{"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"}