This volume consists of refereed research and expository articles by both plenary and other speakers at the International Conference on Algebra and Applications held at Ohio University in June 2008, to honor S.K. Jain on his 70th birthday. The articles are on a wide variety of areas in classical ring theory and module theory, such as rings satisfying polynomial identities, rings of quotients, group rings, homological algebra, injectivity and its generalizations, etc. Included are also applications of ring theory to problems in coding theory and in linear algebra.
Author(s): Sergio R. López-Permouth, Dinh Van Huynh
Series: Trends in Mathematics
Edition: 1st Edition.
Publisher: Birkhäuser Basel
Year: 2010
Language: English
Pages: 356
Cover......Page 1
Trends in Mathematics......Page 3
Advances in
Ring Theory......Page 4
ISBN 9783034602853......Page 5
Table of Contents
......Page 6
Preface......Page 10
1. Introduction......Page 12
2. Notation and terminology......Page 13
3. What is Cogalois theory?......Page 14
4. Basic concepts and results of Cogalois theory......Page 17
The Kneser criterion......Page 18
Cogalois extensions......Page 19
Galois and Cogalois connections......Page 20
Strongly G-Kneser extensions......Page 21
5. Examples of G-Cogalois extensions......Page 22
6.1. Effective degree computation:......Page 23
6.3. Finding all intermediate fields:......Page 24
6.7. When can a positive superposed radical not be decomposed into a finite sum of real numbers of type ± n√i ai , 1< i < r?......Page 25
6.10. Simple radical separable extensions having the USP:......Page 26
References......Page 27
1. Introduction......Page 30
1.1. The skeletons of R-tors, R-Serre and R-op......Page 33
2. The big lattice R-sext......Page 34
2.1. R-sext and R-nat......Page 38
3. The big lattice R-qext......Page 39
4. R-nat and R-conat......Page 40
5. R-sext and R-qext......Page 42
References......Page 46
1. Introduction......Page 48
2.1. Reversibility in group algebras KG......Page 49
2.2. Reversible group rings over commutative rings......Page 50
3. Duo group rings......Page 52
3.1. Duo group algebras......Page 53
3.2. Duo group rings over integral domains......Page 54
4. Graded reversibility in integral group rings......Page 56
References......Page 57
Principally Quasi-Baer Ring Hulls......Page 58
References......Page 71
1. Preliminaries......Page 74
2. Strongly prime ideals in N0(Rn)......Page 75
3. Strongly prime ideals in N0(Rω)......Page 77
References......Page 79
Introduction......Page 80
1. On r-strongly prime ideals......Page 81
2. Weak zero-divisors......Page 83
3. Examples and special rings......Page 87
References......Page 92
1. The theorem......Page 94
References......Page 95
1. Preliminaries......Page 96
2. Topologies on a Boolean ring......Page 98
3. Countably linearly compact Boolean rings......Page 100
4. On minimal topologies......Page 101
5. Intersection of totally bounded topologies......Page 102
6. The Bohr topology on a Boolean ring......Page 103
7. Compact topologies on Boolean rings......Page 105
9. Self-injective Boolean rings......Page 106
10. Zero-dimensional F-spaces......Page 109
11. Necessary conditions for countably compactness......Page 110
12. Basically disconnected spaces......Page 112
13. Open questions......Page 120
References......Page 121
Introduction......Page 124
1. Preliminaries......Page 126
2. Large right ideals......Page 127
3. Mod-R and mod-Q for R ⊂ Q......Page 128
4. Over rings......Page 133
5. Lattices......Page 135
6. Examples......Page 137
References......Page 140
1. Introduction......Page 142
2. Chain rings, Galois rings, and constacyclic codes......Page 144
3. Negacyclic codes of length 2s over GR(2a,m)......Page 147
4. Some classes of constacyclic codes of length 2s over GR(2a,m)......Page 151
References......Page 156
1. Introduction......Page 160
2. Couniformly presented modules......Page 161
3. Epigeny class and lower part......Page 165
4. Weak Krull-Schmidt Theorem......Page 167
5. Kernels of morphisms between indecomposable injective modules......Page 169
