Ideal systems: an introduction to multiplicative ideal theory
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Dekker
1998
|
| Series: | Pure and applied mathematics
211 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 422 S. |
| ISBN: | 0824701860 |
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MARC
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| 245 | 1 | 0 | |a Ideal systems |b an introduction to multiplicative ideal theory |c Franz Halter-Koch |
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Record in the Search Index
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| adam_text | Contents
Preface iii
Part A. GENERAL IDEAL THEORY
1. Monoids and monoid homomorphisms 3
1.1 (Partial) orderings, po groups and o groups 3
1.2 Monoids, groupoids, rings and multiplicatively closed subsets 5
1.3 Homomorphisms, systems of generators and free monoids 6
1.4 (Inner) direct products of monoids 7
1.5 Divisibility in monoids 8
1.6 Congruence relations, factor monoids, reduced monoids 9
1.7 Divisor homomorphisms, saturated and divisor closed subsets 10
1.8 A criterion for saturated submonoids 10
Notes to Chapter 1 11
Exercises for Chapter 1 11
2. Arithmetic of ideal systems 15
2.1 (Weak) ideal systems and r ideals 15
2.2 The s system, the d system and further examples 18
2.3 Products of r ideals 20
2.4 Residuation of r ideals 21
Notes to Chapter 2 22
Exercises for Chapter 2 22
3. Finitary and noetherian ideal systems 25
3.1 The weak ideal system r, 25
3.2 Finitary weak ideal systems: s, d and further examples 27
3.3 Construction of finitary weak ideal systems 28
3.4 Chain conditions in partially ordered sets 29
v
vi CONTENTS
3.5 r noetherian monoids; ACCP 29
3.6 s noetherian cancellative monoids 31
3.7 A criterion for the ACCP 32
Notes to Chapter 3 32
Exercises for Chapter 3 33
4. Monoids of quotients 35
4.1 Definition and elementary properties of quotient monoids 35
4.2 Quotient monoids of cancellative monoids; quotient groupoids 37
4.3 The behavior of s ideals in quotient monoids 39
4.4 Quotients T~lr of weak ideal systems r 40
Notes to Chapter 4 44
Exercises for Chapter 4 45
5. Comparison and mappings of ideal systems 47
5.1 Comparison of weak ideal systems 47
5.2 Pullbacks of weak ideal systems 48
5.3 Reduction of weak ideal systems 50
5.4 (r, r ) homomorphisms and functorial properties 51
5.5 The system of r ideals as a functor 52
Notes to Chapter 5 53
Exercises for Chapter 5 53
6. Prime and primary ideals 55
6.1 Prime r ideals and prime elements 55
6.2 Unions and intersections of prime r ideals 56
6.3 Existence of prime r ideals 56
6.4 r maximal r ideals 57
6.5 r local monoids 58
6.6 Minimal prime ideals and prime divisors 58
6.7 The radical of an r ideal 59
6.8 Primary r ideals and primary elements 61
6.9 Properties of primary r ideals 62
6.10 A special property of finite unions of prime d ideals 62
Notes to Chapter 6 63
Exercises for Chapter 6 63
7. Quotients of primary ideals and primary decompositions 67
7.1 (Reduced) r primary decompositions 67
7.2 Primary decompositions in quotient monoids 69
7.3 Localizations with respect to prime r ideals 71
7.4 Behavior of prime and primary r ideals under localizations 72
7.5 Symbolic powers 72
7.6 Definition and properties of Ass(J) 73
7.7 r irreducible r ideals and r primary decompositions 74
vii
7.8 A laskerian ring is t(d) noetherian 75
Notes to Chapter 7 77
Exercises for Chapter 7 78
8. Strictly noetherian ideal systems 81
8.1 Complete and modular lattices of r ideals 81
8.2 (Almost) r simple r ideals 83
8.3 Characterization of s simple and d simple ideals 85
8.4 (Almost) strictly r noetherian monoids 87
8.5 Almost strictly r noetherian monoids are r laskerian 88
8.6 Krull s Intersection Theorem, first form 89
Notes to Chapter 8 89
