Commutative Algebra

This Second Volume Of Our Treatise On Commutative Algebra Deals Largely With Three Basic Topics, Which Go Beyond The More Or Less Classical Material Of Volume I And Are On The Whole Of A More Advanced Nature And A More Recent Vintage. These Topics Are: (a) Valuation Theory; (b) Theory Of Polynomial And Power Series Rings (including Generalizations To Graded Rings And Modules); (c) Local Algebra. Because Most Of These Topics Have Either Their Source Or Their Best Motivation In Algebraic Geom­ Etry, The Algebro-geometric Connections And Applications Of The Purely Algebraic Material Are Constantly Stressed And Abundantly Scattered Through­ Out The Exposition. Thus, This Volume Can Be Used In Part As An Introduc­ Tion To Some Basic Concepts And The Arithmetic Foundations Of Algebraic Geometry. The Reader Who Is Not Immediately Concerned With Geometric Applications May Omit The Algebro-geometric Material In A First Reading (see Instructions To The Reader, Page Vii), But It Is Only Fair To Say That Many A Reader Will Find It More Instructive To Find Out Immediately What Is The Geometric Motivation Behind The Purely Algebraic Material Of This Volume. The First 8 Sections Of Chapter Vi (including § 5bis) Deal Directly With Properties Of Places, Rather Than With Those Of The Valuation Associated With A Place. These, Therefore, Are Properties Of Valuations In Which The Value Group Of The Valuation Is Not Involved. Valuation Theory -- Polynomial And Power Series Rings -- Local Algebra. By Oscar Zariski, Pierre Samuel.