J. Korean Math. Soc. 2013; 50(5): 1051-1066
Printed September 1, 2013
https://doi.org/10.4134/JKMS.2013.50.5.1051
Copyright © The Korean Mathematical Society.
Jun Zhang, Fanggui Wang, and Hwankoo Kim
Sichuan Normal University, Sichuan Normal University, Hoseo University
By utilizing known characterizations of $w$-Noetherian rings in terms of injective modules, we give more characterizations of $w$-Noether\-ian 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 $\mathscr{F}_{w,f}(R)$-injective $w$-module is injective; if and only if every GV-torsion-free $R$-module admits an $i$-decomposition.
Keywords: $GV$-torsion-free module, $w$-module, $w$-simple module, $w$-Noetherian ring, injective module
MSC numbers: Primary 13A15, 13C11, 13D99, 13E99
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