Idealization and Theorems of D. D. Anderson

Abstract

All rings are commutative with identity and all modules are unital. Anderson proved that a submodule N of an R-module M is multiplication (resp. join principal) if and only if 0₍₊₎ N is a multiplication (resp. join principal) ideal or R(M). The idealization of M. In this article we develop more fully the tool of idealization of a module, particularly in the context of multiplication modules, generalizing Anderson's theorems and discussing the behavior under idealization of some ideals and some submodules associated with a module.

Keywords:

• Flat module
• Idealization
• Join principal submodule
• Multiplication module
• Projective module
• Pure submodule

Mathematics Subject Classification:

• 13C13
• 13C05
• 13A15

ACKNOWLEDGMENTS

The author is grateful to Prof. S. Veldsman for his helpful comments.

Notes

Communicated by I. Swanson.

Dedicated to Heinz Lüneburg on the occasion of his 70th birthday.