For a horizontally linked module, over a commutative semiperfect Noetherian ring R, the connections of its invariants reduced grade, Gorenstein dimension and depth are studied. It is shown that under certain conditions the depth of a horizontally linked module is equal to the reduced grade of its linked module. The connection of the Serre condition (Sn) on an R-module of finite Gorenstein dimension with the vanishing of the local cohomology groups of its linked module is discussed.(M, R) is determined in terms of λM for any horizontally linked module M .As a consequence, we determine the reduced grade of a horizontally linked module of finite Gorenstein dimension. Also a characterization of a horizontally linked module to have G-dimension zero is given in Proposition 2.3. In Theorem 2.7 , we give some equivalent conditions for a horizontally linked module M of finite and positive G-dimension such that depth of M is equal to the reduced grade of λM .In section 3, we study the reduced G-perfect modules (Definition 3.1). For a reduced G-3). We also generalize this formula for modules of finite G-dimensions (not necessarily reduced G-perfect) with some conditions on depth of extension modules (Theorem 3.4). In Proposition 3.5, we establish a characterization for stable reduced G-perfect module to be horizontally linked. Key words and phrases. Linkage of modules, Gorenstein dimension, reduced grade, Semidualizing module, G K -dimension. 1. M.T. Dibaei was supported in part by a grant from IPM (No. 90130110). eM −→ M * * −→ Ext 2 R (Tr M, R) −→ 0. Let P α → M be an epimorphism such that P is a projective. The syzygy module of M , denoted by ΩM , is the kernel of α which is unique up to projective equivalence. Thus ΩM is uniquely determined, up to isomorphism, by a projective cover of M . Recall that, over a Gorenstein local ring R, two ideals a and b are said to be linked by a Gorenstein ideal c if c ⊆ a ∩ b, a = c : R b and b = c : R a. Equivalently, the ideals a and b are linked by c if and only if the ideals a/c and b/c of the ring R/c are linked by the zero ideal of R/c. To define linkage for modules, Martsinkovsky and Strooker introduced the operator λ = ΩTr . In [16, Proposition 1], it is shown that ideals a and b are linked by zero ideal if and only if R/a and R/b are related to each other through the operator λ; more precisely, R/a ∼ = λ(R/b) and R/b ∼ = λ(R/a). Definition 1.1. [16, Definition 3] Two R-modules M and N are said to be horizontally linked if M ∼ = λN and N ∼ = λM . Also, M is called horizontally linked (to λM ) if M ∼ = λ 2 M .Having defined the horizontal linkage, the general linkage for modules is defined as follows.
