Steps in Commutative AlgebraCambridge University Press, 2000 - 355 páginas 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. |
Conteúdo
1 Commutative rings and subrings | 1 |
2 Ideals | 18 |
3 Prime ideals and maximal ideals | 37 |
4 Primary decomposition | 61 |
5 Rings of fractions | 80 |
6 Modules | 101 |
7 Chain conditions on modules | 123 |
8 Commutative Noetherian rings | 145 |
11 Canonical forms for square matrices | 208 |
12 Some applications to field theory | 220 |
13 Integral dependence on subrings | 243 |
14 Afflne algebras over fields | 264 |
15 Dimension theory | 288 |
16 Regular sequences and grade | 311 |
17 CohenMacaulay rings | 328 |
| 345 | |
Termos e frases comuns
a₁ algebraically independent Artinian ascending chain canonical form chain of prime Chapter Cohen-Macaulay ring commutative algebra commutative Noetherian ring commutative ring composition series contradiction COROLLARY deduce DEFINITION denote direct sum EXERCISE exists field extension finitely generated R-module follows functor grade Hence indeterminates X1 integral domain irreducible element K[Y₁ Ker ƒ LEMMA Let f Let G Let the situation M-sequence M₁ matrix maximal ideal minimal member minimal polynomial minimal primary decomposition minimal prime ideal module monic multiplicatively closed subset natural ring homomorphism non-empty non-zero finitely non-zerodivisor Note P-primary P₁ polynomial ring precisely primary decomposition primary ideal principal ideal domain Proof proper ideal proved R-algebra R-homomorphisms reader regular local ring relative to F residue class result ring and let ring homomorphism Show Spec(R submodule submodules of G subring Suppose surjective theory unique vector space zerodivisor