COMMUNICATIONS IN ALGEBRA, vol.30, no.4, pp.1575-1589, 2002 (SCI-Expanded)
In this paper we study when a unital right module M over a ring R with identity has a special "small image" property we call (S*): namely, M has (S*) if every submodule N of M contains a direct summand K of M such that every cyclic submodule C of N/K is small (meaning "small in its injective hull E(C)"). If xR is small for every element x of a module M, M is said to be cosingular. In Theorem 4.4 we prove every right R-module satisfies (S*) if and only if every right R-module is the direct sum of an injective module and a cosingular module. Over a right self-injective ring R, every right R-module satisfies (S*) if and only if R is quasi-Frobenius (Theorem 5.5). It follows that over a commutative ring R, every module satisfies (S*) if and only if R is a direct product of a quasi-Frobenius ring and a cosingular ring.