The class of Gorenstein injective modules is covering if and only if it is closed under direct limits

Abstract

We prove that the class of Gorenstein injective modules, $\mathcal{G}\mathcal{I}$, is special precovering if and only if it is covering if and only if it is closed under direct limits. This adds to the list of examples that support Enochs' conjecture:\\ "Every covering class of modules is closed under direct limits".\\ We also give a characterization of the rings for which $\mathcal{G}\mathcal{I}$ is covering: the class of Gorenstein injective left $R$-modules is covering if and only if $R$ is left noetherian, and such that character modules of Gorenstein injective left $R$ modules are Gorenstein flat.