Exercises for Chapter 8 91
9. The intersection theorem and the principal ideal theorem 93
9.1 Krull s Intersection Theorem, second form 93
9.2 Lengths of chains in modular lattices 96
9.3 The theorem of Jordan Holder Dedekind 97
9.4 A criterion for the DCC 98
9.5 Strictly r noetherian r local monoids 98
9.6 The Principal Ideal Theorem in the local case 99
9.7 Preliminary version of the Principal Ideal Theorem 101
9.8 Krull s Principal Ideal Theorem 102
9.9 Generalized Krull s Principal Ideal Theorem 103
Notes to Chapter 9 103
Exercises for Chapter 9 104
Part B. MULTIPLICATIVE IDEAL THEORY
10. Abstract elementary number theory 109
10.1 Greatest common divisors 109
10.2 Arithmetic in GCD monoids 110
10.3 Atoms and atomic monoids 112
10.4 The number of factorizations of an element 113
10.5 Atoms, primes and the uniqueness of factorizations 113
10.6 Arithmetic in free monoids and their quotient groupoids 115
10.7 Characterizations of factorial monoids 116
Notes to Chapter 10 117
Exercises for Chapter 10 117
11. Fractional divisorial ideals 121
11.1 Fractional ideals 121
11.2 Fractional ideals in quotient monoids 123
11.3 Fractional ideals as intersections of localizations 124
viii CONTENTS
11.4 Definition of the v and of the t system 124
11.5 Ideal theory of factorial monoids and GCD monoids 126
11.6 Minimal prime ideals are t ideals 127
11.7 Quotients and inverses of fractional ideals 128
11.8 Characterizations of u ideals 130
Notes to Chapter 11 131
Exercises for Chapter 11 132
12. Invertible ideals and class groups 135
12.1 r invertible fractional r ideals 135
12.2 r invertible fractional r ideals in r local monoids 137
12.3 Local criteria for r invertibility 137
12.4 r invertible fractional r ideals in r semilocal integral domains 138
12.5 Krull s Intersection Theorem for r invertible r ideals 139
12.6 The r class group 139
12.7 Localizations of r class groups 140
Notes to Chapter 12 141
Exercises for Chapter 12 142
13. Arithmetic of invertible and cancellative ideals 145
13.1 Properties of primary r invertible r ideals 145
13.2 Primary r ideals with an r invertible radical 146
13.3 r cancellative r ideals 147
13.4 Characterizations of v invertible i; ideals 148
13.5 Properties of t invertible prime t ideals 149
13.6 Strong w ideals 149
13.7 A criterion for overmonoids to be v noetherian 150
13.8 d finitely generated d cancellative d ideals are d invertible 151
Notes to Chapter 13 152
Exercises for Chapter 13 152
14. Integral closures 155
14.1 Almost integral elements and the complete integral closure 155
14.2 r Lntegral elements and the r closure 157
14.3 r integral elements for r = s and r = d 159
14.4 Construction of ideal systems on certain overmonoids 160
14.5 Properties of the canonical ideal system on the r closure 162
Notes to Chapter 14 163
Exercises for Chapter 14 164
15. Valuation monoids and primary monoids 167
15.1 Valuation monoids: Definition and elementary properties 167
15.2 Existence of valuation overmonoids and overrings 169
15.3 Ideal systems on valuation monoids 171
15.4 Primary monoids 172
ix
15.5 Primary valuation monoids 173
Notes to Chapter 15 174
Exercises for Chapter 15 174
16. Ideal theory of valuation monoids 177
16.1 Powers of s ideals in valuation monoids 177
16.2 Prime and primary s ideals in valuation monoids 178
16.3 The complete integral closure of valuation monoids 179
16.4 Discrete valuation monoids (dv monoids) 180
16.5 A criterion for a localization to be a dv monoid 181
16.6 Non principal s ideals in valuation monoids 182
16.7 Pseudo valuation monoids 182
Notes to Chapter 16 184
Exercises for Chapter 16 184
17. Priifer and Bezout monoids 187
17.1 r Priifer and r Bezout monoids 187
17.2 Characterizations of r Pnifer monoids 189
17.3 Characterizations of Priifer domains 192
