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El. knyga: Simple Extensions with the Minimum Degree Relations of Integral Domains

(Kochi University, Japan)
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Although there are many types of ring extensions, simple extensions have yet to be thoroughly explored in one book. Covering an understudied aspect of commutative algebra, Simple Extensions with the Minimum Degree Relations of Integral Domains presents a comprehensive treatment of various simple extensions and their properties. In particular, it examines several properties of simple ring extensions of Noetherian integral domains.

As experts who have been studying this field for over a decade, the authors present many arguments that they have developed themselves, mainly exploring anti-integral, super-primitive, and ultra-primitive extensions. Within this framework, they study certain properties, such as flatness, integrality, and unramifiedness. Some of the topics discussed include Sharma polynomials, vanishing points, Noetherian domains, denominator ideals, unit groups, and polynomial rings.

Presenting a complete treatment of each topic, Simple Extensions with the Minimum Degree Relations of Integral Domains serves as an ideal resource for graduate students and researchers involved in the area of commutative algebra.

Recenzijos

"All topics are developed in a clear way and illustrated by many examples."

EMS Newsletter, September 2008

Chapter 1 Birational Simple Extensions 1
1.1 The Ring R[ α] intersection R[ α-1]
2
1.2 Anti-Integral Extension and Flat Simple Extensions
5
1.3 The Ring R(Iα) and the Anti-Integrality of α
12
1.4 Strictly Closedness and Integral Extensions
14
1.5 Upper-Prime, Upper-Primary, or Upper-Quasi-Primary Ideals
15
1.6 Some Subsets of Spec(R) in the Birational Case
20
Chapter 2 Simple Extensions of High Degree 23
2.1 Sharma Polynomials
24
2.2 Anti-Integral Elements and Super-Primitive Elements
27
2.3 Integrality and Flatness of Anti-Integral Extensions
33
2.4 Anti-Integrality of α and α-1
37
2.5 Vanishing Points and Blowing-Up Points
42
Chapter 3 Subrings of Anti-Integral Extensions 49
3.1 Extensions R[ α] intersection R[ α-1] of Noetherian Domains R
49
3.2 The Integral Closedness of the Ring R[ α] R[ α-1] (I)
65
3.3 The Integral Closedness of the Ring R[ α] intersectiom R [ α-1] (II)
71
3.4 Extensions of Type R[ β] intersection R[ β-1] with β element of K (α)
80
Chapter 4 Denominator Ideals and Excellent Elements 97
4.1 Denominator Ideals and Flatness (I)
97
4.2 Excellent Elements of Anti-Integral Extensions
99
4.3 Flatness and LCM-Stableness
103
4.4 Some Subsets of Spec(R) in the High Degree Case
108
Chapter 5 Unramified Extensions 111
5.1 Unramifiedness and Etaleness of Super-Primitive Extensions
111
5.2 Differential Modules of Anti-Integral Extensions
114
5.3 Kernels of Derivations on Simple Extensions
120
Chapter 6 The Unit-Groups of Extensions 125
6.1 The Unit-Groups of Anti-Integral Extensions
126
6.2 Invertible Elements of Super-Primitive Ring Extensions
129
Chapter 7 Exclusive Extensions of Noetherian Domains 135
7.1 Subring R[ α] intersection K of Anti-Integral Extensions
136
7.2 Exclusive Extensions and Integral Extensions
143
7.3 An Exclusive Extension Generated by a Super-Primitive Element
145
7.4 Finite Generation of an Intersection R[ α] intersection K over R
155
7.5 Pure Extensions
160
Chapter 8 Ultra-Primitive Extensions and Their Generators 167
8.1 Super-Primitive Elements and Ultra-Primitive Elements
168
8.2 Comparisons of Subrings of Type R[ aα] intersection R[ (aα)-1]
175
8.3 Subrings of Type R[ Hα] intersection R[ (Hα)-1]
183
8.4 A Linear Generator of an Ultra-Primitive Extension R[ α]
189
8.5 Two Generators of Simple Extensions
194
Chapter 9 Flatness and Contractions of Ideals 201
9.1 Flatness of a Birational Extension
201
9.2 Flatness of a Non-Birational Extension
203
9.3 Anti-Integral Elements and Coefficients of its Minimal Polynomial
209
9.4 Denominator Ideals and Flatness (II)
218
9.5 Contractions of Principal Ideals and Denominator Ideals
224
Chapter 10 Anti-Integral Ideals and Super-Primitive Polynomials 233
10.1 Anti-Integral Ideals and Super-Primitive Ideals
234
10.2 Super-Primitive Polynomials and Sharma Polynomials
239
10.3 Anti-Integral, Super-Primitive, or Flat Polynomials
243
Chapter 11 Semi Anti-Integral and Pseudo-Simple Extensions 249
11.1 Anti-Integral Extensions of Polynomial Rings
249
11.2 Subrings of R[ α] Associated with Ideals of R
253
11.3 Semi Anti-Integral Elements
259
11.4 Pseudo-Simple Extensions
262
References 269
Index 275


Kochi University, Japan Okayama University of Science, Japan
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