Covers and envelopes with respect to a semidualizing module

Let R be a commutative ring and C be a semidualizing R-module. For a given class of R-modules Q, we define a class QC by M element of QC -- HomR (C, M) element of Q. We prove that if Q subseteq AC(R) is a Kaplansky class and closed under direct sums, then Q C up tack is special preenveloping. As corollaries, we can show that P n C up tack and F n C up tack are both special preenveloping. Finally, we show that I n C is covering, I n C up tack is enveloping and special preenveloping provided R is Noetherian.