KYUNGPOOK Math. J. 54(2014), 237-247 http://dx.doi.org/10.5666/KMJ.2014.54.2.237 On Strongly Extending Modules S. Ebrahimi Atani * , M. Khoramdel and S. Dolati Pish Hesari Department of Mathematics, University of Guilan , Rasht, Iran e-mail : ebrahimi@guilan.ac.ir, mehdikhoramdel@gmail.com and saboura_dolati@yahoo.com Abstract. The purpose of this paper is to introduce the concept of strongly extending modules which are particular subclass of the class of extending modules, and study some basic properties of this new class of modules. A module M is called strongly extending if each submodule of M is essential in a fully invariant direct summand of M. In this paper we examine the behavior of the class of strongly extending modules with respect to the preservation of this property in direct summands and direct sums and give some proper- ties of these modules, for instance, strongly summand intersection property and weakly co-Hopfian property. Also such modules are characterized over commutative Dedekind domains. 1. Introduction The theory of extending modules has come to play an important role and major contributions to this theory have been made in recent years, providing extensively interesting results on extending properties in the module-theoretical setting. An R-module M is called (strongly FI-) extending if each (fully invariant) submodule is essential in a (fully invariant) direct summand. Now it is natural to ask: When does a module have the property that every submodule is essential in a fully invari- ant direct summand? The main purpose of this paper is to answer this question and investigate these modules. Here is a brief summary of our paper. In fact, we will show that direct summands of a strongly extending module are strongly extending, and some conditions are given to show direct sum of two strongly extending modules is strongly extending. Also we prove that an R-module M is strongly extending if and only if M = Z 2 (M ) ⊕ N for some submodule N of M , where Z 2 (M ) and N are both strongly extending and Hom(K, Z 2 (M )) = 0 for each submodule K of N . We introduce the notion of strongly Rickart modules and use this to show that * Corresponding Author. Received July 15, 2012; revised May 28, 2013; accepted June 17, 2013. 2010 Mathematics Subject Classification: 16D10, 16D80, 16D70, 16D40. Key words and phrases: Extending modules, Strongly extending modules, Strongly Rickart modules. 237