r-Costar Pair of Contravariant Functors

We generalize r-costar module to r-costar pair of contravariant functors between abelian categories. 1. Introduction For a ring , a fixed right -module , and , let fgd-tl( ) denote the class of torsionless right -modules whose -dual are finitely generated over and fg-tl ( ) denote the class of finitely generated torsionless left -modules. is called costar module if is a duality. Costar modules were introduced by Colby and Fuller in [1]. is said to be an r-costar module provided that any exact sequence such that and are -reflexive, remains exact after applying the functor if and only if is -reflexive. The notion of r-costar module was introduced by Liu and Zhang in [2]. We say that a right -module is -finitely -copresented if there exists a long exact sequence such that are positive integers for . The class of all -finitely -copresented modules is denoted by - . We say that a right -module is a finitistic -self-cotilting module provided that any exact sequence such that - and is a positive integer, remains exact after applying the functor and - - . Finitistic -self-cotilting modules were introduced by Breaz in [3]. In [4] Casta？o-Iglesias generalizes the notion of costar module to Grothendieck categories. Pop in [5] generalizes the notion of finitistic -self-cotilting module to finitistic - -cotilting object in abelian categories and he describes a family of dualities between abelian categories. Breaz and Pop in [6] generalize a duality exhibited in [3, Theorem 2.8] to abelian categories. In this work we continue this kind of study and generalizes the notion of r-costar module to r-costar pair of contravariant functors between abelian categories, by generalizing the work in [2]. We use the same technique of proofs of that paper. 2. Preliminaries Let and be additive and contravariant left exact functors between two abelian categories and . It is said that they are adjoint on the right if there are natural isomorphisms for and . Then they induce two natural transformations and defined by and . Moreover the following identities are satisfied for each and : The pair is called a duality if there are functorial isomorphisms and . An object of ( ) is called -reflexive (resp., -reflexive) in case (resp., ) is an isomorphism. By we will denote the full subcategory of all -reflexive objects. As well by we will denote the full subcategory of all -reflexive objects. It is clear that the functors and induce a duality between the categories and . We say that the pair of left exact contravariant functors is r-costar provided that any exact sequence with remains exact