The relation between generalized notions of radical formula of a module

Abstract

In this article, a new radical formula for a module over a commutative ring is introduced by constructing a generalization of radical and envelope and defining an (e, r)-radical formula. Since the set of these radical formulae is not totally ordered, we develop characterizations related to the module structure satisfying some radical formulae and McCasland modules. Furthermore, we show that some relations between those modules hold if and only if they are modules over a local ring. Moreover, we define a fully primal module which is needed for some relations between modules which satisfy some radical formulae and their localizations.

KEYWORDS:

• Envelope of a submodule
• fully primal module
• localization of a module
• prime submodules
• radical formula of a module

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13C13
• 13C05
• 13A15

Acknowledgments

The author would like to thank the anonymous referee for the valuable feedback which has contributed to improving the quality of the manuscript.