Commutative Algebra Volume I

Title
Preface
Contents
I. Introductory concepts
	§ 1. Binary operations
	§ 2. Groups
	§ 3. Subgroups
	§ 4. Abelian groups
	§ 5. Rings
	§ 6. Rings with identity
	§ 7. Powers and multiples
	§ 8. Fields
	§ 9. Subrings and subfields
	§ 10. Transformations and mappings
	§ 11. Group homomorphisms
	§ 12. Ring homomorphisms
	§ 13. Identification of rings
	§ 14. Unique factorization domains
	§ 15. Euclidean domains
	§ 16. Polynomials in one indeterminate
	§ 17. Polynomial rings
	§ 18. Polynomials in several indeterminates
	§ 19. Quotient fields and total quotient rings
	§ 20. Quotient rings with respect to multiplicative systems
	§ 21. Vector spaces
II. Elements of field theory
	§ 1. Field extensions
	§ 2. Algebraic quantities
	§ 3. Algebraic extensions
	§ 4. The characteristic of a field
	§ 5. Separable and inseparable algebraic extension
	§ 6. Splitting fields and normal extensions
	§ 7. The fundamental theorem of Galois theory
	§ 8. Galois fields
	§ 9. The theorem of the primitive element
	§ 10. Field polynomials. Norms and traces
	§ 11. The discriminant
	§ 12. Transcendental extensions
	§ 13. Separably generated fields of algebraic functions
	§ 14. Algebraically closed fields
	§ 15. Linear disjointness and separability
	§ 16. Order of inseparability of a field of algebraic functions
	§ 17. Derivations
III. Ideals and modules
	§ 1. Ideals and modules
	§ 2. Operations on submodules
	§ 3. Operator homomorphisms and difference modules
	§ 4. The isomorphism theorems
	§ 5. Ring homomorphisms and residue class rings
	§ 6. The order of a subset of a module
	§ 7. Operations on ideals
	§ 8. Prime and maximal ideals
	§ 9. Primary ideals
	§ 10. Finiteness conditions
	§ 11. Composition series
	§ 12. Direct sums
	§ 12bis. Infinite direct sums
	§ 13. Comaximal ideals and direct sums of ideals
	§ 14. Tensor products of rings
	§ 15. Free joins of integral domains (or of fields)
IV. Noetherian rings
	§ 1. Definitions. The Hilbert basis theorem
	§ 2. Rings with descending chain condition
	§ 3. Primary rings
	§ 3bis. Alternative method for studying the rings with d.c.c.
	§ 4. The Lasker-Noether decomposition theorem
	§ 5. Uniqueness theorems
	§ 6. Application to zero-divisors and nilpotent elements
	§ 7. Application to the intersection of the powers of an ideal
	§ 8. Extended and contracted ideals
	§ 9. Quotient rings
	§ 10. Relations between ideals in R and ideals in R_M
	§ 11. Examples and applications of quotient rings
	§ 12. Symbolic powers
	§ 13. Length of an ideal
	§ 14. Prime ideals in noetherian rings
	§ 15. Principal ideal rings
	§ 16. Irreducible ideals
	Appendix: Primary representation in Noetherian modules
V. Dedekind domains, classical ideal theory
	§ 1. Integral elements
	§ 2. Integrally dependent rings
	§ 3. Integrally closed rings
	§ 4. Finiteness theorems
	§ 5. The conductor of an integral closure
	§ 6. Characterizations of Dedekind domains
	§ 7. Further properties of Dedekind domains
	§ 8. Extensions of Dedekind domains
	§ 9. Decomposition of prime ideals in extensions of Dedekind domains
	§ 10. Decomposition group, inertia group, and ramification groups
	§ 11. Different and discriminant
	§ 12. Application to quadratic fields and cyclotomic fields
	§ 13. A theorem of Kummer
Index of notations
Index of definitions