{"paper":{"title":"Adjoining Idempotents to a Commutative Ring preprint version","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"R. Raphael, W.D. Burgess","submitted_at":"2026-06-19T22:05:04Z","abstract_excerpt":"Everything takes place in the category of commutative unitary rings. For a fixed ring $R$, $\\alg{R}$ is the class of $R$-algebras and $\\igr{R}$ the subclass of idempotent generated $R$-algebras. Following Bezhanishvili et al and their study of Specker and locally Specker $R$-algebras, this paper studies the interplay of properties of $R$ and $A\\in \\igr{R}$ (both as rings and as $R$-modules). Examples: (1) If $R\\sbq A\\in \\igr{R}$ and $R$ is weak Baer (aka p.p.\\ ring) and $A$ is ring essential over $R$, then $A$ is weak Baer and locally Specker. (2) If $R$ is semiprime and all the idempotents of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21782","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/2606.21782/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"}