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
5 - Principally Injective and FP 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 ring R is right mininjective if R-linear maps aR → R extend to RR → RR whenever aR is a simple right ideal. It is natural to enquire about the rings for which this condition is satisfied for all principal right ideals aR (for example if R is regular or right self-injective). These rings, called right principally injective (or right P-injective), play a central role in injectivity theory. An example is given of a right P-injective ring that is not left P-injective. If R is right P-injective it is shown that Zr = J, that R is directly finite if and only if monomorphisms RR → RR are epic, and that R has the ACC on right annihilators if and only if it is left artinian. If R is right P-injective and right Kasch then Sι ⊆ essRR. A semiperfect, right P-injective ring R in which Sr ⊆ess RR is called a right GPF ring. Hence the right PF rings are precisely the right self-injective, right GPF rings, and these right GPF rings exhibit many of the properties of the right PF rings: They admit a Nakayama permutation, they are right and left kasch, they are left finitely cogenerated, Sr = Sι is essential on both sides, and Zr = J = Zι.
Unlike mininjectivity, being right P-injective is not a Morita invariant. In fact, Mn(R) is right P-injective implies that R is right n-injective.
- Type
- Chapter
- Information
- Quasi-Frobenius Rings , pp. 95 - 129Publisher: Cambridge University PressPrint publication year: 2003