MODULES WITH SMALL CYCLIC SUBMODULES IN THEIR INJECTIVE HULLS

ABSTRACT

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.

ACKNOWLEDGMENTS

This paper was written while the author was visiting the University of Glasgow. The author wishes to thank Professor Patrick F. Smith for many useful discussions during the preparation of this paper. The author also wishes to thank the members of the Department of Mathematics for their kind hospitality, the Scientific and Technical Research Council of Turkey (TÜBITAK) for their financial support, the referee and Professor S. Wiegand (University of Nebraska) for their valuable suggestions.