Two Open Questions On Goldie Extending Modules

Abstract

Five open questions on Goldie extending modules were posed by Akalan et al. [1]. The first one and the second one are considered in this article. It is shown that a 𝒢-extending module M with (C ₃) is 𝒢⁺-extending. Moreover, let M = M ₁ ⊕ M ₂ be a direct sum of 𝒢-extending modules, where M satisfies (C ₃) and M ₁ is M ₂-ojective (or M ₂ is M ₁-ojective), then M is 𝒢-extending. Other sufficient conditions for a direct sum of 𝒢-extending modules to be 𝒢-extending are obtained.

Key Words:

• Duo module
• Ejective module
• Goldie extending module

2000 Mathematics Subject Classification:

• Primary 16D70
• Secondary 16D50, 16D40

ACKNOWLEDGMENTS

The authors thank Gary F. Birkenmeier for valuable suggestions, especially for Example 2.2. The authors wish to thank the referee for very careful reading of the manuscript and also for his/her suggestions.

Notes

Communicated by T. Albu.