Description
This book surveys more than 125 years of aspects of associative
algebras, especially ring and module theory. It is the first to
probe so extensively such a wealth of historical development.
Moreover, the author brings the reader up to date, in particular
through his report on the subject in the second half of the
twentieth century.
Included in the book are certain categorical properties from
theorems of Frobenius and Stickelberger on the primary
decomposition of finite Abelian groups; Hilbert's basis theorem
and his Nullstellensatz, including the modern formulations of the
latter by Krull, Goldman, and others; Maschke's theorem on the
representation theory of finite groups over a field; and the
fundamental theorems of Wedderburn on the structure of finite
dimensional algebras and finite skew fields and their extensions
by Braver, Kaplansky, Chevalley, Goldie, and others. A special
feature of the book is the in-depth study of rings with chain
condition on annihilator ideals pioneered by Noether, Artin, and
Jacobson and refined and extended by many later mathematicians.
Two of the author's prior works, Algebra: Rings, Modules and
Categories, I and II (Springer-Verlag, 1973), are devoted to the
development of modern associative algebra and ring and module
theory. Those works serve as a foundation for the present survey,
which includes a bibliography of over 1,600 references and is
exhaustively indexed.
In addition to the mathematical survey, the author gives candid
and descriptive impressions of the last half of the twentieth
century in "Part II: Snapshots of Some Mathematical Friends
and Places". Beginning with his teachers and fellow graduate
students at the University of Kentucky and at Purdue, Faith
discusses his Fulbright-Nato Postdoctoral at Heidelberg and at
the Institute for Advanced Study (IAS) at Princeton, his year as
a visiting scholar at Berkeley, and the many acquaintances he met
there and in subsequent travels in India, Europe, and most
recently, Barcelona.
Contents
Part I. An array of twentieth century associative algebra
Direct product and sums of rings and modules and the structure of
fields
Introduction to ring theory: Schur's lemma and semisimple rings,
prime and primitive rings, nil, prime and Jacobson radicals
Direct sum decompositions of projective and injective modules
Direct product decompositions of von Neumann regular rings and
self-injective rings
Direct sums of cyclic modules
When injectives are flat: Coherent FP-injective rings
Direct decompositions and dual generalizations of Noetherian
rings
Completely decomposable modules and the Krull-Schmidt-Azumaya
theorem
Polynomial rings over Vamosian and Kerr rings, valuation rings
and Prufer rings
Isomorphic polynomial rings and matrix rings
Group rings and Maschke's theorem revisited
Maximal quotient rings
Morita duality and dual rings
Krull and global dimensions
Polynomial identities and PI-rings
Unions of primes, prime avoidance, associated prime ideals, ACC
on irreducible ideals and annihilator ideals in commutative rings
Dedekind's theorem on the independence of automorphisms revisited
Part II. Snapshots of some mathematical friends and places
Snapshots of some mathematical friends and places
Index to part II (snapshots)
Bibliography
Register of names
Index of terms and authors of theorems
Details:
Series: Mathematical Surveys and Monographs, Volume: 65
Publication Year: 2004
ISBN: 0-8218-3672-2
Paging: approximately 512 pp.
Binding: Softcover
Description
This book is a collection of articles written by leading
researchers in information theory stemming from the DIMACS
Workshop on Network Information held at Rutgers University (Piscataway,
NJ). The articles focus on problems concerning efficient and
reliable communication in multi-terminal settings. Information
theory has recently attracted renewed attention because of key
developments spawning challenging research problems.
The material is divided into four parts: "Information Theory
for Sources", which concentrates on network source coding
problems; "Information Theory for Channels", where
channels, rather than sources, are central to the problem; "Information
Theory for Sources and Channels", which addresses both
source and channel coding; and "Coding", which deals
with more practical issues. Mathematicians using applications
such as wireless cellular and LAN data services, ad hoc networks
and sensor networks will benefit from the developments outlined
in these sections. The book is suitable for graduate students and
research mathematicians interested in communications and network
information theory.
Contents
Part I. Information theory for sources
A. Faridi, K. Sayrafian-Pour, M. Alasti, and A. Ephremides --
Source coding and parallel routing
S. A. Savari -- Compressing a representation of events in a
concurrent system
P. Viswanath -- Sum rate of a class of multiterminal Gaussian
source coding problems
F. M. J. Willems and T. Kalker -- Coding theorems for reversible
embedding
Part II. Information theory for channels
A. S. Cohen and R. Zamir -- Unbounded loss in writing on dirty
paper is possible
R. J. La and V. Anatharam -- A game-theoretic look at the
Gaussian multiaccess channel
X. Liu and R. Srikant -- Bounds on the sum timing capacity of
single-server queues with multiple input and output terminals
S. Raj, E. Telatar, and D. Tse -- Job scheduling and multiple
access
D. Tuninetti and S. Shamai (Shitz) -- Fading Gaussian broadcast
channels with state information at the receivers
L.-L. Xie and P. R. Kumar -- Wireless network information theory
W. Yu -- The structure of least-favorable noise in Gaussian
vector broadcast channels
Part III. Information theory for sources and channels
J. Barros and S. D. Servetto -- Coding theorems for the sensor
reachback problem with partially cooperating nodes
M. Effros, M. Medard, T. Ho, S. Ray, D. Karger, R. Koetter, and B.
Hassibi -- Linear network codes: A unified framework for source,
channel, and network coding
M. Gastpar -- On source-channel communication in networks
S. S. Pradhan and K. Ramchandran -- Duality in multi-user source
and channel coding
Part IV. Coding
G. Caire, S. Shamai, and S. Verdu -- Noiseless data compression
with low-density parity-check codes
S. N. Diggavi, N. Al-Dhahir, and A. R. Calderbank -- Diversity
embedding in multiple antenna communications
E. Erkip, A. Sendonaris, A. Stefanov, and B. Aazhang --
Cooperative communication in wireless systems
E. Soljanin, R. Liu, and P. Spasojevic -- Hybrid ARQ with random
transmission assignments
J. K. Wolf -- An information-theoretic approach to bit-stuffing
for network protocols
Details:
Series: DIMACS: Series in Discrete Mathematics and Theoretical
Computer Science,Volume: 66
Publication Year: 2004
ISBN: 0-8218-3467-3
Paging: approximately 352 pp.
