Noncommutative Algebra

Noncommutative Algebra

About This Book This book is meant to be used by beginning graduate students.

Author: Benson Farb

Publisher: Springer Science & Business Media

ISBN: 9781461208891

Category: Mathematics

Page: 226

View: 667

About This Book This book is meant to be used by beginning graduate students. It covers basic material needed by any student of algebra, and is essential to those specializing in ring theory, homological algebra, representation theory and K-theory, among others. It will also be of interest to students of algebraic topology, functional analysis, differential geometry and number theory. Our approach is more homological than ring-theoretic, as this leads the to many important areas of mathematics. This ap student more quickly proach is also, we believe, cleaner and easier to understand. However, the more classical, ring-theoretic approach, as well as modern extensions, are also presented via several exercises and sections in Chapter Five. We have tried not to leave any gaps on the paths to proving the main theorem- at most we ask the reader to fill in details for some of the sideline results; indeed this can be a fruitful way of solidifying one's understanding.
Categories: Mathematics

Geometric Models for Noncommutative Algebras

Geometric Models for Noncommutative Algebras

The volume is based on a course, ``Geometric Models for Noncommutative Algebras'' taught by Professor Weinstein at Berkeley.

Author: Ana Cannas da Silva

Publisher: American Mathematical Soc.

ISBN: 0821809520

Category: Mathematics

Page: 184

View: 485

The volume is based on a course, ``Geometric Models for Noncommutative Algebras'' taught by Professor Weinstein at Berkeley. Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces, for example, the commutative algebras associated to affine algebraic varieties, differentiable manifolds, topological spaces, and measure spaces. In this work, the authors discuss several types of geometric objects (in the usual sense of sets with structure) that are closely related to noncommutative algebras. Central to the discussion are symplectic and Poisson manifolds, which arise when noncommutative algebras are obtained by deforming commutative algebras. The authors also give a detailed study of groupoids (whose role in noncommutative geometry has been stressed by Connes) as well as of Lie algebroids, the infinitesimal approximations to differentiable groupoids. Featured are many interesting examples, applications, and exercises. The book starts with basic definitions and builds to (still) open questions. It is suitable for use as a graduate text. An extensive bibliography and index are included.
Categories: Mathematics

Commutative Algebra and Noncommutative Algebraic Geometry

Commutative Algebra and Noncommutative Algebraic Geometry

This book surveys fundamental current topics in these two areas of research, emphasising the lively interaction between them. Volume 1 contains expository papers ideal for those entering the field.

Author:

Publisher:

ISBN: 9781107065628

Category:

Page:

View: 879

Categories:

Noncommutative Algebraic Geometry

Noncommutative Algebraic Geometry

This book provides a comprehensive introduction to the interactions between noncommutative algebra and classical algebraic geometry.

Author: Gwyn Bellamy

Publisher: Cambridge University Press

ISBN: 9781107129542

Category: Mathematics

Page:

View: 327

This book provides a comprehensive introduction to the interactions between noncommutative algebra and classical algebraic geometry.
Categories: Mathematics

Introduction to Noncommutative Algebra

Introduction to Noncommutative Algebra

Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings.

Author: Matej Brešar

Publisher: Springer

ISBN: 9783319086934

Category: Mathematics

Page: 199

View: 189

Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.
Categories: Mathematics

Computational Noncommutative Algebra and Applications

Computational Noncommutative Algebra and Applications

Proceedings of the NATO Advanced Study Institute, on Computatoinal
Noncommutative Algebra and Applications, Il Ciocco, Italy, 6-19 July 2003 Jim
Byrnes, Gerald Ostheimer. Acknowledgments The authors would like to thank Jim
Byrnes ...

Author: Jim Byrnes

Publisher: Springer Science & Business Media

ISBN: 9781402023071

Category: Mathematics

Page: 428

View: 725

Categories: Mathematics

Noncommutative Algebra and Geometry

Noncommutative Algebra and Geometry

This volume is mainly devoted to the contributions related to the European Science Foundation workshop, organized under the framework of noncommuntative geometry and i

Author: Corrado De Concini

Publisher: CRC Press

ISBN: 9781420028102

Category: Mathematics

Page: 272

View: 639

A valuable addition to the Lecture Notes in Pure and Applied Mathematics series, this reference results from a conference held in St. Petersburg, Russia, in honor of Dr. Z. Borevich. This volume is mainly devoted to the contributions related to the European Science Foundation workshop, organized under the framework of noncommuntative geometry and integrated in the Borevich meeting. The topics presented, including algebraic groups and representations, algebraic number theory, rings, and modules, are a timely distillation of recent work in the field. Featuring a wide range of international experts as contributors, this book is an ideal reference for mathematicians in algebra and algebraic geometry.
Categories: Mathematics

Introduction to Noncommutative Algebra

Introduction to Noncommutative Algebra

This book provides an elementary introduction to noncommutative rings and algebras.

