Abstract
We define a module M to be -extending if for each exact submodule X of M there exists a direct summand D of M such that
is essential in both X and D. We investigate
-extending modules and locate the implications between the other extending properties. We study decomposition theory and extensions for
-extending concept. We show that if a ring is right
-extending, then so its essential overring. It is shown that
-extending property is inherited by its rational hull. We provide examples by making special choices of left exact preradicals.
2020 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgements
The authors would like to express their thanks to the referee for her/his careful reading and useful suggestions on this paper.