Ring and module theory

✍ Scribed by Albu T., et al. (eds.)

• Publisher: Birkhauser
• Year: 2010
• Tongue: English
• Leaves: 209
• Series: Trends in Mathematics
• Category: Library

✦ Synopsis

The volume consists of a collection of invited research papers and expository/survey articles, many of which were presented at the International Conference on Ring and Module Theory held at Hacettepe University in Ankara, Turkey, in August 2008. The selected papers and articles examine wide ranging and cutting edge developments in various areas of algebra, including ring theory, module theory, and commutative algebra. There is also a section devoted to open problems to motivate further research.

✦ Table of Contents

Cover......Page 1
Trends in Mathematics......Page 3
Ring and
Module Theory......Page 4
ISBN 9783034600064......Page 5
Table of Contents......Page 6
Preface......Page 8
1. Introduction......Page 10
The Classical Hopkins-Levitzki Theorem (H-LT)......Page 11
Extensions of the H-LT......Page 12
2. The Relative H-LT......Page 13
Hereditary torsion theories......Page 14
Relativization......Page 15
Quotient categories and the Gabriel-Popescu Theorem......Page 16
Absolutization......Page 17
A latticial strategy......Page 18
Lattice background......Page 19
The H-LT and Dual H-LT for arbitrary modular lattices......Page 20
The H-LT for upper continuous modular lattices......Page 21
Relative H-LT =⇒ Absolute H-LT......Page 22
The Faith’s counter version of the Relative H-LT......Page 23
Absolute H-LT ⇐⇒ Classical H-LT......Page 24
6. The Absolute and Relative Dual H-LT......Page 25
The definition of the Krull dimension of a poset......Page 26
The definition of the dual Krull dimension of a poset......Page 27
A Krull dimension-like extension of the Absolute H-LT......Page 28
Localization of modular lattices......Page 29
Sketch of the proof of Theorem 7.3......Page 31
8. Four open problems......Page 32
References......Page 33
1. Introduction......Page 36
2. Brief history of hulls......Page 40
3. Definitions of a ring hull......Page 43
4. Existence and uniqueness of ring hulls......Page 48
5. Transference between R and overrings......Page 61
6. How does Q(R) determine R?......Page 64
7. Hulls of ring extensions......Page 66
8. Modules with FI-extending hulls......Page 68
9. Applications to rings with involution......Page 71
References......Page 76
1. Introduction and preliminaries......Page 82
2. Proof of Theorem 1.1......Page 84
3. Grobner-Shirshov basis of An......Page 86
References......Page 89
1. Introduction......Page 92
3. The isomorphism problem......Page 93
3.1. Preliminary results......Page 94
4. The main result......Page 96
4.1. A special case......Page 97
References......Page 99
Introduction......Page 100
1. Frobenius and symmetric algebras......Page 104
2. Character algebras of semisimple Hopf algebras......Page 108
3. C(H) is a commutative algebra......Page 111
4. Factorizable Hopf algebras......Page 118
References......Page 122
1. Introduction......Page 124
2. Equivalent conditions to r.w.r.......Page 125
3. Examples and constructions......Page 127
4. Related conditions......Page 129
References......Page 131
2. Regular homomorphisms and the total......Page 134
4. The radicals for HomR(A,M)......Page 135
5. Regular substructures of Hom......Page 136
References......Page 137
1. Definitions, notations and preliminaries......Page 138
2. Main results......Page 140
References......Page 143
1. Introduction......Page 144
2. FP-injective complexes......Page 145
References......Page 150
1. Introduction......Page 152
2. T-projective dimension......Page 153
3. Tnm-injective modules and Tnm-projective modules......Page 155
References......Page 157
1. Subrings of Artinian rings......Page 158
2. Additive rank functions......Page 161
3. Chain conditions......Page 165
4. Modules with the direct sum condition......Page 168
References......Page 171
1. Preliminaries and summary of results......Page 174
2. (α, β)-higher derivations......Page 176
3. Higher differentiation invariance......Page 177
4. Extending derivation to different modules of quotients......Page 179
5. Symmetric modules of quotients......Page 180
6. Torsion theory that is not differential......Page 182
References......Page 183
0. Introduction......Page 184
1. Bad behavior......Page 186
2. Affine domains of dimension two......Page 188
3. Polynomial rings over semilocal one-dimensional domains......Page 194
4. Two-dimensional power series rings......Page 196
References......Page 200
Participants......Page 204