Free ideal rings and localization in general rings / P.M. Cohn.

Format:

• Book

Author/Creator:

• Cohn, P. M. (Paul Moritz)

Contributor:

• Alumni and Friends Memorial Book Fund.

Series:

• New mathematical monographs ; 3.
• New mathematical monographs ; 3

Language:

English

Subjects (All):

• Rings (Algebra).
• Ideals (Algebra).

Physical Description:

xxii, 572 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge, UK ; New York : Cambridge University Press, 2006.

Summary:

• 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. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization, which is treated for general rings but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends with a historical note.

Contents:

• Terminology, notation and conventions used xvi
• 0 Generalities on rings and modules 1
• 0.1 Rank conditions on free modules 1
• 0.2 Matrix rings and the matrix reduction functor 7
• 0.3 Projective modules 12
• 0.4 Hermite rings 19
• 0.5 The matrix of definition of a module 25
• 0.6 Eigenrings and centralizers 33
• 0.7 Rings of fractions 37
• 0.8 Modules over Ore domains 47
• 0.9 Factorization in commutative integral domains 52
• 1 Principal ideal domains 60
• 1.1 Skew polynomial rings 60
• 1.2 The division algorithm 66
• 1.3 Principal ideal domains 73
• 1.4 Modules over principal ideal domains 77
• 1.5 Skew Laurent polynomials and Laurent series 86
• 1.6 Iterated skew polynomial rings 98
• 2 Firs, semifirs and the weak algorithm 107
• 2.1 Hereditary rings 107
• 2.2 Firs and [alpha]-firs 110
• 2.3 Semifirs and n-firs 113
• 2.4 The weak algorithm 124
• 2.5 Monomial K-bases in filtered rings and free algebras 131
• 2.6 The Hilbert series of a filtered ring 141
• 2.7 Generators and relations for GE[subscript 2](R) 145
• 2.8 The 2-term weak algorithm 153
• 2.9 The inverse weak algorithm 156
• 2.10 The transfinite weak algorithm 171
• 2.11 Estimate of the dependence number 176
• 3 Factorization in semifirs 186
• 3.1 Similarity in semifirs 186
• 3.2 Factorization in matrix rings over semifirs 192
• 3.3 Rigid factorizations 199
• 3.4 Factorization in semifirs: a closer look 207
• 3.5 Analogues of the primary decomposition 214
• 4 Rings with a distributive factor lattice 225
• 4.1 Distributive modules 225
• 4.2 Distributive factor lattices 231
• 4.3 Conditions for a distributive factor lattice 237
• 4.4 Finite distributive lattices 243
• 4.5 More on the factor lattice 247
• 4.6 Eigenrings 251
• 5 Modules over firs and semifirs 263
• 5.1 Bound and unbound modules 264
• 5.2 Duality 269
• 5.3 Positive and negative modules over semifirs 272
• 5.4 The ranks of matrices 281
• 5.5 Sylvester domains 290
• 5.6 Pseudo-Sylvester domains 300
• 5.7 The factorization of matrices over semifirs 304
• 5.8 A normal form for matrices over a free algebra 311
• 5.9 Ascending chain conditions 320
• 5.10 The intersection theorem for firs 326
• 6 Centralizers and subalgebras 331
• 6.1 Commutative subrings and central elements in 2-firs 331
• 6.2 Bounded elements in 2-firs 340
• 6.3 2-Firs with prescribed centre 351
• 6.4 The centre of a fir 355
• 6.5 Free monoids 357
• 6.6 Subalgebras and ideals of free algebras 367
• 6.7 Centralizers in power series rings and in free algebras 374
• 6.8 Invariants in free algebras 379
• 6.9 Galois theory of free algebras 387
• 6.10 Automorphisms of free algebras 396
• 7 Skew fields of fractions 410
• 7.1 The rational closure of a homomorphism 411
• 7.2 The category of R-fields and specializations 418
• 7.3 Matrix ideals 428
• 7.4 Constructing the localization 437
• 7.5 Fields of fractions 444
• 7.6 Numerators and denominators 455
• 7.7 The depth 466
• 7.8 Free fields and the specialization lemma 474
• 7.9 Centralizers in the universal field of fractions of a fir 482
• 7.10 Determinants and valuations 491
• 7.11 Localization of firs and semifirs 500
• 7.12 Reversible rings 511
• A Lattice theory 519
• B Categories and homological algebra 524
• C Ultrafilters and the ultraproduct theorem 538.

Notes:

Includes bibliographical references (pages 540-565) and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0521853370

OCLC:

61176552

Publisher Number:

9780521853378 (hbk.)

Online:

The Alumni and Friends Memorial Book Fund Home Page