{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:7FUJFEHHFOYNKAZV2PAPRBWWRA","short_pith_number":"pith:7FUJFEHH","schema_version":"1.0","canonical_sha256":"f9689290e72bb0d50335d3c0f886d6881b5796e72bc6effd3c8226bf2e37d8b4","source":{"kind":"arxiv","id":"2506.08537","version":1},"attestation_state":"computed","paper":{"title":"On ideals of product of commutative rings and their applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.RA","authors_text":"Ali Rezaie Aliabad, Foad Obeidavi, Mehdi Badie","submitted_at":"2025-06-10T07:59:41Z","abstract_excerpt":"In this paper, leveraging the recent achievements of researchers, we have revisited the family of ideals of product of commutative rings. We demonstrate that if $ \\{ R_\\alpha \\}_{\\alpha \\in A} $ is an infinite family of rings, then $ \\left| Max \\left( \\prod_{\\alpha \\in A} R_\\alpha \\right) \\right| \\geqslant 2^{2^{|A|}} $. Notably, if these rings are local then the equality holds. We establish that $ Max(R_\\alpha) $ is homeomorphic to a closed subset of $ Max \\left( \\prod_{\\alpha \\in A} R_\\alpha \\right) $, for each $ \\alpha \\in A $. Additionally, we show that $ Max(R) $ is disconnected \\ff $ R $"},"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":"2506.08537","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2025-06-10T07:59:41Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"9567ebcd72d16102bdbc7fd2a08ca7070e3c672157a08420f4690215a445db44","abstract_canon_sha256":"32f4493509d3ec04a4d21fd09228243f1f11c2f38493bf39b70ca2a042b67921"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:18:51.463115Z","signature_b64":"RLqD8vJ28sUruujLd4Kh151urcdQOp19Iy6QSDVjpRjJeJOnDOVbhYUhhpSuwCcCCz5odRL2CExrc7YWcXRuCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9689290e72bb0d50335d3c0f886d6881b5796e72bc6effd3c8226bf2e37d8b4","last_reissued_at":"2026-07-05T11:18:51.462578Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:18:51.462578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On ideals of product of commutative rings and their applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.RA","authors_text":"Ali Rezaie Aliabad, Foad Obeidavi, Mehdi Badie","submitted_at":"2025-06-10T07:59:41Z","abstract_excerpt":"In this paper, leveraging the recent achievements of researchers, we have revisited the family of ideals of product of commutative rings. We demonstrate that if $ \\{ R_\\alpha \\}_{\\alpha \\in A} $ is an infinite family of rings, then $ \\left| Max \\left( \\prod_{\\alpha \\in A} R_\\alpha \\right) \\right| \\geqslant 2^{2^{|A|}} $. Notably, if these rings are local then the equality holds. We establish that $ Max(R_\\alpha) $ is homeomorphic to a closed subset of $ Max \\left( \\prod_{\\alpha \\in A} R_\\alpha \\right) $, for each $ \\alpha \\in A $. Additionally, we show that $ Max(R) $ is disconnected \\ff $ R $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08537","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.08537/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2506.08537","created_at":"2026-07-05T11:18:51.462636+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.08537v1","created_at":"2026-07-05T11:18:51.462636+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08537","created_at":"2026-07-05T11:18:51.462636+00:00"},{"alias_kind":"pith_short_12","alias_value":"7FUJFEHHFOYN","created_at":"2026-07-05T11:18:51.462636+00:00"},{"alias_kind":"pith_short_16","alias_value":"7FUJFEHHFOYNKAZV","created_at":"2026-07-05T11:18:51.462636+00:00"},{"alias_kind":"pith_short_8","alias_value":"7FUJFEHH","created_at":"2026-07-05T11:18:51.462636+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/7FUJFEHHFOYNKAZV2PAPRBWWRA","json":"https://pith.science/pith/7FUJFEHHFOYNKAZV2PAPRBWWRA.json","graph_json":"https://pith.science/api/pith-number/7FUJFEHHFOYNKAZV2PAPRBWWRA/graph.json","events_json":"https://pith.science/api/pith-number/7FUJFEHHFOYNKAZV2PAPRBWWRA/events.json","paper":"https://pith.science/paper/7FUJFEHH"},"agent_actions":{"view_html":"https://pith.science/pith/7FUJFEHHFOYNKAZV2PAPRBWWRA","download_json":"https://pith.science/pith/7FUJFEHHFOYNKAZV2PAPRBWWRA.json","view_paper":"https://pith.science/paper/7FUJFEHH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.08537&json=true","fetch_graph":"https://pith.science/api/pith-number/7FUJFEHHFOYNKAZV2PAPRBWWRA/graph.json","fetch_events":"https://pith.science/api/pith-number/7FUJFEHHFOYNKAZV2PAPRBWWRA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7FUJFEHHFOYNKAZV2PAPRBWWRA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7FUJFEHHFOYNKAZV2PAPRBWWRA/action/storage_attestation","attest_author":"https://pith.science/pith/7FUJFEHHFOYNKAZV2PAPRBWWRA/action/author_attestation","sign_citation":"https://pith.science/pith/7FUJFEHHFOYNKAZV2PAPRBWWRA/action/citation_signature","submit_replication":"https://pith.science/pith/7FUJFEHHFOYNKAZV2PAPRBWWRA/action/replication_record"}},"created_at":"2026-07-05T11:18:51.462636+00:00","updated_at":"2026-07-05T11:18:51.462636+00:00"}