Modules whose classical prime submodules are intersections of maximal submodules
Bull. Korean Math. Soc. 2014 Vol. 51, No. 1, 253-266
https://doi.org/10.4134/BKMS.2014.51.1.253
Printed January 1, 2014
Marzieh Arabi-Kakavand and Mahmood Behboodi
Isfahan University of Technology, Institute for Research in Fundamental Sciences (IPM)
Abstract : Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that {\it classical prime} submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings $R$ over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings $R$ for which all finitely generated $R$-modules are classical Hilbert.
Keywords : Hilbert ring, Hilbert module, classical prime submodule
MSC numbers : 13C10, 13C13
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