Let [Formula: see text] be a semiperfect commutative Noetherian ring with identity and [Formula: see text] be semidualizing [Formula: see text]-modules. We study the theory of linkage with respect to [Formula: see text] for modules of finite [Formula: see text]-dimension. For a module which is horizontally linked with respect to [Formula: see text], the connections of its invariants reduced grade with respect to [Formula: see text], [Formula: see text]-dimension and depth are discussed. Along the way, we provide a characterization of horizontally linked modules with respect to [Formula: see text] of [Formula: see text]-dimension zero.

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We contribute to the theory of G-dimension relative to a semidualizing module [Formula: see text], in connection to the properties of a module being totally [Formula: see text]-reflexive, [Formula: see text]-[Formula: see text]-torsionless, and [Formula: see text]-[Formula: see text]-syzygy, where [Formula: see text] is an integer. We extend several known results, and for an integer [Formula: see text] we introduce the class of [Formula: see text]-[Formula: see text]-Gorenstein rings. We also consider [Formula: see text]-duals and initiate the study of [Formula: see text]-valued derivation modules over a local ring [Formula: see text] of low depth; if [Formula: see text] and [Formula: see text] is a three-dimensional Gorenstein domain, we find a bound for the number of generators and we propose a question when [Formula: see text] is general.

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In their investigation of horizontal linkage of modules of finite Gorenstein dimension over a commutative, Noetherian, semiperfect (e.g., local) ring, Dibaei and Sadeghi introduced the class of reduced G-perfect modules, making use of Bass' concept of reduced grade. A few years later, the same authors extended this class by considering the relative property of reduced G C -perfection, where C is a semidualizing module, and studied linkage even further. In the present paper, we contribute to their theory and also generalize results of Auslander and Bridger as well as of Martsinkovsky and Strooker. Our investigation includes, for example, when reduced G C -perfection is preserved by relative Auslander transpose, and how to numerically characterize horizontally linked modules under suitable conditions. Along the way, we show how to produce reduced G C -perfect modules that are also C-k-torsionless (for a given integer k ≥ 0) but fail to be G C -perfect, and moreover we illustrate that, in contrast to the usual grade, the relative reduced grade does depend on the choice of C.