ABSTRACT
A ring is quasi-Frobenius if and only if every injective right R-module is projective. Thus, the ring has the property that each right -module embeds in a free module if and only if the ring is a quasi-Frobenius ring. A ring is called right FGF whenever every finitely generated right R-module embeds in a free module. This chapter reviews the "FGF problem" which asks whether every right FGF ring is quasi-Frobenius. It is convenient to point out that the FGF problem is a strictly one-sided question, for if the ring is left and right FGF, then Ring is already quasi-Frobenius. If R is a ring such that every cyclic right R-module, and every cyclic left R-module embeds in a free module, then R is quasi-Frobenius. R is right CF whenever every cyclic right R-module embeds in a free module. All rings are associative and with identity, and Mod-R denotes the category of right R-modules.
