Let C be a semidualizing module for a commutative ring R. It is shown that the :IC-injective dimension has the ability to detect the regularity of R as well as the Pc-projective dimension. It is proved that if D is dualizing for a Noetherian ring R such that idR(D) = n 〈 ∞, then :ID-idR(F) ≤ n for every flat R-module F. This extends the result due to Enochs and Jenda. Finally, over a Noetherian ring R, it is shown that if M is a pure submodule of an R-module N, then/TC-idR(M) ≤ IC-idR(N). This generalizes the result of Enochs and Holm.