{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6VFGXCZ4PJ7VHP3BBOYY7BN2PE","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":"c22312584aa99527a606a2b7b5cfcebe4e91bf44a68125a49b81d59255902baf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-05-25T08:33:49Z","title_canon_sha256":"f8826ef90d04d7f6bdc51828e8c5dfce19bfe1faf6e54c404b0d15a0483936a6"},"schema_version":"1.0","source":{"id":"1505.06564","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.06564","created_at":"2026-05-18T02:03:44Z"},{"alias_kind":"arxiv_version","alias_value":"1505.06564v1","created_at":"2026-05-18T02:03:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06564","created_at":"2026-05-18T02:03:44Z"},{"alias_kind":"pith_short_12","alias_value":"6VFGXCZ4PJ7V","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6VFGXCZ4PJ7VHP3B","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6VFGXCZ4","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:f80dcade55e8466939b6ed31e83229c104c45bc7d45544b063fa20831cbc4b05","target":"graph","created_at":"2026-05-18T02:03:44Z","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 article, 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 $bm\\in N$. We introduce the concept of \"classical 2-absorbing submodules\" as a generalization of \"classical prime submodules.\" We say that a proper submodule $N$ of $M$ is a classical 2-absorbing submodule if whenever $a,b,c\\in R$ and $m\\in M$ with $abcm\\in N$, then $abm\\in N$ or $acm\\in N$ or $bcm\\in N$.","authors_text":"Hojjat Mostafanasab, Kursat Hakan Oral, Unsal Tekir","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-05-25T08:33:49Z","title":"Classical 2-absorbing submodules of modules over commutative rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06564","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":"sha256:7be54c86822da8c3eeee7ee1b4b494673a8287288709d3f2af513afb48c2d3bc","target":"record","created_at":"2026-05-18T02:03:44Z","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":"c22312584aa99527a606a2b7b5cfcebe4e91bf44a68125a49b81d59255902baf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-05-25T08:33:49Z","title_canon_sha256":"f8826ef90d04d7f6bdc51828e8c5dfce19bfe1faf6e54c404b0d15a0483936a6"},"schema_version":"1.0","source":{"id":"1505.06564","kind":"arxiv","version":1}},"canonical_sha256":"f54a6b8b3c7a7f53bf610bb18f85ba7923146a82b9eaddc92a8e3bf5830894f9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f54a6b8b3c7a7f53bf610bb18f85ba7923146a82b9eaddc92a8e3bf5830894f9","first_computed_at":"2026-05-18T02:03:44.592834Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:03:44.592834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bENvJmnhmWmaaYWZRkFZMY4k9Wev/ceGwQCCemRDEHINqXgwozS3SlM9ZEMp6EED8Ba2IsSbz69NhcPN1rqJAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:03:44.593352Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.06564","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7be54c86822da8c3eeee7ee1b4b494673a8287288709d3f2af513afb48c2d3bc","sha256:f80dcade55e8466939b6ed31e83229c104c45bc7d45544b063fa20831cbc4b05"],"state_sha256":"fde3d6aff2553b3e19b50bbd1d5ffff0f9dce08ee7238241df8d7292550f4e40"}