The book generalizes the well-known regularity of ring elements to regularity of homomorphisms in module categories, and further to regularity of morphisms in any category. Regular homomorphisms are characterized in terms of decompositions of domain and codomain, and numerous other results are presented. While the theory is well developed in module categories many questions remain about generalizations and extensions to other categories. The book only requires the knowledge of a basic course in modern algebra. It is written clearly, with great detail, and is accessible to students and researchers alike.
Author(s): Friedrich Kasch, Adolf Mader
Edition: 1
Year: 2009
Language: English
Pages: 181
Cover......Page 1
Series: Frontiers in Mathematics......Page 3
Regularity and Substructures of Hom......Page 4
Copyright - ISBN: 9783764399894......Page 5
Contents......Page 6
Preface......Page 8
1. Notation......Page 18
2. Rings and Modules......Page 19
3. Abelian Groups......Page 23
1. Definition and Characterization......Page 28
2. Partially Invertible Homomorphisms and Quasi-Inverses......Page 32
3. Regular Homomorphisms Generate Projective Direct Summands......Page 36
4. Existence and Properties of Reg(A,M)......Page 38
5. The Transfer Rule......Page 42
6. Inherited Regularity......Page 43
7. Appendix: Various Formulas......Page 56
1. Reg(A,M) $\neq$ 0......Page 58
2. Structure Theorems......Page 59
1. Fundamental Results......Page 66
2.1 Basic Properties......Page 69
2.2 Partially Invertible Objects are Quasi-Inverses......Page 70
3. Regular Elements Generate Projective Direct Summands......Page 71
4. Remarks on the Literature......Page 73
5. The Transfer Rule......Page 74
1. Iterated Endomorphism Rings......Page 76
2. Definitions and Characterizations......Page 77
3. Largest Regular Submodules......Page 82
4. The Transfer Rule for S-Regularity......Page 83
1. U-Regularity; Definition and Existence of U-Reg(A,M)......Page 86
2. U-Regularity in Modules......Page 89
3. Semiregularity for Modules......Page 90
4. Semiregularity for Hom......Page 93
1. Substructures of Hom......Page 98
2. Properties of Δ(A,M) and ∇(A,M)......Page 102
3. The Special Case Hom[sub(R)](R,M)......Page 104
4. Further Internal Properties of Δ(M)......Page 107
5. Non-singular Modules......Page 111
6. A Correspondenc Between Submodules of Hom[sub(R)](A,M) and Ideals of End(M[sub(R)])......Page 112
7. Correspondences for Modules......Page 117
2. Hom(A,M) and Regularity......Page 120
3. Mixed Groups......Page 132
4. Regularity in Endomorphism Rings of Mixed Groups......Page 144
1. Regularity in Preadditive Categories......Page 148
2. Preadditive Categories......Page 149
3. The Quasi-Isomorphism Category of Torsion-free Abelian Groups......Page 154
4. Regularity in \mathbb{Q}A......Page 167
4.2 Constructing the Group......Page 170
4.3 Computing the Quasi-Endomorphism Ring......Page 171
5. Regularity in the Category of Groups......Page 172
Bibliography......Page 174
E......Page 178
M......Page 179
R......Page 180
Z......Page 181
