Commutative Algebra

Description

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