It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $\left(1\right)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $\left(2\right)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings....

Let $\Lambda =\left(\begin{array}{cc}A& M\\ 0& B\end{array}\right)$ be an Artin algebra. In view of the characterization of finitely generated Gorenstein injective $\Lambda$-modules under the condition that $M$ is a cocompatible $(A,B)$-bimodule, we establish a recollement of the stable category $\overline{\mathrm{Ginj}\left(\Lambda \right)}$. We also determine all strongly complete injective resolutions and all strongly Gorenstein injective modules over $\Lambda$.