Dual of bass numbers and dualizing modules

Let R be a Noetherian ring and let C be a semidualizing R-module. In this paper, we impose various conditions on C to be dualizing. For example, as a generalization of Xu [21 Xu, J. (1995). Minimal injective and flat resolutions of modules over Gorenstein rings. J. Algebra 175:451–477.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 3.2], we show that C is dualizing if and only if for an R-module M, the necessary and su?cient condition for M to be C-injective is that π_i(𝔭,M) = 0 for all 𝔭∈Spec (R) and all i≠ht (𝔭), where π_i is the invariant dual to the Bass numbers defined by Enochs and Xu [8 Enochs, E., Xu, J. (1997). On invariants dual to the Bass numbers. Proc. Am. Math. Soc. 125:951–960.[Crossref], [Web of Science ®] , [Google Scholar]].