A balance result over generalized coherent rings

We introduce the notion of generalized coherent rings and characterize the Gorenstein global dimension of such rings. As applications, we obtain that the invariants $\operatorname{spli}(R)$ and $\operatorname{silp}(R)$ coincide for a commutative generalized coherent ring $R$, where $\operatorname{spli}(R)$ (resp. $\operatorname{silp}(R)$) denotes the supremum of the projective (resp. injective) dimensions of all injective (resp. projective) $R$-modules.