Strict Mittag-Leffler modules over Gorenstein injective modules

In this paper, we study the conditions under which a module is a strict Mittag-Leffler module over the class GI of Gorenstein injective modules. To this aim, we introduce the notion of GI-projective modules and prove that over noetherian rings, if a module can he expressed as the direct limit of finitely presented GI-projective modules, then it is a strict Mittag-Leffler module over GI. As applications, we prove that if R is a two-sided noetherian ring, then GI is a covering class closed under pure submodules if and only if every injective module is strict Mittag-Leffler over GI.