Bulletin of the
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Bull. Korean Math. Soc. 2014; 51(1): 253-266

Printed January 1, 2014

https://doi.org/10.4134/BKMS.2014.51.1.253

Copyright © The Korean Mathematical Society.

Modules whose classical prime submodules are intersections of maximal submodules

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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