  We introduce and investigate the notion of $\gc$-projective modules over
(possibly non-noetherian) commutative rings, where $C$ is a semidualizing
module. This extends Holm and J{\o}rgensen's notion of $C$-Gorenstein
projective modules to the non-noetherian setting and generalizes projective and
Gorenstein projective modules within this setting. We then study the resulting
modules of finite $\gc$-projective dimension, showing in particular that they
admit $\gc$-projective approximations, a generalization of the maximal
Cohen-Macaulay approximations of Auslander and Buchweitz. Over a local
(noetherian) ring, we provide necessary and sufficient conditions for a
$G_C$-approximation to be minimal.
