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Abstract Algebra | Ring (Mathematics) | Field (Mathematics)

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  Algebra I:Commutative Algebra Alexander SchmittBerlin, Winter 2012  /  2013 1  i  Table of Contents Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiI Basic Theory of Rings and their Ideals . . . . . . . . . . . . . . . . . . . . . 1 I.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1I.2 Ideals and Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . 8I.3 Zero Divisors, Nilpotent Elements, and Units . . . . . . . . . . . . . . . 11I.4 Prime Ideals and Maximal Ideals . . . . . . . . . . . . . . . . . . . . . . 12I.5 Irreducible Elements and Prime Elements . . . . . . . . . . . . . . . . . 19I.6 Factorial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22I.7 The Nilradical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31I.8 Operations on Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32I.9 Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 II Noetherian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 II.1 Chain Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49II.2 Artinian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54II.3 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54II.4 Primary Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 III The Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 III.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73III.2 Finite Ring Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 91III.3 The Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97III.4 Noether Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 101III.5 Normal Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 IV Dimension Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 IV.1 Krull Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117IV.2 The Going-Up Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 118IV.3 The Transcendence Degree of a Field . . . . . . . . . . . . . . . . . . . . 122IV.4 The Dimension of an Algebraic Variety . . . . . . . . . . . . . . . . . . 123IV.5 Krull’s Principal Ideal Theorem . . . . . . . . . . . . . . . . . . . . . . . 130i  Preface IV.6 Embedding Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 133IV.7 Singular Points of Algebraic Varieties . . . . . . . . . . . . . . . . . . . 135IV.8 Regularity and Normality . . . . . . . . . . . . . . . . . . . . . . . . . . 148 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 ii
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