7 - FGF Rings

Summary

A theorem of Faith and Walker asserts that a ring R is quasi-Frobenius if and only if every injective right R-module is projective and hence that every right module over a quasi-Frobenius ring embeds in a free module. There is an open problem here. If we call a ring R a right FGF ring if every finitely generated right R-module can be embedded in a free right R-module, it is not known if the following assertion is true:

The FGF-Conjecture. Every right FGF ring is quasi-Frobenius

Here are four important results on the conjecture:

1. Every left Kasch, right FGF ring is quasi-Frobenius.

2. Every right self-injective, right FGF ring is quasi-Frobenius.

3. Every right perfect, right FGF ring is quasi-Frobenius.

4. Every right CS, right FGF ring is quasi-Frobenius.

We prove all these assertions; in fact we capture all of (1), (2), and (3) in Theorem 7.19: If M_n(R) is a right C2 ring for each n ≥ 1 and every 2-generated right R-module embeds in a free module then R is quasi-Frobenius. This theorem also implies that the FGF-conjecture is true for right FP-injective rings, and it reformulates the conjecture by showing that it suffices to prove that every right FGF ring is a right C2 ring. Furthermore, the theorem shows that the conjecture is true for semiregular rings with Z_r = J. We call these rings right weakly continuous, and investigate their basic properties.