Essential extensions with the c-extending property

Abstract

Let (Formula presented.) be a nonempty subset of the set of submodules of a module M. Then M is called a (Formula presented.) -extending module if for each X in (Formula presented.) 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 (Formula presented.) -extending module is not (Formula presented.) -extending. In this paper, our goal is to show how to construct essential extensions of a module which are (Formula presented.) 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 (Formula presented.) -extending in terms of such a set of representatives. Moreover we show that several well known types of (Formula presented.) -extending conditions (e.g., extending and FI-extending) transfer from module to its rational hull. © 2016 NISC (Pty) Ltd.