Approximation Theorems in Commutative Algebra : Classical and Categorical Methods

By: J. Alajbegovic, J. Mockor

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Algebra

Preface
Classical Methods
Approximation Theorems for Valuations on Fields	p. 3
Introduction to Valuation Theory	p. 3
Partially ordered groups	p. 3
Valuations on fields	p. 6
Approximation Theorems for Krull Valuations	p. 11
Weak approximation theorems	p. 11
Approximation theorems	p. 15
Approximation theorems and upper classes	p. 18
Approximation theorems for Prufer domains	p. 25
Approximation theorems for Prufer rings of Krull type	p. 29
Extensions of approximation theorems	p. 34
Applications in Topological Rings	p. 37
Valuations on Commutative Rings	p. 49
Basic Properties of the Manis Valuation	p. 50
Rings of quotients	p. 50
Large quotient rings	p. 51
Definition of the Manis valuation	p. 52
Comparison of valuations	p. 57
Some useful inequalities	p. 64
R-Prufer Rings	p. 66
Definition of R-Prufer rings	p. 66
R-Prufer valuation rings	p. 69
Valuations with the Inverse Property	p. 72
Definition and basic properties	p. 72
Comparison of valuations with the inverse property	p. 75
Inequalities for valuations with the inverse property	p. 77
The independence of valuations	p. 80
R-Prufer rings and the inverse property	p. 85
Approximation Theorems	p. 89
Compatibility conditions	p. 90
Approximation theorem in the neighborhood of zero	p. 92
General approximation theorem	p. 98
R-Prufer rings and families of valuations	p. 110
Ordered Groups and Homomorphisms	p. 127
Groups of Divisibility	p. 127
Lattice-ordered groups	p. 127
Groups of divisibility	p. 135
Groups with the Theory of Divisors	p. 146
Approximation Theorems for Multistructures	p. 159
Introduction to Multirings	p. 160
Basic facts about m-rings	p. 160
m-valuations	p. 167
Approximation Theorem for Multirings	p. 175
Introduction to d-Groups	p. 187
Approximation Theorems for d-Groups	p. 200
Categorical Methods
Categorical Logic	p. 211
Topoi and Sheaves	p. 211
Interpretation of Logic in Categories	p. 230
The syntax of L	p. 230
The semantic of L	p. 231
Interpretations as subobjects	p. 235
Interpretations as morphisms	p. 237
Relations between interpretations	p. 239
Canonical language and its interpretation	p. 247
Axioms Valid in Interpretations of Logic in Categories	p. 250
Example of completeness	p. 250
Examples of valid sequents	p. 253
Approximation Theorems in Categories	p. 263
Models of a Theory of Approximation Theorems	p. 263
Approximation theorems in algebras	p. 263
Approximation theorems in categories	p. 268
Approximation Theorems and Sheaves	p. 277
[epsilon]-sheaves	p. 277
[epsilon]-sheaves over cofinal subsets	p. 281
Relations between Approximation Theorems	p. 288
Derived approximation theorems	p. 288
Construction of derived approximation theorems	p. 301
Examples	p. 304
Bibliography	p. 307
Index of Notation	p. 317
Index	p. 327
Table of Contents provided by Blackwell. All Rights Reserved.

Approximation Theorems in Commutative Algebra : Classical and Categorical Methods

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