On $G$-$(n,d)$-rings and $n$-coherent rings

Abstract

Let $n$ and $d$ be non-negative integers. We introduce the concept of $strongly$ $(n,d)$-$injective$ modules to characterize $n$-coherent rings. For a right perfect ring $R$, it is shown that $R$ is right $n$-coherent if and only if every right $R$-module has a strongly $(n,d)$-injective (pre)cover for some non-negative integer $d \leq n$. We also provide equivalent conditions for an $(n,d)$-ring being $n$-coherent. Then we investigate the so-called $right$ $G$-$(n,d)$-$rings$, over which every $n$-presented right module has Gorenstein projective dimension at most $d$. Finally, we prove a Gorenstein analogue of Costa's first conjecture.