Essentially Retractable Modules.

We call a module mr essentially retractable if homr (m,n ) (not equal) 0 for all essential submodules n of m. for a right fbn ring r, it is shown that: (i) a nonzero module r m is retractable (in the sense that homr (m,n) (not equal) 0 for all nonzero n =< mr ) if and only if certain factor modules of m are essentially retractable nonsingular modules over r modulo their annihilators. (ii) a non-zero module r m is essentially retractable if and only if there exists a prime ideal p (element of) ass (mr) such that homr (m,n) (not equal) 0. over semiprime right nonsingular rings, a nonsingular essentially retractable module is precisely a module with nonzero 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 {ui} i(element of) i and r-homomorphisms m (right arrow) (alpha) i(element of)i ui(right arrow) (beta) m with (beta)(alpha) (not equal)0.