Goldie extending property on the class of exact submodules

Abstract

We define a module M to be $G^{\mathrm{ }r}$-extending if for each exact submodule X of M there exists a direct summand D of M such that $X\cap D$ is essential in both X and D. We investigate $G^{\mathrm{ }r}$-extending modules and locate the implications between the other extending properties. We study decomposition theory and extensions for $G^{\mathrm{ }r}$-extending concept. We show that if a ring is right $G^{\mathrm{ }r}$-extending, then so its essential overring. It is shown that $G^{\mathrm{ }r}$-extending property is inherited by its rational hull. We provide examples by making special choices of left exact preradicals.

Keywords:

• ES-module
• exact submodule
• Goldie-extending module
• left exact preradical
• rational hull

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D10
• 16D50
• Secondary: 16D40

Acknowledgements

The authors would like to express their thanks to the referee for her/his careful reading and useful suggestions on this paper.