Injective modules over w-Noetherian rings, II
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By utilizing known characterizations of w-Noetherian rings in terms of injective modules, we give more characterizations of w-Noetherian rings. More precisely, we show that a commutative ring R is w-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 w-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 Fw,f(R)-injective w-module is injective; if and only if every GV-torsion-free R-module admits an i-decomposition.
Abstract
By utilizing known characterizations of w-Noetherian rings in terms of injective modules, we give more characterizations of w-Noetherian rings. More precisely, we show that a commutative ring R is w-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 w-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 Fw,f(R)-injective w-module is injective; if and only if every GV-torsion-free R-module admits an i-decomposition.
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