Idempotent and Nilpotent Submodules of Multiplication Modules

Abstract

All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate multiplication von Neumann regular modules. For this reason we introduce the concept of nilpotent submodules generalizing nilpotent ideals and then prove that a faithful multiplication module is von Neumann regular if and only if it has no nonzero nilpotent elements and its Krull dimension is zero. We also give a new characterization for the radical of a submodule of a multiplication module and show in particular that the radical of any submodule of a Noetherian multiplication module is a finite intersection of prime submodules.

Key Words:

• Idempotent submodule
• Multiplication module
• Nilpotent submodule
• Pure submodule
• von Neumann regular module

2000 Mathematics Subject Classification:

• 13C13
• 13C05
• 13A15

ACKNOWLEDGMENT

The author is grateful to the referee for helpful suggestions which have resulted in an improvement to the article.

Notes

Communicated by I. Swanson.