Author: Linsen Chou

Publisher:

ISBN: 1681171880

Category:

Page: 286

View: 280

A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions. The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting. In mathematics, spaces, which are geometric in nature, can be related to numerical functions on them. In general, such functions will form a commutative ring. For instance, one may take the ring C(X) of continuous complex-valued functions on a topological space X. In many cases, we can recover X from C(X), and therefore it makes some sense to say that X has commutative topology. The dream of noncommutative geometry is to generalize this duality to the duality between noncommutative algebras, or sheaves of noncommutative algebras, or sheaf-like noncommutative algebraic or operator-algebraic structures and geometric entities of certain kind, and interact between the algebraic and geometric description of those via this duality. Regarding that the commutative rings correspond to usual affine schemes, and commutative C*-algebras to usual topological spaces, the extension to noncommutative rings and algebras requires non-trivial generalization of topological spaces, as "non-commutative spaces". This book provides an elementary introduction to noncommutative rings and algebras.
Categories:

Arithmetic Fundamental Groups and Noncommutative Algebra

Arithmetic Fundamental Groups and Noncommutative Algebra

This book offers a complete overview of developments made over the last decade. The papers in this volume examine the geometry of moduli spaces of curves with a function on them.

Author: Karen Redrobe Beckman

Publisher: American Mathematical Soc.

ISBN: 9780821820360

Category: Mathematics

Page: 569

View: 250

The arithmetic and geometry of moduli spaces and their fundamental groups are a very active research area. This book offers a complete overview of developments made over the last decade. The papers in this volume examine the geometry of moduli spaces of curves with a function on them. The main players in Part 1 are the absolute Galois group $G_{\mathbb Q}$ of the algebraic numbers and its close relatives. By analyzing how $G_{\mathbb Q}$ acts on fundamental groups defined by Hurwitz moduli problems, the authors achieve a grand generalization of Serre's program from the 1960s.Papers in Part 2 apply $\theta$-functions and configuration spaces to the study of fundamental groups over positive characteristic fields. In this section, several authors use Grothendieck's famous lifting results to give extensions to wildly ramified covers. Properties of the fundamental groups have brought collaborations between geometers and group theorists. Several Part 3 papers investigate new versions of the genus 0 problem. In particular, this includes results severely limiting possible monodromy groups of sphere covers. Finally, Part 4 papers treat Deligne's theory of Tannakian categories and arithmetic versions of the Kodaira-Spencer map. This volume is geared toward graduate students and research mathematicians interested in arithmetic algebraic geometry.
Categories: Mathematics

Topics in Noncommutative Algebra

Topics in Noncommutative Algebra

The book assumes some undergraduate-level knowledge of algebra and analysis, but apart from that is self-contained. Part II of the monograph is devoted to the proofs of the algebraic background.

Author: Andrea Bonfiglioli

Publisher: Springer

ISBN: 9783642225970

Category: Mathematics

Page: 539

View: 919

Motivated by the importance of the Campbell, Baker, Hausdorff, Dynkin Theorem in many different branches of Mathematics and Physics (Lie group-Lie algebra theory, linear PDEs, Quantum and Statistical Mechanics, Numerical Analysis, Theoretical Physics, Control Theory, sub-Riemannian Geometry), this monograph is intended to: fully enable readers (graduates or specialists, mathematicians, physicists or applied scientists, acquainted with Algebra or not) to understand and apply the statements and numerous corollaries of the main result, provide a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view and notation, provide a thorough historical background of the results, together with unknown facts about the effective early contributions by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff and Dynkin, give an outlook on the applications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type) and quickly enable the reader, through a description of the state-of-art and open problems, to understand the modern literature concerning a theorem which, though having its roots in the beginning of the 20th century, has not ceased to provide new problems and applications. The book assumes some undergraduate-level knowledge of algebra and analysis, but apart from that is self-contained. Part II of the monograph is devoted to the proofs of the algebraic background. The monograph may therefore provide a tool for beginners in Algebra.
Categories: Mathematics

New Trends in Noncommutative Algebra

New Trends in Noncommutative Algebra

This volume contains the proceedings of the conference ``New Trends in Noncommutative Algebra'', held at the University of Washington, Seattle, in August 2010, in honor of Ken Goodearl's 65th birthday.

