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Approximation Theorems in Commutative Algebra : Classical and Categorical Methods : Mathematics and Its Applications - J. Alajbegovic

Approximation Theorems in Commutative Algebra : Classical and Categorical Methods

Mathematics and Its Applications

Hardcover ISBN: 9780792319481
Number Of Pages: 330

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Various types of approximation theorems are frequently used in general commutative algebra, and they have been found to be useful tools in valuation theory, the theory of Abelian lattice ordered groups, multiplicative ideal theory, etc. Part 1 of this volume is devoted to the investigation of approximation theorems from a classical point of view. The chapters of this part deal with fields and rings, partly ordered groups, and with multirings and d-groups. Part II investigates approximation theorems from a general, categorical point of view. This part is essentially self-contained and requires only a basic knowledge of category theory and first-order logic. For researchers and graduate students of commutative algebra, category theory, as well as applications of logic.

Classical Methods
Approximation Theorems for Valuations on Fieldsp. 3
Introduction to Valuation Theoryp. 3
Partially ordered groupsp. 3
Valuations on fieldsp. 6
Approximation Theorems for Krull Valuationsp. 11
Weak approximation theoremsp. 11
Approximation theoremsp. 15
Approximation theorems and upper classesp. 18
Approximation theorems for Prufer domainsp. 25
Approximation theorems for Prufer rings of Krull typep. 29
Extensions of approximation theoremsp. 34
Applications in Topological Ringsp. 37
Valuations on Commutative Ringsp. 49
Basic Properties of the Manis Valuationp. 50
Rings of quotientsp. 50
Large quotient ringsp. 51
Definition of the Manis valuationp. 52
Comparison of valuationsp. 57
Some useful inequalitiesp. 64
R-Prufer Ringsp. 66
Definition of R-Prufer ringsp. 66
R-Prufer valuation ringsp. 69
Valuations with the Inverse Propertyp. 72
Definition and basic propertiesp. 72
Comparison of valuations with the inverse propertyp. 75
Inequalities for valuations with the inverse propertyp. 77
The independence of valuationsp. 80
R-Prufer rings and the inverse propertyp. 85
Approximation Theoremsp. 89
Compatibility conditionsp. 90
Approximation theorem in the neighborhood of zerop. 92
General approximation theoremp. 98
R-Prufer rings and families of valuationsp. 110
Ordered Groups and Homomorphismsp. 127
Groups of Divisibilityp. 127
Lattice-ordered groupsp. 127
Groups of divisibilityp. 135
Groups with the Theory of Divisorsp. 146
Approximation Theorems for Multistructuresp. 159
Introduction to Multiringsp. 160
Basic facts about m-ringsp. 160
m-valuationsp. 167
Approximation Theorem for Multiringsp. 175
Introduction to d-Groupsp. 187
Approximation Theorems for d-Groupsp. 200
Categorical Methods
Categorical Logicp. 211
Topoi and Sheavesp. 211
Interpretation of Logic in Categoriesp. 230
The syntax of Lp. 230
The semantic of Lp. 231
Interpretations as subobjectsp. 235
Interpretations as morphismsp. 237
Relations between interpretationsp. 239
Canonical language and its interpretationp. 247
Axioms Valid in Interpretations of Logic in Categoriesp. 250
Example of completenessp. 250
Examples of valid sequentsp. 253
Approximation Theorems in Categoriesp. 263
Models of a Theory of Approximation Theoremsp. 263
Approximation theorems in algebrasp. 263
Approximation theorems in categoriesp. 268
Approximation Theorems and Sheavesp. 277
[epsilon]-sheavesp. 277
[epsilon]-sheaves over cofinal subsetsp. 281
Relations between Approximation Theoremsp. 288
Derived approximation theoremsp. 288
Construction of derived approximation theoremsp. 301
Examplesp. 304
Bibliographyp. 307
Index of Notationp. 317
Indexp. 327
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792319481
ISBN-10: 0792319486
Series: Mathematics and Its Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 330
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.5  x 2.0
Weight (kg): 0.69