{"paper":{"title":"Separable functors and firm modules","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Firm modules over nonunital rings support separable functors and a locally unital Maschke theorem for group rings.","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Patrik Lundstr\\\"om","submitted_at":"2026-02-13T19:37:52Z","abstract_excerpt":"We develop a theory of separable ring extensions and separable functors for nonunital rings in the setting of firm modules. We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the category of firm modules over a nonunital ring supplies a sufficiently rich and well-behaved setting in which the classical notions of separability and semisimplicity can be defined and proved without additional hidden restrictions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Develops nonunital analogues of functorial separability and semisimplicity for firm modules and applies them to a locally unital Maschke theorem for group rings.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Firm modules over nonunital rings support separable functors and a locally unital Maschke theorem for group rings.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"759e992d396cba3f31ee8833f95212b726cffeb14c6c3da48b440969baaa0cce"},"source":{"id":"2602.13417","kind":"arxiv","version":2},"verdict":{"id":"424a5d56-5250-4ed8-867c-d8f667b5741d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T22:35:34.759045Z","strongest_claim":"We prove nonunital analogues of classical results on functorial separability and semisimplicity, and apply these results to obtain a locally unital version of Maschke's theorem for group rings.","one_line_summary":"Develops nonunital analogues of functorial separability and semisimplicity for firm modules and applies them to a locally unital Maschke theorem for group rings.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the category of firm modules over a nonunital ring supplies a sufficiently rich and well-behaved setting in which the classical notions of separability and semisimplicity can be defined and proved without additional hidden restrictions.","pith_extraction_headline":"Firm modules over nonunital rings support separable functors and a locally unital Maschke theorem for group rings."},"references":{"count":26,"sample":[{"doi":"","year":1987,"title":"P. N. ´Anh and L. M´ arki, Morita equivalence for rings without identity, Tsukuba J. Math.11(1987), 1–16","work_id":"f75d1049-93f0-47e7-ab8b-90b78c055ba5","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1946,"title":"Baer, Absolute retracts in group theory, Bull","work_id":"bf35611e-ac7a-4f1f-899c-024472d13480","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"G. B¨ ohm and J. G´ omez-Torrecillas, Firm monads and firm Frobenius algebras, Bull. Math. Soc. Sci. Math. Roumanie56(2013), 281–298","work_id":"d4044f2a-81ee-4010-80d1-4faf22e65e28","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"T. Brzezi´ nski, L. Kadison and R. Wisbauer, On coseparable and biseparable corings, in:Hopf Algebras in Noncommutative Geometry and Physics, Pure Appl. Math., vol. 239, Marcel Dekker, New York, 2005","work_id":"42f93225-ad06-4aa7-8576-280149a84bde","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"J. Cala, P. Lundstr¨ om and H. Pinedo, Object-unital groupoid graded rings, crossed products and separability, Comm. Algebra49(2021), 1676–1696","work_id":"ba6f8e09-bdb9-40e2-a99d-5e10fd007e21","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":26,"snapshot_sha256":"fb10b427e1409c485ed84fc9fc04663296a66533351acb442e8c8d356fb18d14","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"e74df066eb963f6dde2bc95e2323dab89a16302262006832a5f3d115e33f0f84"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}