ec46750323f57e00e5d33c1b17ac0a99a40dff5f3bd31c9e094","sha256:d3779d12231beec8831e7efc008518d8296e797b1a66afed08f228351e99bd2a"],"state_sha256":"30c660105d8a33942a405db195822ce138ac712229d94fe713f2b13d104bf920"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vRECEOA+dgmLPNXY9J/n32xS2naBeYAflcJ2W3OC6Gsm88Njq5buIJei8upIvM0rVAnZe1Bzk4nOWOULZ8JqBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T11:02:57.034466Z","bundle_sha256":"fefb5cac7ff5939b48778b18e54f17650253f94981d5c9d4514f6d521e5f4dbe"}}