{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:KBO4G3TGFRB6LYCIABLZAVX62H","short_pith_number":"pith:KBO4G3TG","schema_version":"1.0","canonical_sha256":"505dc36e662c43e5e04800579056fed1d3141621f5cba9d1a2cfe666d666ae2f","source":{"kind":"arxiv","id":"1507.08981","version":1},"attestation_state":"computed","paper":{"title":"$\\phi$-classical prime submodules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Esra Sengelen Sevim, Hojjat Mostafanasab, Sakineh Babaei, Unsal Tekir","submitted_at":"2015-07-30T12:43:16Z","abstract_excerpt":"In this paper, all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a classical prime submodule, if for each $m\\in M$ and elements $a,b\\in R$, $abm\\in N$ implies that $am\\in N$ or $bm\\in N$. Let $\\phi:S(M)\\to S(M)\\cup{\\emptyset}$ be a function where $S(M)$ is the set of all submodules of $M$. We introduce the concept of \"$\\phi$-classical prime submodules\". A proper submodule $N$ of $M$ is a $\\phi$-classical prime submodule if whenever $a,b\\in R$ and $m\\in M$ with $abm\\in N\\backslash\\phi(N)$, 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":"1507.08981","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-07-30T12:43:16Z","cross_cats_sorted":[],"title_canon_sha256":"203f2e2b6a7dc348f5300e7125a0f6db3a5c5ef89f4700bc2d3ae1fb3515a2aa","abstract_canon_sha256":"285db8c77929b13ca8fa99b214835b90c63b5c90963c05e7267e0361a477c2ce"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:02.473302Z","signature_b64":"x6BKRKFMyTKvF5pV+7a0SkSvZAb5g1m8qd55eO4lFqgWwyq1Vuz2lHLInTEM7JGmlRkexDCrSZHKdAW98DxWAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"505dc36e662c43e5e04800579056fed1d3141621f5cba9d1a2cfe666d666ae2f","last_reissued_at":"2026-05-18T01:36:02.472845Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:02.472845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\phi$-classical prime submodules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Esra Sengelen Sevim, Hojjat Mostafanasab, Sakineh Babaei, Unsal Tekir","submitted_at":"2015-07-30T12:43:16Z","abstract_excerpt":"In this paper, all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a classical prime submodule, if for each $m\\in M$ and elements $a,b\\in R$, $abm\\in N$ implies that $am\\in N$ or $bm\\in N$. Let $\\phi:S(M)\\to S(M)\\cup{\\emptyset}$ be a function where $S(M)$ is the set of all submodules of $M$. We introduce the concept of \"$\\phi$-classical prime submodules\". A proper submodule $N$ of $M$ is a $\\phi$-classical prime submodule if whenever $a,b\\in R$ and $m\\in M$ with $abm\\in N\\backslash\\phi(N)$, then $am\\in N$ or $bm\\in N$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08981","kind":"arxiv","version":1},"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":"1507.08981","created_at":"2026-05-18T01:36:02.472920+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.08981v1","created_at":"2026-05-18T01:36:02.472920+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08981","created_at":"2026-05-18T01:36:02.472920+00:00"},{"alias_kind":"pith_short_12","alias_value":"KBO4G3TGFRB6","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"KBO4G3TGFRB6LYCI","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"KBO4G3TG","created_at":"2026-05-18T12:29:27.538025+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/KBO4G3TGFRB6LYCIABLZAVX62H","json":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H.json","graph_json":"https://pith.science/api/pith-number/KBO4G3TGFRB6LYCIABLZAVX62H/graph.json","events_json":"https://pith.science/api/pith-number/KBO4G3TGFRB6LYCIABLZAVX62H/events.json","paper":"https://pith.science/paper/KBO4G3TG"},"agent_actions":{"view_html":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H","download_json":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H.json","view_paper":"https://pith.science/paper/KBO4G3TG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.08981&json=true","fetch_graph":"https://pith.science/api/pith-number/KBO4G3TGFRB6LYCIABLZAVX62H/graph.json","fetch_events":"https://pith.science/api/pith-number/KBO4G3TGFRB6LYCIABLZAVX62H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H/action/storage_attestation","attest_author":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H/action/author_attestation","sign_citation":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H/action/citation_signature","submit_replication":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H/action/replication_record"}},"created_at":"2026-05-18T01:36:02.472920+00:00","updated_at":"2026-05-18T01:36:02.472920+00:00"}