Steps in Commutative Algebra
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This introductory account of commutative algebra is aimed at advanced undergraduates and first year graduate students. Assuming only basic abstract algebra, it provides a good foundation in commutative ring theory, from which the reader can proceed to more advanced works in commutative algebra and algebraic geometry. The style throughout is rigorous but concrete, with exercises and examples given within chapters, and hints provided for the more challenging problems used in the subsequent development. After reminders about basic material on commutative rings, ideals and modules are extensively ...
This introductory account of commutative algebra is aimed at advanced undergraduates and first year graduate students. Assuming only basic abstract algebra, it provides a good foundation in commutative ring theory, from which the reader can proceed to more advanced works in commutative algebra and algebraic geometry. The style throughout is rigorous but concrete, with exercises and examples given within chapters, and hints provided for the more challenging problems used in the subsequent development. After reminders about basic material on commutative rings, ideals and modules are extensively discussed, with applications including to canonical forms for square matrices. The core of the book discusses the fundamental theory of commutative Noetherian rings. Affine algebras over fields, dimension theory and regular local rings are also treated, and for this second edition two further chapters, on regular sequences and Cohen-Macaulay rings, have been added. This book is ideal as a route into commutative algebra.
Table of contents:
Prefaces to the 1st and 2nd editions; 1. Commutative rings and subrings; 2. Ideals; 3. Prime ideals and maximal ideals; 4. Primary decomposition; 5. Rings of fractions; 6. Modules; 7. Chain conditions on modules; 8. Commutative Noetherian rings; 9. More module theory; 10. Modules over principal ideal domains; 11. Canonical forms for square matrices; 12. Some applications to field theory; 13. Integral dependence on subrings; 14. Affine algebras over fields; 15. Dimension theory; 16. Regular sequences and grade; 17. Cohen-Macaulay rings; Bibliography; Index.
This introductory account of commutative algebra is aimed at students with a background only in basic algebra. It is thus suitable for first-year graduate students and advanced undergraduates, with a style which is throughout rigorous but concrete. This new edition contains additional chapters on regular sequences and on Cohen-Macaulay rings.
Introductory account of commutative algebra, aimed at students with a background in basic algebra.
Table of contents:
Prefaces to the 1st and 2nd editions; 1. Commutative rings and subrings; 2. Ideals; 3. Prime ideals and maximal ideals; 4. Primary decomposition; 5. Rings of fractions; 6. Modules; 7. Chain conditions on modules; 8. Commutative Noetherian rings; 9. More module theory; 10. Modules over principal ideal domains; 11. Canonical forms for square matrices; 12. Some applications to field theory; 13. Integral dependence on subrings; 14. Affine algebras over fields; 15. Dimension theory; 16. Regular sequences and grade; 17. Cohen-Macaulay rings; Bibliography; Index.
This introductory account of commutative algebra is aimed at students with a background only in basic algebra. It is thus suitable for first-year graduate students and advanced undergraduates, with a style which is throughout rigorous but concrete. This new edition contains additional chapters on regular sequences and on Cohen-Macaulay rings.
Introductory account of commutative algebra, aimed at students with a background in basic algebra.