# Monads and comonads in module categories - Mathematics > Rings and Algebras

Abstract: Let $A$ be a ring and $\M A$ the category of $A$-modules. It is well known inmodule theory that for any $A$-bimodule $B$, $B$ is an $A$-ring if and only ifthe functor $-\otimes A B: \M A\to \M A$ is a monad or triple.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 categoriesof 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 rightadjoint endofunctors $\Hom AB,-$ and $\Hom A\C,-$ are a comonad and amonad, respectively, but the corresponding comodule categories did not findmuch attention so far. The category of $\Hom AB,-$-comodules is isomorphic tothe category of $B$-modules, while the category of $\Hom A\C,-$-modulescalled $\C$-contramodules by Eilenberg and Moore need not be equivalent tothe category of $\C$-comodules.The purpose of this paper is to investigate these categories and theirrelationships based on some observations of the categorical background. Thisleads to a deeper understanding and characterisations of algebraic structuressuch as corings, bialgebras and Hopf algebras. For example, it turns out thatthe categories of $\C$-comodules and $\Hom A\C,-$-modules are equivalentprovided $\C$ is a coseparable coring. Furthermore, a bialgebra$H$ over a commutative ring $R$ is a Hopf algebra if and only if $\Hom RH-$is a Hopf bimonad on $\M R$ and in this case the categories of $H$-Hopf modulesand mixed $\Hom RH,-$-bimodules are both equivalent to $\M R$.

Author: Gabriella Böhm, Tomasz Brzezinski, Robert Wisbauer

Source: https://arxiv.org/