We study the concepts of the (C)-projective and the I-C-injective dimensions of a module in the noncommutative case, weakening the condition of C being semidualizing. We give the relations between these dimensions and the C-relative Gorenstein dimensions (G(C)-projective and G(C)-injective dimensions) of the module. Finally, we compare, in some circumstances, the global (C)-projective dimension of a ring and the global dimension of the endomorphisms ring of C.
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