Free Ideal Rings and Localization in General Rings

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.

Author: P. M. Cohn

Publisher: Cambridge University Press

ISBN: 1139454994

Category: Mathematics

Page:

View: 200

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.

Free Ideal Rings and Localization in General Rings

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: P. M. Cohn

Publisher: Cambridge University Press

ISBN: 9780521853378

Category: Mathematics

Page: 594

View: 998

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.

Advances in Rings and Modules

[1] J. A. Beachy, Inversive localization at semiprime Goldie ideals, Manuscripta Math. 34 (1981), no. 2-3, 211–239. ... MR0364324 [7] P. M. Cohn, Free ideal rings and localization in general rings, New Mathematical Monographs ...

Author: Sergio R. López-Permouth

Publisher: American Mathematical Soc.

ISBN: 1470435551

Category: Modules (Algebra)

Page: 283

View: 349

This volume, dedicated to Bruno J. Müller, a renowned algebraist, is a collection of papers that provide a snapshot of the diversity of themes and applications that interest algebraists today. The papers highlight the latest progress in ring and module research and present work done on the frontiers of the topics discussed. In addition, selected expository articles are included to give algebraists and other mathematicians, including graduate students, an accessible introduction to areas that may be outside their own expertise.

Purity Spectra and Localisation

P. M. Cohn, Progress in free associative algebras, Israel J. Math, 19 (1974), 109— 15 1 . P. M. Cohn, The affine scheme of a general ring, ... P. M. Cohn, Free Ideal Rings and Localization in General Rings, New Mathematical Monographs, ...

Author: Mike Prest

Publisher: Cambridge University Press

ISBN: 1139643894

Category: Mathematics

Page:

View: 857

It is possible to associate a topological space to the category of modules over any ring. This space, the Ziegler spectrum, is based on the indecomposable pure-injective modules. Although the Ziegler spectrum arose within the model theory of modules and plays a central role in that subject, this book concentrates specifically on its algebraic aspects and uses. The central aim is to understand modules and the categories they form through associated structures and dimensions, which reflect the complexity of these, and similar, categories. The structures and dimensions considered arise particularly through the application of model-theoretic and functor-category ideas and methods. Purity and associated notions are central, localisation is an ever-present theme and various types of spectrum play organising roles. This book presents a unified, coherent account of material which is often presented from very different viewpoints and clarifies the relationships between these various approaches.

Multiplicative Ideal Theory and Factorization Theory

P.M. Cohn, Free Ideal Rings and Localization in General Rings, vol. 3, New Mathematical Monographs (Cambridge University Press, Cambridge, 2006) 40. C.W.Curtis,I.Reiner,Methods of Representation Theory, vol. II.

Author: Scott Chapman

Publisher: Springer

ISBN: 331938855X

Category: Mathematics

Page: 407

View: 572

This book consists of both expository and research articles solicited from speakers at the conference entitled "Arithmetic and Ideal Theory of Rings and Semigroups," held September 22–26, 2014 at the University of Graz, Graz, Austria. It reflects recent trends in multiplicative ideal theory and factorization theory, and brings together for the first time in one volume both commutative and non-commutative perspectives on these areas, which have their roots in number theory, commutative algebra, and algebraic geometry. Topics discussed include topological aspects in ring theory, Prüfer domains of integer-valued polynomials and their monadic submonoids, and semigroup algebras. It will be of interest to practitioners of mathematics and computer science, and researchers in multiplicative ideal theory, factorization theory, number theory, and algebraic geometry.

Rings with Polynomial Identities and Finite Dimensional Representations of Algebras

[Bor69] A. Borel, Linear algebraic groups, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. ... MR279123 [Coh06] P. M. Cohn, Free ideal rings and localization in general rings, New Mathematical Monographs, vol.

Author: Eli Aljadeff

Publisher: American Mathematical Soc.

ISBN: 1470451743

Category: Education

Page: 630

View: 503

A polynomial identity for an algebra (or a ring) A A is a polynomial in noncommutative variables that vanishes under any evaluation in A A. An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike. The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.

An Invitation to General Algebra and Universal Constructions

92 P. M. Cohn, Free ideal rings and localization in general rings. New Mathematical Monographs, 3., Cambridge University Press, 2006. MR 2007k:16020. 92 Thomas Delzant and Misha Gromov, Courbure mésoscopique et théorie de la toute ...

Author: George M. Bergman

Publisher: Springer

ISBN: 3319114786

Category: Mathematics

Page: 572

View: 200

Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book.

Foundations of Free Noncommutative Function Theory

Free ideal rings and localization in general rings. Cambridge University Press, Cambridge, 2006. New Mathematical Monographs 3. K. R. Davidson and D. R. Pitts, Invariant subspaces and hyper-reflexivity for the free semigroup algebras, ...

Author: Dmitry S. Kaliuzhnyi-Verbovetskyi

Publisher: American Mathematical Soc.

ISBN: 1470416972

Category: Mathematics

Page: 183

View: 529

In this book the authors develop a theory of free noncommutative functions, in both algebraic and analytic settings. Such functions are defined as mappings from square matrices of all sizes over a module (in particular, a vector space) to square matrices over another module, which respect the size, direct sums, and similarities of matrices. Examples include, but are not limited to, noncommutative polynomials, power series, and rational expressions. Motivation and inspiration for using the theory of free noncommutative functions often comes from free probability. An important application area is "dimensionless" matrix inequalities; these arise, e.g., in various optimization problems of system engineering. Among other related areas are those of polynomial identities in rings, formal languages and finite automata, quasideterminants, noncommutative symmetric functions, operator spaces and operator algebras, and quantum control.

Noncommutative Rational Series with Applications

Free Rings and Their Relations, volume 19 of London Mathematical Society Monographs. Academic Press. 27, 198 Cohn, P. M. (2006). Free Ideal Rings and Localization in General Rings, volume 3 of New Mathematical Monographs.

Author: Jean Berstel

Publisher: Cambridge University Press

ISBN: 0521190223

Category: Mathematics

Page: 248

View: 301

A modern account of the subject and its applications. Excellent resource for those working in algebra and theoretical computer science.

Geometry Topology and Dynamics in Negative Curvature

Math. France, Paris, 1976. Tim Austin. Rational group ring elements with kernels having irrational dimension. Proc. London Math. ... Free ideal rings and localization in general rings, volume 3 of New Mathematical Monographs.

Author: C. S. Aravinda

Publisher: Cambridge University Press

ISBN: 1316539180

Category: Mathematics

Page:

View: 774

The ICM 2010 satellite conference 'Geometry, Topology and Dynamics in Negative Curvature' afforded an excellent opportunity to discuss various aspects of this fascinating interdisciplinary subject in which methods and techniques from geometry, topology, and dynamics often interact in novel and interesting ways. Containing ten survey articles written by some of the leading experts in the field, this proceedings volume provides an overview of important recent developments relating to negative curvature. Topics covered include homogeneous dynamics, harmonic manifolds, the Atiyah Conjecture, counting circles and arcs, and hyperbolic buildings. Each author pays particular attention to the expository aspects, making the book particularly useful for graduate students and mathematicians interested in transitioning from other areas via the common theme of negative curvature.