Commutative Algebra through Exercises

Preface
Notation
Contents
Part I Theory
	Chapter 1 Rings
		1.1 Rings and Ideals
		1.2 Homomorphisms and Quotient Rings
		1.3 Nilradical, Jacobson Radical and Local Rings
		1.4 Extension and Contraction of Ideals
		1.5 The Chinese Remainder Theorem
		1.6 Factorization in Integral Domains: PID and UFD
	Chapter 2 The Ring K[x1, . . . , xn]
		2.1 Monomial Ideals and ε-subsets
		2.2 Monomial Orderings
		2.3 Division in K[x1, . . . , xn]
		2.4 Gröbner Bases: First Properties
		2.5 Buchberger’s Algorithm
		2.6 Minimal and Reduced Gröbner Bases
		2.7 Some Applications
			2.7.1 Lexicographic Orderings and Elimination
	Chapter 3 Affine Algebraic Varieties
		3.1 Definitions and First Properties
		3.2 The Resultant
		3.3 The Extension Theorem
		3.4 Hilbert’s Nullstellensatz
		3.5 Systems of Polynomial Equations
		3.6 Appendix: the Zariski Topology
	Chapter 4 Modules
		4.1 Modules and Submodules
		4.2 Module Homomorphisms
		4.3 Free Modules
		4.4 Direct Sum and Direct Product of Modules
		4.5 Nakayama’s Lemma
		4.6 Categories and Functors
		4.7 Exact Sequences
			4.7.1 The Functors Hom A(•,N) and HomA(M, •)
			4.7.2 Split Sequences
			4.7.3 Snake Lemma
		4.8 Projective Modules
		4.9 Modules over a PID
			4.9.1 Smith Normal Form
			4.9.2 Structure Theorems for Finitely Generated Modules
		4.10 Appendix: the Rational Canonical Form and the Jordan Form
	Chapter 5 Tensor Product
		5.1 Universal Property of Tensor Product
		5.2 Tensor Product as a Functor
		5.3 Extension of Scalars
	Chapter 6 Localization
		6.1 Rings of Fractions
		6.2 Modules of Fractions
		6.3 The Functor
		6.4 Local Properties
		6.5 Appendix: the Saturation of a Multiplicative Subset
	Chapter 7 Noetherian and Artinian Rings. Primary Decomposition
		7.1 Noetherian and Artinian Modules
		7.2 Noetherian Rings and Primary Decomposition
		7.3 Artinian Rings
Part II Exercises
	Chapter 8 Rings and Ideals
	Chapter 9 Polynomials, Gröbner Bases, Resultant, and Varieties
	Chapter 10 Modules
		10.1 Modules, Submodules and Homomorphisms
		10.2 Exact Sequences and Projective Modules
		10.3 Modules over a PID and Smith Normal Form
	Chapter 11 Tensor Product
	Chapter 12 Localization
	Chapter 13 Noetherian and Artinian Modules
	Chapter 14 True or False?
	Chapter 15 Review Exercises
Part III Proofs and Solutions
	Chapter 16 Proofs of Theoretical Results
		16.1 Chapter 1
		16.2 Chapter 2
		16.3 Chapter 3
		16.4 Chapter 4
		16.5 Chapter 5
		16.6 Chapter 6
		16.7 Chapter 7
	Chapter 17 Solutions to the Exercises
		17.1 Chapter 8
		17.2 Chapter 9
		17.3 Chapter 10
		17.4 Chapter 11
		17.5 Chapter 12
		17.6 Chapter 13
		17.7 Chapter 14
		17.8 Chapter 15
References
Index