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We introduce the concept of \"weakly classical prime submodules.\" A proper submodule $N$ of $M$ is a weakly classical prime submodule if whenever $a,b\\in R$ and $m\\in M$ with $0\\neq abm\\in N$, then $am\\in N$ or $bm\\in N$."},"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":"1505.06730","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-05-25T12:32:28Z","cross_cats_sorted":[],"title_canon_sha256":"dbe93b20ae2584d89cd385e6a8567fbabe1a0f78f6bd51e1ab9f563a0dd083e4","abstract_canon_sha256":"51c7e5820acdbf0924bdd83315e8d052704789e83f5e49a7593925b7e4e11755"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:23.597845Z","signature_b64":"vbyGXSWh9ECaMopJwjeVwEOwGpPtMmIkrsEjZXSwcreJ0JZyB3Jqtvmpnq0ATNSH4B370WuIO2llrWgz1IFrCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebef0bc908129f5b7541e1ed23fd9badb99f22a31c6d7fccc26d170608e5a57d","last_reissued_at":"2026-05-18T01:15:23.597230Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:23.597230Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weakly classical prime submodules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Hojjat Mostafanasab, Kursat Hakan Oral, Unsal Tekir","submitted_at":"2015-05-25T12:32:28Z","abstract_excerpt":"In this paper, all rings are commutative with nonzero identity. 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We introduce the concept of \"weakly classical prime submodules.\" A proper submodule $N$ of $M$ is a weakly classical prime submodule if whenever $a,b\\in R$ and $m\\in M$ with $0\\neq abm\\in N$, then $am\\in N$ or $bm\\in N$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06730","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":"1505.06730","created_at":"2026-05-18T01:15:23.597325+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.06730v2","created_at":"2026-05-18T01:15:23.597325+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06730","created_at":"2026-05-18T01:15:23.597325+00:00"},{"alias_kind":"pith_short_12","alias_value":"5PXQXSIICKPV","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"5PXQXSIICKPVW5KB","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"5PXQXSII","created_at":"2026-05-18T12:29:05.191682+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/5PXQXSIICKPVW5KB4HWSH7M3VW","json":"https://pith.science/pith/5PXQXSIICKPVW5KB4HWSH7M3VW.json","graph_json":"https://pith.science/api/pith-number/5PXQXSIICKPVW5KB4HWSH7M3VW/graph.json","events_json":"https://pith.science/api/pith-number/5PXQXSIICKPVW5KB4HWSH7M3VW/events.json","paper":"https://pith.science/paper/5PXQXSII"},"agent_actions":{"view_html":"https://pith.science/pith/5PXQXSIICKPVW5KB4HWSH7M3VW","download_json":"https://pith.science/pith/5PXQXSIICKPVW5KB4HWSH7M3VW.json","view_paper":"https://pith.science/paper/5PXQXSII","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.06730&json=true","fetch_graph":"https://pith.science/api/pith-number/5PXQXSIICKPVW5KB4HWSH7M3VW/graph.json","fetch_events":"https://pith.science/api/pith-number/5PXQXSIICKPVW5KB4HWSH7M3VW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5PXQXSIICKPVW5KB4HWSH7M3VW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5PXQXSIICKPVW5KB4HWSH7M3VW/action/storage_attestation","attest_author":"https://pith.science/pith/5PXQXSIICKPVW5KB4HWSH7M3VW/action/author_attestation","sign_citation":"https://pith.science/pith/5PXQXSIICKPVW5KB4HWSH7M3VW/action/citation_signature","submit_replication":"https://pith.science/pith/5PXQXSIICKPVW5KB4HWSH7M3VW/action/replication_record"}},"created_at":"2026-05-18T01:15:23.597325+00:00","updated_at":"2026-05-18T01:15:23.597325+00:00"}