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Let be a commutative Noetherian local ring of Krull dimension d, and let C be a semidualizing R-module. In this paper, it is shown that if R is complete, then C is a dualizing module if and only if the top local cohomology module of , has finite G _C -injective dimension. This generalizes a recent result due to Yoshizawa, where the ring is assumed to be complete Cohen-Macaulay.