ON REFLEXIVE MODULES OVER COMMUTATIVE RINGS

Let R be a commutative ring and U an R-module. The aimof this paper is to study the duality between U-reflexive (pre)envelopesand U-reflexive (pre)covers of R-modules. 1. Introduction and preliminariesLet Rbe a commutative ring and Uan R-module. For R-modules MandN, Hom(M,N) means Hom R (M,N), and Ext n (M,N) means Ext nR (M,N) foran integer n≥ 1. For an R-module M, Hom(M,U) is called the dual moduleof Mwith respect to Uand denoted by M ∗ . For a homomorphism fbetweenR-modules, we put f ∗ = Hom(f,U). Let δ M : M→ M ∗∗ via δ M (x)(f) = f(x)for any x∈ Mand f∈ M ∗ be the canonical evaluation homomorphism. If δ M is an isomorphism, then Mis called a U-reflexive module. We denote by R U the class of U-reflexive modules.Let Rbe a Noetherian ring. A finitely generated R-module Mis said to haveGorensteindimension (abbr. G-dimension) zero[2]if M∼=Hom(Hom(M,R),R)(i.e., M is R-reflexive); and Ext i (M,R) = 0 = Ext i (Hom(M,R),R) for alli≥ 1. We denote by G(R) the class of R-modules having G-dimension zero.Let F be a class of R-modules and M an R-module. A homomorphismφ : M → F with F ∈ F is called an F-preenvelope of M [10] if for anyhomomorphism f: M→ F