GENERALIZED MATLIS DUALITY AND LINEAR COMPACTNESS

Abstract

ABSTRACT Let R be a commutative ring. We show that a finitely generated R-module M is reflexive (equivalently, linearly compact) if and only if R/annR(M) is reflexive (equivalently, linearly compact). If R is a quasi-noetherian ring, we prove that an R-module M is reflexive if and only if R/annR(M) is linearly compact and M has a finitely generated submodule N such that M/N is finitely cogenerated. These properly generalize results of Belshoff, Enochs and Garcia Rozas. A ring extension S of R is considered and we show that under some minor condition an S-module is reflexive if and only if it is reflexive as an R-module. Throughout this paper, rings are commutative with identity and modules are unitary. The terminology and notations not defined here can be found in Anderson and Fuller [1]. Let R be a ring and be the minimal injective cogenerator of the category of R-modules, where is an irredundant set of representatives of the simple R-modules. If M is an R-module we let . Following Belshoff, Enochs and Garcia Rozas [2], we call M a (Matlis) reflexive R-module if the evaluation map is an isomorphism. Note that eM is always injective, since U is a cogenerator of R-modules. We recall that an R-module M is linearly compact if every finitely solvable family (where each ) is solvable. For the convenience of the reader we record some basic properties of linearly compact modules and reflexive modules as follows (see [[6], Sec. 3] and [2]): (1) Every artinian module is linearly compact; (2) an infinite direct sum of non-zero modules is not linearly compact (reflexive); (3) If N is a submodule of a module M, then M is linearly compact (reflexive) if and only if both N and M/N are linearly compact (reflexive); consequently, a finite direct sum of modules is linearly compact (reflexive) if and only if each summand is linearly compact (reflexive); and (4) If M is an R-module and IM=0 for an ideal I of R, then M is linearly compact (reflexive) as an R-module if and only if it is linearly compact (reflexive) as an (R/I)-module. This paper consists of two sections. In Sec. 1, we show that the following are equivalent for a finitely generated R-module M: (1) M is reflexive; (2) M is linearly compact; (3) is linearly compact; (4) is reflexive. If R is a quasi-noetherian ring, i.e., the ideal annR(M) is finitely generated for each finitely cogenerated reflexive R-module M, we show that an R-module M is reflexive if and only if R/annR(M) is linearly compact and M has a finitely generated submodule N such that M/N is finitely cogenerated. In Sec. 2, we consider a ring extension S of R. We prove that is the minimal injective cogenerator of S-modules and an S-module M is reflexive as an S-module if and only if M is reflexive as an R-module, provided that one of the following three conditions is satisfied: (1) S is finitely generated as an R-module, (2) S is linearly compact as an R-module, or (3) S=RQ is a split extension with Q nilpotent.