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Essential extensions with the -extending property

Published in: Quaestiones Mathematicae
Volume 39, issue 1, 2016 , pages: 59–64
DOI: 10.2989/16073606.2015.1024186
Author(s): Gary F. BirkenmeierDepartment of Mathematics, USA, Adnan TercanHacettepe University, Turkey, Canan C YücelDepartment of Mathematics, Turkey


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.

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