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
4 - Min-CS 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
In this chapter, we consider the class of left min-CS rings (for which every minimal left ideal is essential in a direct summand) and show that this weak injectivity property is useful in obtaining semiperfect rings. Indeed, it is proved in Theorem 4.8 that if R is left min-CS, then the dual of every simple right R-module is simple, if and only if R is semiperfect with Sι = Sr and soc(Re) is simple and essential for every local idempotent e of R. The hypotheses of Theorem 4.8 are the weakest known conditions of this type that imply that R is semiperfect.
If we strengthen the left min-CS hypothesis in Theorem 4.8 by requiring that each closed left ideal with simple essential socle be a direct summand of RR (R is left strongly min-CS), we obtain a class of rings that satisfies many of the characteristic properties of left PF rings. If instead of assuming in Theorem 4.8 that the duals of simple right R-modules are simple we suppose, more generally, that R is right Kasch, then we obtain a larger class of rings that still retains many of these properties: It is shown in Theorem 4.10 that R is left CS and right Kasch if and only if it is semiperfect and left continuous with Sr ⊆essRR.
- Type
- Chapter
- Information
- Quasi-Frobenius Rings , pp. 78 - 94Publisher: Cambridge University PressPrint publication year: 2003