Advances in Ring Theory (eBook)

Title Page 3
Copyright Page 4
Table of Contents 5
Preface 8
Applications of Cogalois Theory to Elementary Field Arithmetic 9
1. Introduction 9
2. Notation and terminology 10
3. What is Cogalois theory? 11
4. Basic concepts and results of Cogalois theory 14
G-Radical extensions 15
G-Kneser extensions 15
The Kneser criterion 15
Cogalois extensions 16
Galois and Cogalois connections 17
Strongly G-Kneser extensions 18
G-Cogalois extensions 19
5. Examples of G-Cogalois extensions 19
6. Applications to elementary field arithmetic 20
6.1. Effective degree computation: 20
6.2. Exhibiting extension basis: 21
6.3. Finding all intermediate fields: 21
6.4. Primitive element: 22
6.5. When is a sum of radicals of positive rational numbers a rational number? 22
6.6. When can a positive algebraic number a be written as a finite sum of real numbers of type ± nvi ai , 1< i <
6.7. When can a positive superposed radical not be decomposed into a finite sum of real numbers of type ± nvi ai , 1< i <
6.8. When is a rational combination of powers from a given set of radicals of positive rational numbers itself a radical of a positive rational number? 23
6.9. Radical extensions of prime exponent: 23
6.10. Simple radical separable extensions having the USP: 23
7. Other applications 24
7.1. Binomial ideals and Grobner bases 24
7.2. Hecke’s systems of ideal numbers 24
References 24
On Big Lattices of Classes of R-modules Defined by Closure Properties 26
1. Introduction 26
1.1. The skeletons of R-tors, R-Serre and R-op 29
2. The big lattice R-sext 30
2.1. R-sext and R-nat 34
3. The big lattice R-qext 35
3.1. R-qext and R-conat 36
4. R-nat and R-conat 36
5. R-sext and R-qext 38
References 42
Reversible and Duo Group Rings 44
1. Introduction 44
2. Reversibility in group rings 45
2.1. Reversibility in group algebras KG 45
2.2. Reversible group rings over commutative rings 46
2.3. Minimal reversible group rings 48
3. Duo group rings 48
3.1. Duo group algebras 49
3.2. Duo group rings over integral domains 50
4. Graded reversibility in integral group rings 52
References 53
Principally Quasi-Baer Ring Hulls 54
References 67
Strongly Prime Ideals of Near-rings of Continuous Functions 69
1. Preliminaries 69
2. Strongly prime ideals in N0(Rn) 70
3. Strongly prime ideals in N0(R.) 72
References 74
Elements of Minimal Prime Ideals in General Rings 75
Introduction 75
1. On r-strongly prime ideals 76
2. Weak zero-divisors 78
3. Examples and special rings 82
References 87
On a Theorem of Camps and Dicks 88
1. The theorem 88
References 89
Applications of the Stone Duality in the Theory of Precompact Boolean Rings 90
1. Preliminaries 90
2. Topologies on a Boolean ring 92
3. Countably linearly compact Boolean rings 94
4. On minimal topologies 95
5. Intersection of totally bounded topologies 96
6. The Bohr topology on a Boolean ring 97
7. Compact topologies on Boolean rings 99
8. Dense ideals of a ring 100
9. Self-injective Boolean rings 100
10. Zero-dimensional F-spaces 103
11. Necessary conditions for countably compactness 104
12. Basically disconnected spaces 106
13. Open questions 114
References 115
Over Rings and Functors 117
Introduction 117
1. Preliminaries 119
2. Large right ideals 120
3. Mod-R and mod-Q for R . Q 121
4. Over rings 126
5. Lattices 128
6. Examples 130
References 133
On Some Classes of Repeated-root Constacyclic Codes of Length a Power of 2 over Galois Rings 134
1. Introduction 134
2. Chain rings, Galois rings, and constacyclic codes 136
3. Negacyclic codes of length 2s over GR(2a,m) 139
4. Some classes of constacyclic codes of length 2s over GR(2a,m) 143
References 148
Couniformly Presented Modules and Dualities 151
1. Introduction 151
2. Couniformly presented modules 152
3. Epigeny class and lower part 156
4. Weak Krull-Schmidt Theorem 158
5. Kernels of morphisms between indecomposable injective modules 160
6. A further duality between epigeny classes and monogeny classes 162
References 165
Semiclassical Limits of Quantized Coordinate Rings 167
0. Introduction 167
1. Quantized coordinate rings 169
1.1. Quantum SL2. 169
1.2. Quantum matrices, quantum SLn and GLn. 170
1.3. Quantum affine spaces. 170
1.4. Quantized coordinate rings of semisimple groups. 171
1.5. Additional examples. 171
1.6. Limits of families of algebras. 171
1.7. An older example: the Weyl algebra. 172
2. Semiclassical limit constructions 172
2.1. Semiclassical limits: commutative fibre version. 172
2.2. Examples 173
2.3. Multiparameter examples. 174
2.4. Semiclassical limits: filtered/graded version. 175
2.5. Bridging the two constructions. 175
2.6. Example: enveloping algebras. 176
3. Symplectic leaves 177
3.1. Poisson algebras. 177
3.2. Symplectic leaves in Poisson manifolds. 177
3.3. Poisson bivector fields. 177
3.4. Poisson varieties. 178
3.5. Smooth Poisson varieties as manifolds. 178
