Idempotent monads and ☆-functors

For an associative ring R, let P be an R-module with S = End(R)(P). C. Menini and A. Orsatti posed the question of when the related functor Hom(R)(P, -) (with left adjoint P circle times(s) -) induces an equivalence between a subcategory of M-R closed under factor modules and a subcategory of M-S closed under submodules. They observed that this is precisely the case if the unit of the adjunction is an epimorphism and the counit is a monomorphism. A module P inducing these properties is called a star-module. The purpose of this paper is to consider the corresponding question for a functor G : B -> A between arbitrary categories. We call G a star-functor if it has a left adjoint F : A -> B such that the unit of the adjunction is an extrema( epimorphism and the counit is an extremal monommphism. In this case (F, G) is an idempotent pair of functors and induces an equivalence between the category A(GF) of modules for the monad CF and the category B-FG of comodules for the comonad FG. Moreover, B-FG = Fix(FG) is closed under factor objects in B, A(GF) = Fix(GF) is closed under subobjects in A. (C) 2010 Elsevier B.V. All rights reserved.