Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. There is also a full account of localization which is treated for general rings but the features arising in firs are given special attention.
Author(s): P. M. Cohn
Series: New Mathematical Monographs
Publisher: Cambridge University Press
Year: 2006
Language: English
Pages: 596
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Note to the reader......Page 16
Terminology, notation and conventions used......Page 18
List of special notation......Page 22
0.1 Rank conditions on free modules......Page 25
0.2 Matrix rings and the matrix reduction functor......Page 31
0.3 Projective modules......Page 36
0.4 Hermite rings......Page 43
0.5 The matrix of definition of a module......Page 49
0.6 Eigenrings and centralizers......Page 57
0.7 Rings of fractions......Page 61
0.8 Modules over Ore domains......Page 71
0.9 Factorization in commutative integral domains......Page 76
Notes and comments on Chapter 0......Page 82
1.1 Skew polynomial rings......Page 84
1.2 The division algorithm......Page 90
1.3 Principal ideal domains......Page 97
1.4 Modules over principal ideal domains......Page 101
1.5 Skew Laurent polynomials and Laurent series......Page 110
1.6 Iterated skew polynomial rings......Page 122
Notes and comments on Chapter 1......Page 129
2.1 Hereditary rings......Page 131
2.2 Firs and Alpha-firs......Page 134
2.3 Semifirs and n-firs......Page 137
2.4 The weak algorithm......Page 148
2.5 Monomial K-bases in filtered rings and free algebras......Page 155
2.6 The Hilbert series of a filtered ring......Page 165
2.7 Generators and relations for GE(R)......Page 169
2.8 The 2-term weak algorithm......Page 177
2.9 The inverse weak algorithm......Page 180
2.10 The transfinite weak algorithm......Page 195
2.11 Estimate of the dependence number......Page 200
Notes and comments on Chapter 2......Page 207
3.1 Similarity in semifirs......Page 210
3.2 Factorization in matrix rings over semifirs......Page 216
3.3 Rigid factorizations......Page 223
3.4 Factorization in semifirs: a closer look......Page 231
3.5 Analogues of the primary decomposition......Page 238
Notes and comments on Chapter 3......Page 247
4.1 Distributive modules......Page 249
4.2 Distributive factor lattices......Page 255
4.3 Conditions for a distributive factor lattice......Page 261
4.4 Finite distributive lattices......Page 267
4.5 More on the factor lattice......Page 271
4.6 Eigenrings......Page 275
Notes and comments on Chapter 4......Page 285
5 Modules over firs and semifirs......Page 287
5.1 Bound and unbound modules......Page 288
5.2 Duality......Page 293
5.3 Positive and negative modules over semifirs......Page 296
5.4 The ranks of matrices......Page 305
5.5 Sylvester domains......Page 314
5.6 Pseudo-Sylvester domains......Page 324
5.7 The factorization of matrices over semifirs......Page 328
5.8 A normal form for matrices over a free algebra......Page 335
5.9 Ascending chain conditions......Page 344
5.10 The intersection theorem for firs......Page 350
Notes and comments on Chapter 5......Page 353
6.1 Commutative subrings and central elements in 2-firs......Page 355
6.2 Bounded elements in 2-firs......Page 364
6.3 2-Firs with prescribed centre......Page 375
6.4 The centre of a fir......Page 379
6.5 Free monoids......Page 381
6.6 Subalgebras and ideals of free algebras......Page 391
6.7 Centralizers in power series rings and in free algebras......Page 398
6.8 Invariants in free algebras......Page 403
6.9 Galois theory of free algebras......Page 411
6.10 Automorphisms of free algebras......Page 420
Notes and comments on Chapter 6......Page 431
7 Skew fields of fractions......Page 434
7.1 The rational closure of a homomorphism......Page 435
7.2 The category of R-fields and specializations......Page 442
7.3 Matrix ideals......Page 452
7.4 Constructing the localization......Page 461
7.5 Fields of fractions......Page 468
7.6 Numerators and denominators......Page 479
7.7 The depth......Page 490
7.8 Free fields and the specialization lemma......Page 498
7.9 Centralizers in the universal field of fractions of a fir......Page 506
7.10 Determinants and valuations......Page 515
7.11 Localization of firs and semifirs......Page 524
7.12 Reversible rings......Page 535
Notes and comments on Chapter 7......Page 539
A. Lattice theory......Page 543
B. Categories and homological algebra......Page 548
C. Ultrafilters and the ultraproduct theorem......Page 562
Bibliography and author index......Page 564
Subject index......Page 590
