Modules Whose Classical Prime Submodules Are Intersections of Maximal Submodules

TL;DR

This paper introduces classical Hilbert modules where classical prime submodules are intersections of maximal submodules, explores their properties, and characterizes rings over which all modules are classical Hilbert.

Contribution

It defines classical Hilbert modules, establishes their properties, and characterizes rings where all modules are classical Hilbert, extending the concept of Hilbert rings to modules.

Findings

All co-semisimple modules are classical Hilbert.

All Artinian modules are classical Hilbert.

Over zero-dimensional rings, all modules are classical Hilbert.

Abstract

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.