Introduction To Commutative AlgebraFirst Published in 2018. This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of producing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts such as Zariski-Samuel or Bourbaki. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization. |
Contents
Modules | |
Rings and Modules of Fractions | |
Primary Decomposition | |
Integral Dependence and Valuations | |
Chain Conditions | |
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Common terms and phrases
A-algebra A-module homomorphism a₁ abelian group Artin ring Artinian chain of prime Chapter closure coefficients commutative algebra composition series coprime Corollary Dedekind domain Deduce defined denote dimension discrete valuation ring equation exact sequence example Exercise exists field of fractions finitely generated A-module finitely-generated A-module flat following are equivalent fractional ideal graded ring Hausdorff hence induces injective integral domain integrally closed intersection inverse irreducible isomorphism Jacobson radical kernel lemma Let f m/m² M₁ mapping maximal element maximal ideal minimal primary decomposition minimal prime ideal module multiplicatively closed subset nilpotent nilradical Noetherian local ring Noetherian ring open sets p-primary p₁ polynomial ring power series primary decomposition primary ideals prime ideals belonging Proof Proposition Prove q₁ quotient resp ring and let ring homomorphism SıA satisfies Show Spec submodule subring Suppose surjective tensor product Theorem topology unique x₁ y₁ zero-divisor