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Then the main results of this paper are: (1) $\\mathrm{ir}_M(N) = \\sum_{{\\frak p} \\in \\mathrm{Ass}_R(M/N)} \\dim_{k(\\frak p)} \\mathrm{Soc}(M/N)_{\\frak p} $; (2) For an irredundant primary decomposition of $N = Q_1 \\cap \\cdots \\cap Q_n$, where $Q_i$ is $\\frak p_i$-primary, then $\\mathrm{ir}_M(N) = \\mathrm{i"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1405.1136","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-05-06T03:09:32Z","cross_cats_sorted":[],"title_canon_sha256":"90639af02cd637e31af9902650b882fc9d22d485cf3004ac1d8fb77c7c2af1fe","abstract_canon_sha256":"47ab139f1575ef5a98fc2aec4824b460dcfb3ce8e23f22486278c9e8ec895c03"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:08.497246Z","signature_b64":"UBTOjePMjoZ9Hpzwgu5yPOTGOTG031v90hZGqJ4yrQcC4uE1y9hNd7fbJaXNf6sVW5Kj95zTKjqwXNTEpi9QAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4678c67e9c453508b5ee0dddb3c49367d352a96b72bd224ff1cedb6194090d73","last_reissued_at":"2026-05-18T02:19:08.496768Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:08.496768Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the index of reducibility in Noetherian modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hoang Le Truong, Nguyen Tu Cuong, Pham Hung Quy","submitted_at":"2014-05-06T03:09:32Z","abstract_excerpt":"Let $M$ be a finitely generated module over a Noetherian ring $R$ and $N$ a submodule. The index of reducibility ir$_M(N)$ is the number of irreducible submodules that appear in an irredundant irreducible decomposition of $N$ (this number is well defined by a classical result of Emmy Noether). Then the main results of this paper are: (1) $\\mathrm{ir}_M(N) = \\sum_{{\\frak p} \\in \\mathrm{Ass}_R(M/N)} \\dim_{k(\\frak p)} \\mathrm{Soc}(M/N)_{\\frak p} $; (2) For an irredundant primary decomposition of $N = Q_1 \\cap \\cdots \\cap Q_n$, where $Q_i$ is $\\frak p_i$-primary, then $\\mathrm{ir}_M(N) = \\mathrm{i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1136","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1405.1136","created_at":"2026-05-18T02:19:08.496844+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.1136v2","created_at":"2026-05-18T02:19:08.496844+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1136","created_at":"2026-05-18T02:19:08.496844+00:00"},{"alias_kind":"pith_short_12","alias_value":"IZ4MM7U4IU2Q","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"IZ4MM7U4IU2QRNPO","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"IZ4MM7U4","created_at":"2026-05-18T12:28:33.132498+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IZ4MM7U4IU2QRNPOBXO3HRETM7","json":"https://pith.science/pith/IZ4MM7U4IU2QRNPOBXO3HRETM7.json","graph_json":"https://pith.science/api/pith-number/IZ4MM7U4IU2QRNPOBXO3HRETM7/graph.json","events_json":"https://pith.science/api/pith-number/IZ4MM7U4IU2QRNPOBXO3HRETM7/events.json","paper":"https://pith.science/paper/IZ4MM7U4"},"agent_actions":{"view_html":"https://pith.science/pith/IZ4MM7U4IU2QRNPOBXO3HRETM7","download_json":"https://pith.science/pith/IZ4MM7U4IU2QRNPOBXO3HRETM7.json","view_paper":"https://pith.science/paper/IZ4MM7U4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.1136&json=true","fetch_graph":"https://pith.science/api/pith-number/IZ4MM7U4IU2QRNPOBXO3HRETM7/graph.json","fetch_events":"https://pith.science/api/pith-number/IZ4MM7U4IU2QRNPOBXO3HRETM7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IZ4MM7U4IU2QRNPOBXO3HRETM7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IZ4MM7U4IU2QRNPOBXO3HRETM7/action/storage_attestation","attest_author":"https://pith.science/pith/IZ4MM7U4IU2QRNPOBXO3HRETM7/action/author_attestation","sign_citation":"https://pith.science/pith/IZ4MM7U4IU2QRNPOBXO3HRETM7/action/citation_signature","submit_replication":"https://pith.science/pith/IZ4MM7U4IU2QRNPOBXO3HRETM7/action/replication_record"}},"created_at":"2026-05-18T02:19:08.496844+00:00","updated_at":"2026-05-18T02:19:08.496844+00:00"}