Author: K. R. Goodearl

Publisher: American Mathematical Soc.

ISBN: 9780821852972

Category: Mathematics

Page: 297

View: 160

This volume contains the proceedings of the conference ``New Trends in Noncommutative Algebra'', held at the University of Washington, Seattle, in August 2010, in honor of Ken Goodearl's 65th birthday. The articles reflect the wide interests of Goodearl and will provide researchers and graduate students with an indispensable overview of topics of current interest. Specific fields covered include: noncommutative algebraic geometry, representation theory, Calabi-Yau algebras, quantum algebras and deformation quantization, Poisson algebras, growth of algebras, group algebras, and noncommutative Iwasawa algebras.
Categories: Mathematics

One Sided Prime Ideals in Noncommutative Algebra

One Sided Prime Ideals in Noncommutative Algebra

The goal of this dissertation is to provide noncommutative generalizations of the following theorems from commutative algebra: (Cohen's Theorem) every ideal of a commutative ring R is finitely generated if and only if every prime ideal of R ...

Author: Manuel Lionel Reyes

Publisher: Proquest, UMI Dissertation Publishing

ISBN: 1243778245

Category:

Page: 98

View: 587

The goal of this dissertation is to provide noncommutative generalizations of the following theorems from commutative algebra: (Cohen's Theorem) every ideal of a commutative ring R is finitely generated if and only if every prime ideal of R is finitely generated, and (Kaplansky's Theorems) every ideal of R is principal if and only if every prime ideal of R is principal, if and only if R is noetherian and every maximal ideal of R is principal. We approach this problem by introducing certain families of right ideals in noncommutative rings, called right Oka families, generalizing previous work on commutative rings by T. Y. Lam and the author. As in the commutative case, we prove that the right Oka families in a ring R correspond bijectively to the classes of cyclic right R-modules that are closed under extensions. We define completely prime right ideals and prove the Completely Prime Ideal Principle, which states that a right ideal maximal in the complement of a right Oka family is completely prime. We exploit the connection with cyclic modules to provide many examples of right Oka families. Our methods produce some new results that generalize well-known facts from commutative algebra, and they also recover earlier theorems stating that certain noncommutative rings are domains---namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian. After developing the theory of right Oka families, we proceed to the generalizations of the theorems stated above. Define a right ideal P of a ring R to be cocritical if the module R/P has larger Krull dimension than each of its proper factors. We prove that a ring is right noetherian (resp. a principal right ideal ring) if and only if all of its (essential) cocritical right ideals are finitely generated (resp. principal). We apply our methods to prove that a (left and right) noetherian ring is a principal right ideal ring if and only if all of its maximal right ideals are principal. Examples are provided to show that the left noetherian hypothesis cannot be omitted. Finally, we compare these results with previous generalizations of these theorems, and are able to recover most of these with our methods.
Categories:

Noncommutative Rings

Noncommutative Rings

A cross-section of ideas, techniques and results that give the reader an unparalleled introductory overview of the subject.

Author: I. N. Herstein

Publisher: Cambridge University Press

ISBN: 0883850397

Category: Mathematics

Page: 202

View: 153

A cross-section of ideas, techniques and results that give the reader an unparalleled introductory overview of the subject.
Categories: Mathematics

Arithmetic Fundamental Groups and Noncommutative Algebra

Arithmetic Fundamental Groups and Noncommutative Algebra

Author: Karen Redrobe Beckman

Publisher: American Mathematical Soc.

ISBN: 9780821820360

Category: Mathematics

Page: 569

View: 349

The arithmetic and geometry of moduli spaces and their fundamental groups are a very active research area. This book offers a complete overview of developments made over the last decade. The papers in this volume examine the geometry of moduli spaces of curves with a function on them. The main players in Part 1 are the absolute Galois group $G_{\mathbb Q}$ of the algebraic numbers and its close relatives. By analyzing how $G_{\mathbb Q}$ acts on fundamental groups defined by Hurwitz moduli problems, the authors achieve a grand generalization of Serre's program from the 1960s.Papers in Part 2 apply $\theta$-functions and configuration spaces to the study of fundamental groups over positive characteristic fields. In this section, several authors use Grothendieck's famous lifting results to give extensions to wildly ramified covers. Properties of the fundamental groups have brought collaborations between geometers and group theorists. Several Part 3 papers investigate new versions of the genus 0 problem. In particular, this includes results severely limiting possible monodromy groups of sphere covers. Finally, Part 4 papers treat Deligne's theory of Tannakian categories and arithmetic versions of the Kodaira-Spencer map. This volume is geared toward graduate students and research mathematicians interested in arithmetic algebraic geometry.
Categories: Mathematics

Graduate Algebra

Graduate Algebra

This book is a companion volume to Graduate Algebra: Commutative View (published as volume 73 in this series).

