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The annihilating-ideal graph of $R$ is defined as the graph ${\\Bbb{AG}}(R)$ with the vertex set ${\\Bbb{A}}(R)^*={\\Bbb{A}}\\setminus\\{(0)\\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We investigate commutative rings $R$ whose annihilating-ideal graphs have positive genus $\\gamma(\\Bbb{AG}(R))$. It is shown that if $R$ is an Artinian ring such that $\\gamma(\\Bbb{AG}(R))<\\infty$, then $R$ has finitely many ideals or $(R,\\mathfrak{m})$ is a Gorenstein ring with maximal 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":"1102.4835","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-02-23T20:21:22Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"8e96fe5f166a5daaef7c5660fb10479881c6a63df0a3d23169e1474ae940e4c2","abstract_canon_sha256":"e2962ac3981c8f8522557dc2bc1bb7d0dd769c2a5e3f0981bdbbbe0bf9812c4d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:07.238866Z","signature_b64":"7Fyc569xcy4tQqBX+BemzK7Aj2SxgMa/36XEv0MYBhLujmuyUYtJgX5tQMVPWxxKQC4ST596gq1D7JXXO+4lDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebbfb70932f12d23af99ee0b89ac5ef43ba3fff1711f150b701111f725b52b61","last_reissued_at":"2026-05-18T04:28:07.238298Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:07.238298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rings Whose Annihilating-Ideal Graphs Have Positive Genus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Farid Aliniaeifard, Mahmood Behboodi","submitted_at":"2011-02-23T20:21:22Z","abstract_excerpt":"Let $R$ be a commutative ring and ${\\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\\Bbb{AG}}(R)$ with the vertex set ${\\Bbb{A}}(R)^*={\\Bbb{A}}\\setminus\\{(0)\\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We investigate commutative rings $R$ whose annihilating-ideal graphs have positive genus $\\gamma(\\Bbb{AG}(R))$. It is shown that if $R$ is an Artinian ring such that $\\gamma(\\Bbb{AG}(R))<\\infty$, then $R$ has finitely many ideals or $(R,\\mathfrak{m})$ is a Gorenstein ring with maximal i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.4835","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":"1102.4835","created_at":"2026-05-18T04:28:07.238395+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.4835v1","created_at":"2026-05-18T04:28:07.238395+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.4835","created_at":"2026-05-18T04:28:07.238395+00:00"},{"alias_kind":"pith_short_12","alias_value":"5O73OCJS6EWS","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"5O73OCJS6EWSHL4Z","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"5O73OCJS","created_at":"2026-05-18T12:26:20.644004+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/5O73OCJS6EWSHL4Z5YFYTLC66Q","json":"https://pith.science/pith/5O73OCJS6EWSHL4Z5YFYTLC66Q.json","graph_json":"https://pith.science/api/pith-number/5O73OCJS6EWSHL4Z5YFYTLC66Q/graph.json","events_json":"https://pith.science/api/pith-number/5O73OCJS6EWSHL4Z5YFYTLC66Q/events.json","paper":"https://pith.science/paper/5O73OCJS"},"agent_actions":{"view_html":"https://pith.science/pith/5O73OCJS6EWSHL4Z5YFYTLC66Q","download_json":"https://pith.science/pith/5O73OCJS6EWSHL4Z5YFYTLC66Q.json","view_paper":"https://pith.science/paper/5O73OCJS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.4835&json=true","fetch_graph":"https://pith.science/api/pith-number/5O73OCJS6EWSHL4Z5YFYTLC66Q/graph.json","fetch_events":"https://pith.science/api/pith-number/5O73OCJS6EWSHL4Z5YFYTLC66Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5O73OCJS6EWSHL4Z5YFYTLC66Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5O73OCJS6EWSHL4Z5YFYTLC66Q/action/storage_attestation","attest_author":"https://pith.science/pith/5O73OCJS6EWSHL4Z5YFYTLC66Q/action/author_attestation","sign_citation":"https://pith.science/pith/5O73OCJS6EWSHL4Z5YFYTLC66Q/action/citation_signature","submit_replication":"https://pith.science/pith/5O73OCJS6EWSHL4Z5YFYTLC66Q/action/replication_record"}},"created_at":"2026-05-18T04:28:07.238395+00:00","updated_at":"2026-05-18T04:28:07.238395+00:00"}