抽象代数III

这是MAT8041抽象代数III的随堂笔记：

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看起来有点乱，按章节整理了一下：

Chapter 1 General Structure Theory

1.1 Core Concepts

• A. Fundamental Ring Structures
  • Annihilator and Jacobson Radical
• B. Fundamental Module Structures
  • Module Constructions and Universal Properties
  • Socle and Radical of Module
  • Socle and Radical Series

1.2 Categorical Language

• Category and Hom, Tensor Functors

1.3 Homological Properties of Modules

• Projective, Injective and Flat Modules
• Projective Cover
• Projective Morphism

1.4 Semisimple Theory

• Semisimple Artinian Rings and Algebras
• Idempotents and Module Structure

Chapter 2 Ideal Theory of Integral Domains

2.1 Foundations

• Integrality and Integral Closures
• Torsion (Free) Elements and Fractional Modules

2.2 The Local Picture: Valuation Rings

• Valuation Rings and DVRs

2.3 The Global Picture: Dedekind Domains

• Dedekind Domains

Chapter 3 Application: Introduction to Modular Representation Theory

• p-modular System