On the Prime Spectrum of X-Injective Modules

Abstract

Let R be a commutative ring and M an R-module. The purpose of this article is to introduce a new class of modules over R called X-injective R-modules, where X is the prime spectrum of M. This class contains the family of top modules and that of weak multiplication modules properly. In this article our concern is to extend the properties of multiplication, weak multiplication, and top modules to this new class of modules. Furthermore, for a top module M, we study some conditions under which the prime spectrum of M is a spectral space for its Zariski topology.

Key Words:

• Prime submodule
• Spectral space
• Top module
• Weak multiplication module
• X-Injective module
• Zariski topology

2000 Mathematics Subject Classification:

• 13C13
• 13C99

ACKNOWLEDGMENT

The authors are grateful for the referee's valuable comments and helpful suggestions that considerably improved this article.

Notes

Communicated by I. Swanson.