Binding: Hardcover
Description
Vertex algebras are algebraic objects that encapsulate the
concept of operator product expansion from two-dimensional
conformal field theory. Vertex algebras are fast becoming
ubiquitous in many areas of modern mathematics, with applications
to representation theory, algebraic geometry, the theory of
finite groups, modular functions, topology, integrable systems,
and combinatorics.
This book is an introduction to the theory of vertex algebras
with a particular emphasis on the relationship with the geometry
of algebraic curves. The notion of a vertex algebra is introduced
in a coordinate-independent way, so that vertex operators become
well defined on arbitrary smooth algebraic curves, possibly
equipped with additional data, such as a vector bundle. Vertex
algebras then appear as the algebraic objects encoding the
geometric structure of various moduli spaces associated with
algebraic curves. Therefore they may be used to give a geometric
interpretation of various questions of representation theory.
The book contains many original results, introduces important new
concepts, and brings new insights into the theory of vertex
algebras. The authors have made a great effort to make the book
self-contained and accessible to readers of all backgrounds.
Reviewers of the first edition anticipated that it would have a
long-lasting influence on this exciting field of mathematics and
would be very useful for graduate students and researchers
interested in the subject.
This second edition, substantially improved and expanded,
includes several new topics, in particular an introduction to the
Beilinson-Drinfeld theory of factorization algebras and the
geometric Langlands correspondence.
The book is suitable for graduate students and research
mathematicians interested in representation theory, algebraic
geometry, and mathematical physics. For more information, visit
www.math.berkeley.edu/~frenkel/book.
Contents
Introduction
Definition of vertex algebras
Vertex algebras associated to Lie algebras
Associativity and operator product expansion
Applications of the operator product expansion
Modules over vertex algebras and more examples
Vertex algebra bundles
Action of internal symmetries
Vertex algebra bundles: Examples
Conformal blocks I
Conformal blocks II
Free field realization I
Free field realization II
The Knizhnik-Zamolodchikov equations
Solving the KZ equations
Quantum Drinfeld-Sokolov reduction and $\mathcal{W}$-algebras
Vertex Lie algebras and classical limits
Vertex algebras and moduli spaces I
Vertex algebras and moduli spaces II
Chiral algebras
Factorization
Appendix
Bibliography
Index
List of frequently used notation
Details:
Series: Mathematical Surveys and Monographs, Volume: 88
Publication Year: 2004
ISBN: 0-8218-3674-9
Paging: 400 pp.
Binding: Softcover
Description
The space of all Riemann surfaces (the so-called moduli space)
plays an important role in algebraic geometry and its
applications to quantum field theory. The present book is devoted
to the study of topological properties of this space and of
similar moduli spaces, such as the space of real algebraic
curves, the space of mappings, and also superanalogs of all these
spaces.
The book can be used by researchers and graduate students working
in algebraic geometry, topology, and mathematical physics.
Contents
Introduction
Moduli of Riemann surfaces, Hurwitz type spaces and their
superanalogs
Moduli of real algebraic curves and their superanalogs.
Differentials, spinors, and Jacobians of real curves
Spaces of meromorphic functions on complex and real algebraic
curves
Bibliography
Index
Details:
Series: Translations of Mathematical Monographs, Volume: 225
Publication Year: 2004
ISBN: 0-8218-3594-7
Paging: 160 pp.
Binding: Hardcover
Description
First published in 1991, this book contains the core material for
an undergraduate first course in ring theory. Using the
underlying theme of projective and injective modules, the author
touches upon various aspects of commutative and noncommutative
ring theory. In particular, a number of major results are
highlighted and proved.
The first part of the book, called "Projective Modules",
begins with basic module theory and then proceeds to surveying
various special classes of rings (Wedderburn, Artinian and
Noetherian rings, hereditary rings, Dedekind domains, etc.). This
part concludes with an introduction and discussion of the
concepts of the projective dimension.
Part II, "Polynomial Rings", studies these rings in a
mildly noncommutative setting. Some of the results proved include
the Hilbert Syzygy Theorem (in the commutative case) and the
Hilbert Nullstellensatz (for almost commutative rings).
Part III, "Injective Modules", includes, in particular,
various notions of the ring of quotients, the Goldie Theorems,
and the characterization of the injective modules over Noetherian
rings.
The book contains numerous exercises and a list of suggested
additional reading. It is suitable for graduate students and
researchers interested in ring theory.
Contents
Projective modules
Modules and homomorphisms
Projective modules
Completely reducible modules
Wedderburn rings
Artinian rings
Hereditary rings
Dedekind domains
Projective dimension
Tensor products
Local rings
Polynomial rings
Skew polynomial rings
Grothendieck groups
Graded rings and modules
Induced modules
Syzygy theorem
Patching theorem
Serre conjecture
Big projectives
Generic flatness
Nullstellensatz
Injective modules
Injective modules
Injective dimension
Essential extensions
Maximal ring of quotients
Classical ring of quotients
Goldie rings
Uniform dimension
Uniform injective modules
Reduced rank
Index
Details:
Series: AMS Chelsea Publishing
Publication Year: 2004
ISBN: 0-8218-3680-3
Paging: 306 pp.
Binding: Hardcover