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INJECTIVE MODULES OVER ω-NOETHERIAN RINGS, II

  • Zhang, Jun (Institute of Mathematics and Software Science School of Foreign Languages Sichuan Normal University) ;
  • Wang, Fanggui (Institute of Mathematics and Software Science Sichuan Normal University) ;
  • Kim, Hwankoo (Department of Information Security Hoseo University)
  • Received : 2012.11.23
  • Published : 2013.09.01

Abstract

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.

Keywords

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