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Purely Goldie Extending Modules | ||
| Ibn Al-Haitham Journal For Pure And Applied Science | ||
| Article 1, Volume 28, Issue 2, 2015, Pages 147-154 | ||
| Authors | ||
| Saad A. Al-Saadi; Ikbal A. Omer | ||
| Abstract | ||
| An R-module M is extending if every submodule of M is essential in a direct summand of M. Following Clark, an R-module M is purely extending if every submodule of M is essential in a pure submodule of M. It is clear purely extending is generalization of extending modules. Following Birkenmeier and Tercan, an R-module M is Goldie extending if, for each submodule X of M, there is a direct summand D of Msuch that XβD. In this paper, we introduce and study class of modules which are proper generalization of both the purely extending modules and G-extending modules. We call an R-module M is purely Goldie extending if, for each X≤M, there is a pure submodule P of Msuch that XβP. Many characterizations and properties of purely Goldie extending modules are given. Also, we discuss when a direct sum of purely Goldie extending modules is purely Goldie extending and moreover we give a sufficient condition to make this property of purely Goldie extending modules is valid. | ||
| Keywords | ||
| extending module; purely extending module; purely Goldie extending | ||
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