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

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 ≤ 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.