On the theory and applications of G-dimension with respect to a semidualizing module

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.