{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:KBO4G3TGFRB6LYCIABLZAVX62H","short_pith_number":"pith:KBO4G3TG","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"},"canonical_sha256":"505dc36e662c43e5e04800579056fed1d3141621f5cba9d1a2cfe666d666ae2f","source":{"kind":"arxiv","id":"1507.08981","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08981","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08981v1","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08981","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"pith_short_12","alias_value":"KBO4G3TGFRB6","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KBO4G3TGFRB6LYCI","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KBO4G3TG","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:KBO4G3TGFRB6LYCIABLZAVX62H","target":"record","payload":{"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"},"canonical_sha256":"505dc36e662c43e5e04800579056fed1d3141621f5cba9d1a2cfe666d666ae2f","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"},"source_kind":"arxiv","source_id":"1507.08981","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RajRnF2fYa8Rbo2N48p8dxaGNHLHr3Fbx43WvApCu/Gxm1qury4HU/HoRB15E1JWi6zJ8lfol/Um3T2mnmZ3Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T14:52:47.395555Z"},"content_sha256":"68baedd513aec3d038eb308761463ab937f04c4d56ee876fd2221be5cf668522","schema_version":"1.0","event_id":"sha256:68baedd513aec3d038eb308761463ab937f04c4d56ee876fd2221be5cf668522"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:KBO4G3TGFRB6LYCIABLZAVX62H","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yCWfR5KQIjeKFv3sNC6+v0MmCpL7mx0IhQVuR8/ToP8dmils4Qsk/qolfUlEH4Qsw5HaQv+YhFqfZWIGqk6ICg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T14:52:47.396355Z"},"content_sha256":"d424710cbbcf6ca07d75c5a65aec0bf41a87cc18c2b118da687bfdbb027b902d","schema_version":"1.0","event_id":"sha256:d424710cbbcf6ca07d75c5a65aec0bf41a87cc18c2b118da687bfdbb027b902d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H/bundle.json","state_url":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KBO4G3TGFRB6LYCIABLZAVX62H/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":"2026-05-31T14:52:47Z","links":{"resolver":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H","bundle":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H/bundle.json","state":"https://pith.science/pith/KBO4G3TGFRB6LYCIABLZAVX62H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KBO4G3TGFRB6LYCIABLZAVX62H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:KBO4G3TGFRB6LYCIABLZAVX62H","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":"285db8c77929b13ca8fa99b214835b90c63b5c90963c05e7267e0361a477c2ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-07-30T12:43:16Z","title_canon_sha256":"203f2e2b6a7dc348f5300e7125a0f6db3a5c5ef89f4700bc2d3ae1fb3515a2aa"},"schema_version":"1.0","source":{"id":"1507.08981","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.08981","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"arxiv_version","alias_value":"1507.08981v1","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.08981","created_at":"2026-05-18T01:36:02Z"},{"alias_kind":"pith_short_12","alias_value":"KBO4G3TGFRB6","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"KBO4G3TGFRB6LYCI","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"KBO4G3TG","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:d424710cbbcf6ca07d75c5a65aec0bf41a87cc18c2b118da687bfdbb027b902d","target":"graph","created_at":"2026-05-18T01:36:02Z","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":"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 $bm\\in N$.","authors_text":"Esra Sengelen Sevim, Hojjat Mostafanasab, Sakineh Babaei, Unsal Tekir","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-07-30T12:43:16Z","title":"$\\phi$-classical prime submodules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08981","kind":"arxiv","version":1},"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":"sha256:68baedd513aec3d038eb308761463ab937f04c4d56ee876fd2221be5cf668522","target":"record","created_at":"2026-05-18T01:36:02Z","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":"285db8c77929b13ca8fa99b214835b90c63b5c90963c05e7267e0361a477c2ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-07-30T12:43:16Z","title_canon_sha256":"203f2e2b6a7dc348f5300e7125a0f6db3a5c5ef89f4700bc2d3ae1fb3515a2aa"},"schema_version":"1.0","source":{"id":"1507.08981","kind":"arxiv","version":1}},"canonical_sha256":"505dc36e662c43e5e04800579056fed1d3141621f5cba9d1a2cfe666d666ae2f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"505dc36e662c43e5e04800579056fed1d3141621f5cba9d1a2cfe666d666ae2f","first_computed_at":"2026-05-18T01:36:02.472845Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:02.472845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x6BKRKFMyTKvF5pV+7a0SkSvZAb5g1m8qd55eO4lFqgWwyq1Vuz2lHLInTEM7JGmlRkexDCrSZHKdAW98DxWAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:02.473302Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.08981","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:68baedd513aec3d038eb308761463ab937f04c4d56ee876fd2221be5cf668522","sha256:d424710cbbcf6ca07d75c5a65aec0bf41a87cc18c2b118da687bfdbb027b902d"],"state_sha256":"06d5227e0a4a22fe7e257456800b1412757a3c63324a4e627dc845d4c8421b03"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yRzSnKxPHEl6kQMJtld2U1y0qrBfKDCFlXinAWqCfPluaPIiYSisO541fHK1oOAQVGG03dHqec9yCEmkHHEyCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T14:52:47.400854Z","bundle_sha256":"5aa9fd75e6bae5c979d0ce9dee476c08d2cfcd0c8bcb316ba2232d4da5fedd9d"}}