17.4 Prime and primary r ideals in r Priifer monoids 192
17.5 Bezout domains with finitely many maximal ideals 193
17.6 Generalized GCD monoids 194
Notes to Chapter 17 195
Exercises for Chapter 17 196
18. Essential homomorphisms, GCD homomorphisms and valuations ... 199
18.1 Properties of divisor homomorphisms 199
18.2 Essential homomorphisms and essential overmonoids 200
18.3 GCD homomorphisms 202
18.4 r valuations and r valuation monoids 203
18.5 Z valuation monoids of a GCD monoid 204
Notes to Chapter 18 205
Exercises for Chapter 18 205
19. Lorenzen monoids 207
19.1 The Lorenzen r groupoid and the Lorenzen r monoid 207
19.2 Characterizations of ti Prufer monoids 210
19.3 The ideal systems sa and da 211
19.4 Structure of the Lorenzen r monoid 213
19.5 The Lorenzen r monoid of an r Bezout monoid 215
Notes to Chapter 19 216
Exercises for Chapter 19 216
20. Quasi divisor theories 219
20.1 Universal properties of the Lorenzen r monoid 219
x CONTENTS
20.2 Quasi divisor theories and r Prufer monoids 222
20.3 Divisor homomorphisms into GCD monoids 224
20.4 Existence and uniqueness of quasi divisor theories 224
20.5 Quasi divisor theories and the t class group 225
20.6 The behavior of quasi divisor theories under quotients 226
20.7 Valuation monoids and quasi divisor theories 227
20.8 A further characterization of r Prufer monoids 228
Notes to Chapter 20 229
Exercises for Chapter 20 229
21. Defining systems 233
21.1 Defining systems (of finite character) 233
21.2 Defining systems of localizations 234
21.3 Ideal systems induced by defining systems 234
21.4 Ideal systems induced by quotient monoids 237
21.5 Ideal systems induced by localizations 237
21.6 Defining systems under divisor homomorphisms 239
21.7 The r closure of a monoid and r valuation monoids 240
Notes to Chapter 21 242
Exercises for Chapter 21 243
22. Krull monoids and generalizations 247
22.1 Krull monoids and various generalizations 247
22.2 Essential valuation monoids and quasi divisor theories 248
22.3 GCD monoids of Krull type 249
22.4 Monoids of Krull type and generalized Krull monoids 251
22.5 Ideal theory of weakly Krull monoids 252
22.6 A preliminary characterization of weakly Krull monoids 253
22.7 Characterizations of weakly Krull monoids 254
22.8 Characterizations of Krull monoids 256
Notes to Chapter 22 257
Exercises for Chapter 22 257
23. (Almost) Dedekind and Krull monoids 261
23.1 Definition and characterizations of almost Dedekind monoids 261
23.2 Characterizations of almost Dedekind domains 263
23.3 Definition and characterizations of Dedekind monoids 264
23.4 Further characterizations of Krull monoids 267
23.5 An isomorphism theorem for Krull monoids 268
23.6 A construction principle for Krull monoids 269
23.7 Krull monoids with prescribed class groups 270
23.8 Krull monoids are block monoids 271
Notes to Chapter 23 273
Exercises for Chapter 23 273
xi
24. t noetherian monoids 275
24.1 Characterizations of t noetherian monoids 275
24.2 A finiteness result for t noetherian monoids 277
24.3 ^ irreducible ideals in i noetherian monoids 277
24.4 Maximal associated prime ideals 278
24.5 Characterizations of weakly Krull t noetherian monoids 279
24.6 Subintersections of localizations of t noetherian monoids 279
24.7 Strong and t invertible t maximal t ideals 281
24.8 The t closure of t noetherian monoids 281
Notes to Chapter 24 283
Exercises for Chapter 24 283
25. Approximation theorems 285
25.1 Independence of overmonoids 285
25.2 Orthogonality of overmonoids 286
25.3 The Weak Approximation Theorem 286
25.4 Quasi divisor theories and the Weak Approximation Theorem 288
25.5 The Weak Approximation Theorem for integral domains 289
25.6 The Approximation Theorem 291
Notes to Chapter 25 292
Exercises for Chapter 25 292
26. Divisorial defining systems and class groups 295
