**A Course in Commutative Algebra**

TOC

0 Ring Theory Background

0.1 Prime Avoidance

0.2 Jacobson Radicals, Local Rings, and Other Miscellaneous Results

0.3 Nakayama’s Lemma

1 Primary Decomposition and Associated Primes

1.1 Primary Submodules and Ideals

1.2 Primary Decomposition

1.3 Associated Primes

1.4 Associated Primes and Localization

1.5 The Support of a Module

1.6 Artinian Rings

2 Integral Extensions

2.1 Integral Elements

2.2 Integrality and Localization

2.3 Going Down

3 Valuation Rings

3.1 Extension Theorems

3.2 Properties of Valuation Rings

3.3 Discrete Valuation Rings

4 Completion

4.1 Graded Rings and Modules

4.2 Completion of a Module

4.3 The Krull Intersection Theorem

5 Dimension Theory

5.1 The Calculus of Finite Differences

5.2 Hilbert and Hilbert-Samuel Polynomials

5.3 The Dimension Theorem

5.4 Consequences of the Dimension Theorem

5.5 Strengthening of Noether’s Normalization Lemma

5.6 Properties of Affine k-Algebras

6 Depth

6.1 Systems of Parameters

6.2 Regular Sequences

7 Homological Methods

7.1 Homological Dimension: Projective and Global

7.2 Injective Dimension

7.3 Tor and Dimension

7.4 Application

8 Regular Local Rings

8.1 Basic Definitions and Examples

Download : link