6. A further duality between epigeny classes and monogeny classes......Page 171
References......Page 174
0. Introduction......Page 176
1.1. Quantum SL2.......Page 178
1.3. Quantum affine spaces.......Page 179
1.6. Limits of families of algebras.......Page 180
2.1. Semiclassical limits: commutative fibre version.......Page 181
2.2. Examples......Page 182
2.3. Multiparameter examples.......Page 183
2.5. Bridging the two constructions.......Page 184
2.6. Example: enveloping algebras.......Page 185
3.3. Poisson bivector fields.......Page 186
3.5. Smooth Poisson varieties as manifolds.......Page 187
3.8. Example.......Page 188
4.2. Theorem.......Page 189
4.5. Generic versus non-generic situations.......Page 190
5.1. The algebraic adjoint group.......Page 191
5.6. Example.......Page 192
6.1. Poisson prime ideals.......Page 193
6.3. Poisson-primitive ideals and symplectic cores.......Page 194
7.1. Theorem.......Page 195
7.4. Theorem.......Page 196
8.1. Actions of G and g.......Page 197
8.3. Proposition.......Page 198
8.7. Quasi-homeomorphisms and sauber spaces.......Page 199
8.8. Lemma.......Page 200
9. Modified conjectures for quantized coordinate rings......Page 201
9.2. Remarks......Page 202
9.4. Lemma.......Page 204
9.5. Example.......Page 205
9.7. Example.......Page 206
9.8. Evidence for Conjecture......Page 207
9.9. Example......Page 208
9.10. Example.......Page 210
References......Page 213
1. Introduction......Page 216
2. Commutativity theorems......Page 217
References......Page 221
1. Introduction......Page 224
2. Symplectic structures on projective modules......Page 226
3. Characterization of von Neumann regular matrices......Page 232
4. Von Neumann regular matrices of small size......Page 235
References......Page 237
1. Introduction......Page 240
2. CSL for finite length modules......Page 243
3. CSL for all modules......Page 245
4. Some questions......Page 246
References......Page 247
1. Introduction......Page 250
2. Finite exchange rings......Page 252
2.3 Proposition.......Page 253
2.6 Corollary.......Page 254
3.2 Theorem.......Page 255
3.4 Theorem.......Page 256
3.7 Lemma.......Page 257
3.9 Corollary.......Page 258
4.2 Corollary.......Page 259
4.6 Theorem.......Page 260
4.7 Theorem.......Page 261
4.8 Corollary.......Page 262
5.3 Question.......Page 263
5.8 Question.......Page 264
References......Page 265
1. Preliminaries......Page 268
2. The Lie algebra G2......Page 273
References......Page 279
On the Blowing-up Rings, Arf Rings and Type Sequences......Page 280
0. Introduction......Page 281
1. Preliminaries – notation, definitions and some results......Page 282
2. Numerical invariants of certain monomial curves......Page 286
3. Blowing-up rings and Arf rings......Page 288
4. Examples......Page 291
References......Page 292
1.1. Brief overview of tropical geometry......Page 294
1.2. The semiring structure of the max-plus algebra......Page 296
2. Basic notions......Page 297
2.1. Semirings with ghosts......Page 298
2.2. Supertropical domains and semifields......Page 299
3.1. Polynomials in one indeterminate over a supertropical semifield......Page 301
3.2. Factorization of polynomials in several indeterminates......Page 303
4. Supertropical matrix theory......Page 304
4.2. Supertropical determinants......Page 305
4.3. Adjoints......Page 307
4.6. Solving supertropical equations......Page 308
5. The structure theory of semirings with tangibles and ghosts......Page 309
6.1. The role of ghosts......Page 310
References......Page 311
1. Projective modules......Page 314
2. Idempotent ideals......Page 316
3. Projective modules and idempotent ideals......Page 317
4. Shallow rings......Page 322
5. Group rings......Page 326
6. Group rings of infinite groups......Page 330
7. Ideals with a centralizing sequence of generators......Page 333
References......Page 335
1. Introduction......Page 338
2. Results......Page 339
References......Page 344
On Clean Group Rings......Page 346
2. A sufficient condition......Page 347
3. Unit-regular and strongly π-regular rings......Page 349
4. Abelian clean rings......Page 351
5. Uniquely clean group rings......Page 354
References......Page 355
