Book contents
- Frontmatter
- Contents
- List of Symbols
- Preface
- 1 Background
- 2 Mininjective Rings
- 3 Semiperfect Mininjective Rings
- 4 Min-CS Rings
- 5 Principally Injective and FP Rings
- 6 Simple Injective and Dual Rings
- 7 FGF Rings
- 8 Johns Rings
- 9 A Generic Example
- A Morita Equivalence
- B Perfect, Semiperfect, and Semiregular Rings
- C The Camps–Dicks Theorem
- Questions
- Bibliography
- Index
7 - FGF Rings
Published online by Cambridge University Press: 14 September 2009
- Frontmatter
- Contents
- List of Symbols
- Preface
- 1 Background
- 2 Mininjective Rings
- 3 Semiperfect Mininjective Rings
- 4 Min-CS Rings
- 5 Principally Injective and FP Rings
- 6 Simple Injective and Dual Rings
- 7 FGF Rings
- 8 Johns Rings
- 9 A Generic Example
- A Morita Equivalence
- B Perfect, Semiperfect, and Semiregular Rings
- C The Camps–Dicks Theorem
- Questions
- Bibliography
- Index
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:
Every left Kasch, right FGF ring is quasi-Frobenius.
Every right self-injective, right FGF ring is quasi-Frobenius.
Every right perfect, right FGF ring is quasi-Frobenius.
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 Mn(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 Zr = J. We call these rings right weakly continuous, and investigate their basic properties.
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- Quasi-Frobenius Rings , pp. 164 - 200Publisher: Cambridge University PressPrint publication year: 2003