Modules Whose Classical Prime Submodules Are Intersections of Maximal Submodules

Details Modules Whose Classical Prime Submodules Are Intersections of Maximal Submodules Modules Whose Classical Prime Submodules Are Intersections of Maximal Submodules

2012-02-02

arxiv.org/abs/1202.0385v1

Mathematics

Commutative Algebra

!4 Pages

Scientific paper

Commutative rings in which every prime ideal is the 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 the intersection 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 $R$-modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings $R$ for which all finitely generated $R$-modules are classical Hilbert.

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