On Goldie-Extending Modules with Finite Internal Exchange Property

A module M is said to be G-extending (Goldie extending) if, for any submodule X of M, there exist an essential submodule Y of X and a direct summand M′ of M such that Y is essential in M′. A G-extending module is introduced by Akalan et al. (Commun. Algebra 37, 663–683, 2009). Note that G-extending modules are dual to H-supplemented modules (Keskin Tütüncü et al. Alg. Coll. 18, 915–924, 2011). In this paper, we show some characterizations of G-extending modules and consider generalizations of a relative injectivity. And we apply them to study the open problem “When is a direct sum of G-extending (uniform) modules G-extending?” by Akalan et al. (Commun. Algebra 37, 663–683, 2009). © 2015, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.