Commutative Algebra II
From the Preface: "topics are: (a) valuation theory; (b) theory of polynomial and power series rings (including generalizations to graded rings and modules); (c) local algebra... the algebro-geometric connections and applications of the purely algebraic material are constantly stressed and abundantly scattered throughout the exposition. Thus, this volume can be used in part as an introduction to some basic concepts and the arithmetic foundations of algebraic geometry."
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A-module affine variety algebraic extension assertion associated prime ideal assume belong closure coefficients complete ideal completes the proof contains contracted ideal Corollary decomposition defined degree q denote dimension direct sum directional form exists an element fact finite integral domain finite module finite number follows graded ring Hausdorff space Hence homogeneous elements homogeneous ideal homomorphism implies integral domain integrally closed intersection irreducible irredundant isolated prime ideal isolated subgroup isomorphic k[xlt x2 k[xv x2 kernel Lemma m-topology mapping maximal ideal noetherian ring non-negative polynomial ring power series ring primary prime divisor prime sequence principal ideal Proposition prove quotient field quotient ring rank regular local ring relation residue field respect satisfies semi-local ring submodule subring subset system of parameters Theorem 19 topology transcendence degree transcendental valuation ideals valuation ring value group whence zero divisor