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Let be a ring, and a fixed nonnegative integer. An -module is called -injective if for any -module with flat dimension at most . In this paper, we prove first that () is a complete hereditary cotorsion theory, where (resp. ) denotes the class of all -modules with flat dimension at most (resp. -injective -modules). Then we introduce the -injective dimension of a module and -global dimension of a ring. Finally, over rings with weak global dimension , perfect rings, and -hereditary rings, more properties and applications of -injective modules, -injective dimensions of modules and -global dimensions of rings are given.