Essentially Retractable Modules

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