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Author: Hideyuki Matsumura
Publisher: Cambridge University Press
ISBN: 9780521367646
Category : Mathematics
Languages : en
Pages : 338
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Book Description
This book explores commutative ring theory, an important a foundation for algebraic geometry and complex analytical geometry.
Author: Hideyuki Matsumura
Publisher: Cambridge University Press
ISBN: 9780521367646
Category : Mathematics
Languages : en
Pages : 338
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Book Description
This book explores commutative ring theory, an important a foundation for algebraic geometry and complex analytical geometry.
Author: H. Matsumura
Publisher: Cambridge University Press
ISBN: 1107717124
Category : Mathematics
Languages : en
Pages : 338
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Book Description
In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings. More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored. The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject. Exercises are provided at the end of each section and solutions or hints to some of them are given at the end of the book.
Author: Fanggui Wang
Publisher: Springer
ISBN: 9811033374
Category : Mathematics
Languages : en
Pages : 714
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Book Description
This book provides an introduction to the basics and recent developments of commutative algebra. A glance at the contents of the first five chapters shows that the topics covered are ones that usually are included in any commutative algebra text. However, the contents of this book differ significantly from most commutative algebra texts: namely, its treatment of the Dedekind–Mertens formula, the (small) finitistic dimension of a ring, Gorenstein rings, valuation overrings and the valuative dimension, and Nagata rings. Going further, Chapter 6 presents w-modules over commutative rings as they can be most commonly used by torsion theory and multiplicative ideal theory. Chapter 7 deals with multiplicative ideal theory over integral domains. Chapter 8 collects various results of the pullbacks, especially Milnor squares and D+M constructions, which are probably the most important example-generating machines. In Chapter 9, coherent rings with finite weak global dimensions are probed, and the local ring of weak global dimension two is elaborated on by combining homological tricks and methods of star operation theory. Chapter 10 is devoted to the Grothendieck group of a commutative ring. In particular, the Bass–Quillen problem is discussed. Finally, Chapter 11 aims to introduce relative homological algebra, especially where the related concepts of integral domains which appear in classical ideal theory are defined and investigated by using the class of Gorenstein projective modules. Each section of the book is followed by a selection of exercises of varying degrees of difficulty. This book will appeal to a wide readership from graduate students to academic researchers who are interested in studying commutative algebra.
Author: John J. Watkins
Publisher: Princeton University Press
ISBN: 1400828171
Category : Mathematics
Languages : en
Pages : 228
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Book Description
Topics in Commutative Ring Theory is a textbook for advanced undergraduate students as well as graduate students and mathematicians seeking an accessible introduction to this fascinating area of abstract algebra. Commutative ring theory arose more than a century ago to address questions in geometry and number theory. A commutative ring is a set-such as the integers, complex numbers, or polynomials with real coefficients--with two operations, addition and multiplication. Starting from this simple definition, John Watkins guides readers from basic concepts to Noetherian rings-one of the most important classes of commutative rings--and beyond to the frontiers of current research in the field. Each chapter includes problems that encourage active reading--routine exercises as well as problems that build technical skills and reinforce new concepts. The final chapter is devoted to new computational techniques now available through computers. Careful to avoid intimidating theorems and proofs whenever possible, Watkins emphasizes the historical roots of the subject, like the role of commutative rings in Fermat's last theorem. He leads readers into unexpected territory with discussions on rings of continuous functions and the set-theoretic foundations of mathematics. Written by an award-winning teacher, this is the first introductory textbook to require no prior knowledge of ring theory to get started. Refreshingly informal without ever sacrificing mathematical rigor, Topics in Commutative Ring Theory is an ideal resource for anyone seeking entry into this stimulating field of study.
Author: Bernard R. McDonald
Publisher: CRC Press
ISBN: 1000146464
Category : Mathematics
Languages : en
Pages : 564
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Book Description
This monograph arose from lectures at the University of Oklahoma on topics related to linear algebra over commutative rings. It provides an introduction of matrix theory over commutative rings. The monograph discusses the structure theory of a projective module.
Author: Gilberto Bini
Publisher: Springer Science & Business Media
ISBN: 1461509572
Category : Technology & Engineering
Languages : en
Pages : 181
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Book Description
Foreword by Dieter Jungnickel Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory. This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. The book is a concrete and self-contained introduction to finite commutative local rings, focusing in particular on Galois and Quasi-Galois rings. The reader is provided with an active and concrete approach to the study of the purely algebraic structure and properties of finite commutative rings (in particular, Galois rings) as well as to their applications to coding theory. Finite Commutative Rings and their Applications is the first to address both theoretical and practical aspects of finite ring theory. The authors provide a practical approach to finite rings through explanatory examples, thereby avoiding an abstract presentation of the subject. The section on Quasi-Galois rings presents new and unpublished results as well. The authors then introduce some applications of finite rings, in particular Galois rings, to coding theory, using a solid algebraic and geometric theoretical background.
Author: Michael F. Atiyah
Publisher: CRC Press
ISBN: 0429973268
Category : Mathematics
Languages : en
Pages : 140
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Book Description
First Published in 2018. This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of producing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts such as Zariski-Samuel or Bourbaki. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization.
Author: S.T. Chapman
Publisher: Springer Science & Business Media
ISBN: 1475731809
Category : Mathematics
Languages : en
Pages : 477
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Book Description
Commutative Ring Theory emerged as a distinct field of research in math ematics only at the beginning of the twentieth century. It is rooted in nine teenth century major works in Number Theory and Algebraic Geometry for which it provided a useful tool for proving results. From this humble origin, it flourished into a field of study in its own right of an astonishing richness and interest. Nowadays, one has to specialize in an area of this vast field in order to be able to master its wealth of results and come up with worthwhile contributions. One of the major areas of the field of Commutative Ring Theory is the study of non-Noetherian rings. The last ten years have seen a lively flurry of activity in this area, including: a large number of conferences and special sections at national and international meetings dedicated to presenting its results, an abundance of articles in scientific journals, and a substantial number of books capturing some of its topics. This rapid growth, and the occasion of the new Millennium, prompted us to embark on a project aimed at presenting an overview of the recent research in the area. With this in mind, we invited many of the most prominent researchers in Non-Noetherian Commutative Ring Theory to write expository articles representing the most recent topics of research in this area.
Author: K.P.S. Bhaskara Rao
Publisher: CRC Press
ISBN: 0203218876
Category : Mathematics
Languages : en
Pages : 193
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Book Description
The theory of generalized inverses of real or complex matrices has been expertly developed and documented. But the generalized inverses of matrices over rings have received comprehensive treatment only recently. In this book, the author, who contributed to the research and development of the theory, explains his results. The subject of generalized inverses of matrices over rings has now reached a state suitable for a comprehensive treatment - this book provides just that, for mathematicians, algebraists and control theorists.
Author: Marco Fontana
Publisher: CRC Press
ISBN: 9780203910627
Category : Mathematics
Languages : en
Pages : 524
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Book Description
Featuring presentations from the Fourth International Conference on Commutative Algebra held in Fez, Morocco, this reference presents trends in the growing area of commutative algebra. With contributions from nearly 50 internationally renowned researchers, the book emphasizes innovative applications and connections to algebraic number theory, geome
