Modules whose classical prime submodules are intersections of maximal submodules
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Commutative rings in which every prime ideal is an inter- section of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are clas- sical 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. Further- more, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.
Abstract
Commutative rings in which every prime ideal is an inter- section of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are clas- sical 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. Further- more, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.
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