Addison-Wesley Series in Mathematics, 1969

Commutative Algebra

"Commutative algebra is essentially the study of commutative rings. Roughly speaking, it has developed from two sources: (1) algebraic geometry and (2) algebraic number theory. In (1) the prototype of the rings studied is the ring $ k[x_1, \ldots x_n] $ of polynomials in several variables over a field k; in (2) it is the ring $ \mathbb{Z} $ of rational integers. Of these two the algebro-geometric case is the more far-reaching and, in its modern development by Grothendieck, it embraces much of algebraic number theory. Commutative algebra is now one of the foundation stones of this new algebraic geometry. It provides the complete local tools for the subject in much the same way as differential analysis provides the tools for differential geometry."

Chapter 1 : Rings and IdealsEdit

Rings and Ring HomomorphismsEdit

Ideals. Quotient Rings.Edit

Zero-divisors. Nilpotent Elements. Units.Edit

Prime ideals and maximal ideals.Edit

Nilradical and Jacobson RadicalEdit

Operations on idealsEdit

  • Proposition 1.10: The notation $ \prod \mathfrak{a}_i $ denotes the product operation of the ring, not a direct product of rings.

Extension and contractionEdit

In general, $ f(f^{-1}(X)) \subseteq X \subseteq f^{-1}(f(X)). $

  • Proposition 1.17: Note that f is not necessarily surjective.

Chapter 2: ModulesEdit

Chapter 3: Rings and Modules of FractionsEdit

Chapter 4: Primary DecompositionEdit

Chapter 5: Integral Dependence and ValuationsEdit

Chapter 6: Chain ConditionsEdit

Chapter 7: Noetherian RingsEdit

Chapter 8: Artin RingsEdit

Chapter 9: Discrete Valuation Rings and Dedekind DomainsEdit

Chapter 10: CompletionsEdit

Chapter 11: Dimension TheoryEdit