Addison-Wesley Series in Mathematics, 1969
"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.