3.6. Symplectic leaves in singular Poisson varieties. 179
3.7. Example. 179
3.8. Example. 179
4. The Orbit Method from Lie theory 180
4.1. The Orbit Method. 180
4.2. Theorem. 180
4.3. Example. 181
4.4. A general principle. 181
4.5. Generic versus non-generic situations. 181
5. Limitations of the Orbit Method for solvable Lie algebras 182
5.1. The algebraic adjoint group. 182
5.2. Prime and primitive spectra. 183
5.3. The Dixmier map. 183
5.4. Theorem. 183
5.5. Algebraic versus non-algebraic cases. 183
5.6. Example. 183
6. Poisson ideal theory and symplectic cores 184
6.1. Poisson prime ideals. 184
6.2. Lemma. 185
6.3. Poisson-primitive ideals and symplectic cores. 185
6.4. Example. 186
7. Symplectic cores versus symplectic leaves 186
7.1. Theorem. 186
7.2. Lemma. 187
7.3. Lemma. 187
7.4. Theorem. 187
8. Symplectic cores versus primitive ideals for solvable Lie algebras 188
8.1. Actions of G and g. 188
8.2. Lemma. 189
8.3. Proposition. 189
8.4. Corollary. 190
8.5. Theorem. 190
8.6. The extended Dixmier map. 190
8.7. Quasi-homeomorphisms and sauber spaces. 190
8.8. Lemma. 191
8.9. Lemma. 192
8.10. Theorem. 192
8.11. Theorem. 192
9. Modified conjectures for quantized coordinate rings 192
9.1. Primitive spectrum conjecture for quantized coordinate rings. 193
9.2. Remarks 193
9.3. Lemma. 195
9.4. Lemma. 195
9.5. Example. 196
9.6. Some computational tools 197
9.7. Example. 197
9.8. Evidence for Conjecture 198
9.9. Example 199
9.10. Example. 201
References 204
On Unit-Central Rings 207
1. Introduction 207
2. Commutativity theorems 208
3. Open problems 212
References 212
Symplectic Modules and von Neumann Regular Matrices over Commutative Rings 215
1. Introduction 215
2. Symplectic structures on projective modules 217
3. Characterization of von Neumann regular matrices 223
4. Von Neumann regular matrices of small size 226
References 228
Extensions of Simple Modules and the Converse of Schur’s Lemma 230
1. Introduction 230
2. CSL for finite length modules 233
3. CSL for all modules 235
4. Some questions 236
References 237
Report on Exchange Rings 239
1. Introduction 239
2. Finite exchange rings 241
2.1 Theorem. 242
2.2 Corollary. 242
2.3 Proposition. 242
2.4 Proposition. 243
2.5 Theorem. 243
2.6 Corollary. 243
3. Topological rings 244
3.1 Theorem. 244
3.2 Theorem. 244
3.3 Corollary. 245
3.4 Theorem. 245
3.5 Theorem. 246
3.6 Lemma. 246
3.7 Lemma. 246
3.8 Theorem. 247
3.9 Corollary. 247
3.10 Theorem. 248
4. Internal exchange property 248
4.1 Lemma. 248
4.2 Corollary. 248
4.3 Lemma. 249
4.4 Proposition. 249
4.5 Lemma. 249
4.6 Theorem. 249
4.7 Theorem. 250
4.8 Corollary. 251
4.9 Proposition. 252
5. Questions 252
5.1 Question. 252
5.2 Lemma. 252
5.3 Question. 252
5.4 Proposition. 253
5.5 Corollary. 253
5.6 Question. 253
5.7 Question. 253
5.8 Question. 253
5.9 Question. 254
References 254
Filtrations in Semisimple Lie Algebras, III 256
1. Preliminaries 256
2. The Lie algebra G2 261
References 267
On the Blowing-up Rings, Arf Rings and Type Sequences 268
0. Introduction 269
1. Preliminaries – notation, definitions and some results 270
2. Numerical invariants of certain monomial curves 274
3. Blowing-up rings and Arf rings 276
4. Examples 279
References 280
A Guide to Supertropical Algebra 282
1. Introduction 282
1.1. Brief overview of tropical geometry 282
1.2. The semiring structure of the max-plus algebra 284
2. Basic notions 285
2.1. Semirings with ghosts 286
2.2. Supertropical domains and semifields 287
3. Polynomials and their roots 289
3.1. Polynomials in one indeterminate over a supertropical semifield 289
3.2. Factorization of polynomials in several indeterminates 291
3.3. A tropical version of the Hilbert Nullstellensatz 292
4. Supertropical matrix theory 292
4.1. Review of matrix theory over the max-plus algebra 293
4.2. Supertropical determinants 293
4.3. Adjoints 295
4.4. The Hamilton-Cayley theorem 296
4.5. Tropical dependence 296
4.6. Solving supertropical equations 296
4.7. The resultant of two polynomials 297
5. The structure theory of semirings with tangibles and ghosts 297
6. Conclusions and directions for further research 298
6.1. The role of ghosts 298
6.2. Structure theory 299
6.3. Linear algebra 299
6.4. Category theory 299
6.5. Multiple ghost layers 299
References 299
Projective Modules, Idempotent Ideals and Intersection Theorems 302
1. Projective modules 302
2. Idempotent ideals 304
3. Projective modules and idempotent ideals 305
4. Shallow rings 310
5. Group rings 314
6. Group rings of infinite groups 318
7. Ideals with a centralizing sequence of generators 321
References 323
On Ef-extending Modules and Rings with Chain Conditions 326
1. Introduction 326
2. Results 327
References 332
On Clean Group Rings 334
1. A brief review 335
2. A sufficient condition 335
3. Unit-regular and strongly p-regular rings 337
4. Abelian clean rings 339
5. Uniquely clean group rings 342
References 343