Steps in Commutative Algebra

Introductory account of commutative algebra, aimed at students with a background in basic algebra.

Rodney Y. Sharp (Author)

9780521646239, Cambridge University Press

Paperback, published 4 January 2001

368 pages
23 x 16.1 x 2.4 cm, 0.54 kg

'… Sharp is an excellent guide, clearly aiming never to leave his readers floundering … This standard of care for his readers is maintained throughout the book … this is a superb guide to an attractive and important area of mathematics, and one from which I will derive pleasure as a retirement project. But it will never be an easy ride.' John Baylis, The Mathematical Gazette

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.

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.

Subject Areas: Algebra [PBF]