Essential extensions with the -extending property

Let be a nonempty subset of the set of submodules of a module M . Then M is called a -extending module if for each X in there exists a direct summand D of M such that X is essential in D. In general, it is known that an essential extension of a -extending module is not -extending. In this paper, our goal is to show how to construct essential extensions of a module which are extending by using a set of representatives of a certain equivalence relation on the set of all idempotent endomorphisms of the injective hull of the module. Also we characterize when the rational hull of a module is -extending in terms of such a set of representatives. Moreover we show that several well known types of -extending conditions (e.g., extending and FI-extending) transfer from module to its rational hull.