Modules Whose Classical Prime Submodules Are Intersections of Maximal Submodules

Abstract

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 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.

Modules Whose Primary-Like Submodules Are Intersection of Maximal Submodules

“…A generalization of commutative Hilbert rings to modules was extended in [1] and [12]. In this paper, we extend the notion of commutative Hilbert rings to modules via primary-like submodules and study some properties of PH modules.…”