Embeddings in Free Modules and Artinian Rings

Abstract

It is well known that if R is a ring such that each right R-module Ž . embeds in a free module, then R is a quasi-Frobenius QF ring. However, if the embedding is restricted to finitely generated right R-modules, then the situation is much more difficult to handle. A ring R is called right FGF whenever every finitely generated right R-module embeds in a free module and the so-called FGF problem which asks whether a right FGF ring is Ž w x w QF remains open see 5 for a discussion of this question and 7, 8, 10, 12, x . 13, 16 for more recent results . w x w x In 9 Menal used the counting arguments developed by Osofsky in 11 to prove that if R is a ring such that each cyclic right R-module embeds in Ž . a free module R is then called a right CF ring and the injective envelope Ž . E R is projective, then R is already a QF ring. Thus, given a ring R, R there exists a cardinal c with the property, that if every c-generated right R-module embeds in a free module, then R is QF. However, Menal