{"paper":{"title":"Monads and comonads in module categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.RA","authors_text":"Gabriella B\\\"ohm, Robert Wisbauer, Tomasz Brzezinski","submitted_at":"2008-04-09T11:24:37Z","abstract_excerpt":"Let $A$ be a ring and $\\M_A$ the category of $A$-modules. It is well known in module theory that for any $A $-bimodule $B$, $B$ is an $A$-ring if and only if the functor $-\\otimes_A B: \\M_A\\to \\M_A$ is a monad (or triple).\n  Similarly, an $A $-bimodule $\\C$ is an $A$-coring provided the functor $-\\otimes_A\\C:\\M_A\\to \\M_A$ is a comonad (or cotriple). The related categories of modules (or algebras) of $-\\otimes_A B$ and comodules (or coalgebras) of $-\\otimes_A\\C$ are well studied in the literature. On the other hand, the right adjoint endofunctors $\\Hom_A(B,-)$ and $\\Hom_A(\\C,-)$ are a comonad a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.1460","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"}