Author: Louis Halle Rowen

Publisher: American Mathematical Soc.

ISBN: 9780821841532

Category: Mathematics

Page: 648

View: 184

This book is a companion volume to Graduate Algebra: Commutative View (published as volume 73 in this series). The main and most important feature of the book is that it presents a unified approach to many important topics, such as group theory, ring theory, Lie algebras, and gives conceptual proofs of many basic results of noncommutative algebra. There are also a number of major results in noncommutative algebra that are usually found only in technical works, such as Zelmanov's proof of the restricted Burnside problem in group theory, word problems in groups, Tits's alternative in algebraic groups, PI algebras, and many of the roles that Coxeter diagrams play in algebra. The first half of the book can serve as a one-semester course on noncommutative algebra, whereas the remaining part of the book describes some of the major directions of research in the past 100 years. The main text is extended through several appendices, which permits the inclusion of more advanced material, and numerous exercises. The only prerequisite for using the book is an undergraduate course in algebra; whenever necessary, results are quoted from Graduate Algebra: Commutative View.
Categories: Mathematics

An Invitation to Noncommutative Geometry

An Invitation to Noncommutative Geometry

A walk in the noncommutative garden / A. Connes and M. Marcolli -- Renormalization of noncommutative quantum field theory / H. Grosse and R. Wulkenhaar -- Lectures on noncommutative geometry / M. Khalkhali -- Noncommutative bundles and ...

Author: Masoud Khalkhali

Publisher: World Scientific

ISBN: 9789812814333

Category: Mathematics

Page: 515

View: 124

A walk in the noncommutative garden / A. Connes and M. Marcolli -- Renormalization of noncommutative quantum field theory / H. Grosse and R. Wulkenhaar -- Lectures on noncommutative geometry / M. Khalkhali -- Noncommutative bundles and instantons in Tehran / G. Landi and W. D. van Suijlekom -- Lecture notes on noncommutative algebraic geometry and noncommutative tori / S. Mahanta -- Lectures on derived and triangulated categories / B. Noohi -- Examples of noncommutative manifolds: complex tori and spherical manifolds / J. Plazas -- D-branes in noncommutative field theory / R. J. Szabo
Categories: Mathematics

Graduate Algebra

Graduate Algebra

Grobner bases can be found in another appendix. Exercises provide a further extension of the text. The book can be used both as a textbook and as a reference source.

Author: Louis Halle Rowen

Publisher: American Mathematical Soc.

ISBN: 9780821805701

Category: Mathematics

Page: 438

View: 571

This book is an expanded text for a graduate course in commutative algebra, focusing on the algebraic underpinnings of algebraic geometry and of number theory. Accordingly, the theory of affine algebras is featured, treated both directly and via the theory of Noetherian and Artinian modules, and the theory of graded algebras is included to provide the foundation for projective varieties. Major topics include the theory of modules over a principal ideal domain, and its applications to matrix theory (including the Jordan decomposition), the Galois theory of field extensions, transcendence degree, the prime spectrum of an algebra, localization, and the classical theory of Noetherian and Artinian rings. Later chapters include some algebraic theory of elliptic curves (featuring the Mordell-Weil theorem) and valuation theory, including local fields. One feature of the book is an extension of the text through a series of appendices. This permits the inclusion of more advanced material, such as transcendental field extensions, the discriminant and resultant, the theory of Dedekind domains, and basic theorems of rings of algebraic integers. An extended appendix on derivations includes the Jacobian conjecture and Makar-Limanov's theory of locally nilpotent derivations. Grobner bases can be found in another appendix. Exercises provide a further extension of the text. The book can be used both as a textbook and as a reference source.
Categories: Mathematics

A First Course in Noncommutative Rings

A First Course in Noncommutative Rings

Aimed at the novice rather than the connoisseur and stressing the role of examples and motivation, this text is suitable not only for use in a graduate course, but also for self-study in the subject by interested graduate students.

Author: Tsit-Yuen Lam

Publisher: Springer Science & Business Media

ISBN: 9781441986160

Category: Mathematics

Page: 388

View: 135

Aimed at the novice rather than the connoisseur and stressing the role of examples and motivation, this text is suitable not only for use in a graduate course, but also for self-study in the subject by interested graduate students. More than 400 exercises testing the understanding of the general theory in the text are included in this new edition.
Categories: Mathematics