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.