This page intentionally left blank Free Ideal Rings and Localization in General Rings 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 in a historical note. paul cohn is Emeritus Professor of Mathematics at the University of London and Honorary Research Fellow at University College London. NEW MATHEMATICAL MONOGRAPHS Editorial Board B´ela Bollob´as William Fulton Frances Kirwan Peter Sarnak Barry Simon For information about Cambridge University Press mathematics publications visit http://publishing.cambridge.org/stm/mathematics Already published in New Mathematical Monographs: Representation Theory of Finite Reductive Groups Marc Cabanes, Michel Enguehard Harmonic Measure John B. Garnett, Donald E. Marshall Heights in Diophantine Geometry Enrico Bombieri, Walter Gubler Free Ideal Rings and Localization in General Rings P. M. COHN Department of Mathematics University College London Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge ,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521853378 © Cambridge University Press 2006 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2006 - ---- eBook (NetLibrary) - --- eBook (NetLibrary) - ---- hardback - --- hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To my granddaughters Chasya and Ayala Contents Preface page xi Note to the reader xiv Terminology, notation and conventions used xvi List of special notation xx 0 Generalities on rings and modules 1 0.1 Rank conditions on free modules1 0.2 Matrix rings and the matrix reduction functor7 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 Notes and comments on Chapter 0 58 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 Notes and comments on Chapter 1 105 2 Firs, semifirs and the weak algorithm 107 2.1 Hereditary rings 107 vii viii Contents 2.2 Firs and α-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 GE2(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 Notes and comments on Chapter 2 183 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 Notes and comments on Chapter 3 223 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 Notes and comments on Chapter 4 261 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 Contents ix 5.10 The intersection theorem for firs 326 Notes and comments on Chapter 5 329 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 Notes and comments on Chapter 6 407 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 Notes and comments on Chapter 7 515 Appendix 519 A. Lattice theory 519 B. Categories and homological algebra 524 C. Ultrafilters and the ultraproduct theorem 538 Bibliography and author index 540 Subject index 566 Preface It is not your duty to complete the work, But neither are you free to desist from it. R. Tarphon, Sayings of the Fathers. One of the questions that intrigued me in the 1950s was to find conditions for an embedding of a non-commutative ring in a skew field to be possible. I felt that such an embedding should exist for a free product of skew fields, but there seemed no obvious route. My search eventually led to the notion of a free ideal ring, fir for short, and I was able to prove (i) the free product of skew fields (amalgamating a skew subfield) is a fir and (ii) every fir is embeddable in a skew field. Firs may be regarded as the natural generalization (in the non-commutative case) of principal domains, to which they reduce when commutativity is imposed. The proof of (i) involved an algorithm, which when stated in simple terms, resembled the Euclidean algorithm but depended on a condition of linear dependence. In this form it could be used to characterize free associative algebras, and this ‘weak’ algorithm enables one to develop a theory of free algebras similar to that of a polynomial ring in one variable. Of course free algebras are a special case of firs, and other facts about firs came to light, which were set forth in my book Free Rings and their Relations (a pun and a paradox). It appeared in 1971 and in a second edition in 1985. A Russian translation appeared in 1975. More recently there has been a surprising increase of interest, in many fields of mathematics, in non-commutative theories. In functional analysis there has been a greater emphasis on non-commutative function algebras and quantum groups have been introduced in the study of non-commutative geometry, while quantum physics uses non-commutative probability theory, in which even free associative algebras have made their appearance. The localization developed in Free Rings has also found a use by topologists. All this, and the fact that xi xii Preface many proofs have been simplified, has encouraged me to write a book based on the earlier work, but addressed to a wider audience. Since skew fields play a prominent role, the prefix ‘skew’ will often be left out, so fields are generally assumed to be not necessarily commutative. The central part is Chapter 7, in which non-commutative localization is studied. For any ring R the various homomorphisms into fields are described by their singular kernels, the matrices with a singular image, which form a resemblance to prime ideals and so are called prime matrix ideals. Various classes of rings, such as firs and semifirs, are shown to be embeddable in fields, and an explicit criterion is given for such an embedding of a general ring to be possible, as well as conditions for a universal field of fractions to exist. This is the case for firs, while for free algebras the universal field of fractions can be shown to be ‘free’. The existence of the localization now has a simpler and more direct proof, which is described in Sections 7.1–7.4. It makes only occasional reference to earlier chapters (mainly parts of Chapter 0) and so can be read at any stage.