By utilizing known characterizations of ${omega}$-Noetherian rings in terms of injective modules, we give more characterizations of ${omega}$-Noetherian rings. More precisely, we show that a commutative ring R is ${omega}$-Noetherian if and only if the direct limit of GV -torsion-free injective R-modules is injective; if and only if every R-module has a GV -torsion-free injective (pre)cover; if and only if the direct sum of injective envelopes of ${omega}$-simple R-modules is injective; if and only if the essential extension of the direct sum of GV -torsion-free injective R-modules is the direct sum of GV -torsion-free injective R-modules; if and only if every $mathfrak{F}_{w,f}(R)$-injective ${omega}$-module is injective; if and only if every GV-torsion-free R-module admits an $i$-decomposition.