Abstract
Let R be a ring. A left R-module M is called GI-injective if for any Gorenstein injective left R-module N. It is shown that a left R-module M over any ring R is GI-injective if and only if M is a kernel of a Gorenstein injective precover f: A → B of a left R-module B with A injective. Suppose R is an n-Gorenstein ring, we prove that a left R-module M is GI-injective if and only if M is a direct sum of an injective left R-module and a reduced GI-injective left R-module. Then we investigate GI-injective dimensions of modules and rings. As applications, some new characterizations of the weak (Gorenstein) global dimension of coherent rings are given.
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