26.1 Divisorial defining systems 295
26.2 Criteria for being divisorial 296
26.3 Denning systems of weakly Krull monoids 296
26.4 Approximation properties of Krull monoids 297
26.5 The class group of a defining system 298
26.6 Class groups and approximation properties 301
Notes to Chapter 26 302
Exercises for Chapter 26 302
27. Arithmetical properties of overmonoids 305
27.1 Characterizations of valuation monoids (valuation rings) 305
27.2 Structure of overmonoids of r Priifer monoids 309
27.3 Characterizations of Priifer domains 310
27.4 Overmonoids of monoids of Krull type 311
27.5 Overrings of generalized Krull domains and (almost) Dedekind domains ... 312
Notes to Chapter 27 312
Exercises for Chapter 27 313
Solutions of Exercises 317
Chapter 1 317
Chapter 2 319
xii CONTENTS
Chapter 3 321
Chapter 4 323
Chapter 5 325
Chapter 6 327
Chapter 7 331
Chapter 8 334
Chapter 9 338
Chapter 10 341
Chapter 11 346
Chapter 12 349
Chapter 13 353
Chapter 14 355
Chapter 15 , 358
Chapter 16 360
Chapter 17 363
Chapter 18 366
Chapter 19 369
Chapter 20 370
Chapter 21 374
Chapter 22 378
Chapter 23 382
Chapter 24 385
Chapter 25 388
Chapter 26 390
Chapter 27 394
A guide to results on special integral domains 399
Bibliography 401
List of Symbols 417
Index 419
|
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| indexdate | 2024-07-09T18:19:05Z |
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| spelling | Halter-Koch, Franz 1944- Verfasser (DE-588)117711241 aut Ideal systems an introduction to multiplicative ideal theory Franz Halter-Koch New York [u.a.] Dekker 1998 XII, 422 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 211 Algèbre - Problèmes et exercices ram Anneaux (algèbre) ram Groupe classe Idéaux (algèbre) ram Système idéal Théorie multiplicative idéal Ideals (Algebra) Rings (Algebra) Multiplikative Idealtheorie (DE-588)4348597-2 gnd rswk-swf Multiplikative Idealtheorie (DE-588)4348597-2 s DE-604 Pure and applied mathematics 211 (DE-604)BV000001885 211 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008081740&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Halter-Koch, Franz 1944- Ideal systems an introduction to multiplicative ideal theory Pure and applied mathematics Algèbre - Problèmes et exercices ram Anneaux (algèbre) ram Groupe classe Idéaux (algèbre) ram Système idéal Théorie multiplicative idéal Ideals (Algebra) Rings (Algebra) Multiplikative Idealtheorie (DE-588)4348597-2 gnd |
| subject_GND | (DE-588)4348597-2 |
| title | Ideal systems an introduction to multiplicative ideal theory |
| title_auth | Ideal systems an introduction to multiplicative ideal theory |
| title_exact_search | Ideal systems an introduction to multiplicative ideal theory |
| title_full | Ideal systems an introduction to multiplicative ideal theory Franz Halter-Koch |
| title_fullStr | Ideal systems an introduction to multiplicative ideal theory Franz Halter-Koch |
| title_full_unstemmed | Ideal systems an introduction to multiplicative ideal theory Franz Halter-Koch |
| title_short | Ideal systems |
| title_sort | ideal systems an introduction to multiplicative ideal theory |
| title_sub | an introduction to multiplicative ideal theory |
| topic | Algèbre - Problèmes et exercices ram Anneaux (algèbre) ram Groupe classe Idéaux (algèbre) ram Système idéal Théorie multiplicative idéal Ideals (Algebra) Rings (Algebra) Multiplikative Idealtheorie (DE-588)4348597-2 gnd |
| topic_facet | Algèbre - Problèmes et exercices Anneaux (algèbre) Groupe classe Idéaux (algèbre) Système idéal Théorie multiplicative idéal Ideals (Algebra) Rings (Algebra) Multiplikative Idealtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008081740&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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