In this paper, we prove that the Bass class with respect to a semidualizing bimodule C contains all FP-injective S-modules. We introduce the definition of C-FP-injective modules, and give some characterizations of right coherent rings in terms of the C-flat S-modules and C-FP-injective S^op-modules. We discuss when every S-module has an C-flat preenvelope which is epic (or monic). In addition, we investigate the left and right -resolutions of R-modules by left derived functors Ext_n(-, -) over a left Noetherian ring S. As applications, some new characterizations of left perfect rings are induced by these modules associated with C. A few classical results of these rings are obtained as corollaries.