Essentially Retractable Modules

Abstract

We call a module <em>essentially retractable </em>if Hom_R for all essential submodules <em>N</em> of <em>M</em>. For a right FBN ring <em>R</em>, it is shown that: (i) A non-zero module is <em>retractable</em> (in the sense that Hom_R for all non-zero ) if and only if certain factor modules of <em>M</em> are essentially retractable nonsingular modules over <em>R</em> modulo their annihilators. (ii) A non-zero module is essentially retractable if and only if there exists a prime ideal such that Hom_R. Over semiprime right nonsingular rings, a nonsingular essentially retractable module is precisely a module with non-zero dual. Moreover, over certain rings <em>R</em>, including right FBN rings, it is shown that a nonsingular module <em>M</em> with enough uniforms is essentially retractable if and only if there exist uniform retractable <em>R</em>-modules and <em>R</em>-homomorphisms with .