Goldie Extending Modules

Abstract

In this article, we define a module M to be 𝒢-extending if and only if for each X ≤ M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We consider the decomposition theory for 𝒢-extending modules and give a characterization of the Abelian groups which are 𝒢-extending. In contrast to the charac-terization of extending Abelian groups, we obtain that all finitely generated Abelian groups are 𝒢-extending. We prove that a minimal cogenerator for 𝒢od-R is 𝒢-extending, but not, in general, extending. It is also shown that if M is (𝒢-) extending, then so is its rational hull. Examples are provided to illustrate and delimit the theory.

Key Words:

• C ₁₁-module
• C ₁₁-ring
• Extending module
• FI-extending module
• Rational hull
• Uniform module

2000 Mathematics Subject Classification:

• Primary 16D70
• Secondary 16D50, 16D40

Acknowledgments

The authors wish to thank the referee for his/her suggestions which improved this article. The original version of Theorem 3.15 only considered the equivalence of conditions (i) and (iii). The authors are grateful to K. M. Rangaswamy for informing them that conditions (i) and (iii) are equivalent to condition (ii). This work was made possible by TUBITAK (BIDEP) and was carried out during visits of the second author to Hacettepe University in August 2004 and August 2005. The authors wish to thank both organizations for their support. The second author is grateful for the hospitality shown to him by the faculty and staff of Hacettepe University.

Notes

Communicated by T. Albu.