The Spectrum Of A Ring As A Partially Ordered Set
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| Author | : Gregory W. Brumfiel |
| Publisher | : Cambridge University Press |
| Total Pages | : 293 |
| Release | : 1979-12-20 |
| Genre | : Mathematics |
| ISBN | : 052122845X |
The purpose of this unique book is to establish purely algebraic foundations for the development of certain parts of topology. Some topologists seek to understand geometric properties of solutions to finite systems of equations or inequalities and configurations which in some sense actually occur in the real world. Others study spaces constructed more abstractly using infinite limit processes. Their goal is to determine just how similar or different these abstract spaces are from those which are finitely described. However, as topology is usually taught, even the first, more concrete type of problem is approached using the language and methods of the second type. Professor Brumfiel's thesis is that this is unnecessary and, in fact, misleading philosophically. He develops a type of algebra, partially ordered rings, in which it makes sense to talk about solutions of equations and inequalities and to compare geometrically the resulting spaces. The importance of this approach is primarily that it clarifies the sort of geometrical questions one wants to ask and answer about those spaces which might have physical significance.
| Author | : Toma Albu |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 204 |
| Release | : 2011-02-04 |
| Genre | : Mathematics |
| ISBN | : 3034600070 |
This book is a collection of invited papers and articles, many presented at the 2008 International Conference on Ring and Module Theory. The papers explore the latest in various areas of algebra, including ring theory, module theory and commutative algebra.
| Author | : S.T. Chapman |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 477 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 1475731809 |
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 | : Max Dickmann |
| Publisher | : Cambridge University Press |
| Total Pages | : 652 |
| Release | : 2019-03-21 |
| Genre | : Mathematics |
| ISBN | : 1107146720 |
Offers a comprehensive presentation of spectral spaces focussing on their topology and close connections with algebra, ordered structures, and logic.
| Author | : Francisco Miraglia |
| Publisher | : Polimetrica s.a.s. |
| Total Pages | : 517 |
| Release | : 2006 |
| Genre | : Mathematics |
| ISBN | : 8876990356 |
| Author | : Paul-Jean Cahen |
| Publisher | : CRC Press |
| Total Pages | : 489 |
| Release | : 2023-06-14 |
| Genre | : Mathematics |
| ISBN | : 1000946762 |
Presents the proceedings of the Second International Conference on Commutative Ring Theory in Fes, Morocco. The text details developments in commutative algebra, highlighting the theory of rings and ideals. It explores commutative algebra's connections with and applications to topological algebra and algebraic geometry.
| Author | : Marco Fontana |
| Publisher | : Springer |
| Total Pages | : 372 |
| Release | : 2014-07-15 |
| Genre | : Mathematics |
| ISBN | : 1493909258 |
This volume presents a multi-dimensional collection of articles highlighting recent developments in commutative algebra. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non- Noetherian ring theory as well as integer-valued polynomials and functions. Specific topics include: · Homological dimensions of Prüfer-like rings · Quasi complete rings · Total graphs of rings · Properties of prime ideals over various rings · Bases for integer-valued polynomials · Boolean subrings · The portable property of domains · Probabilistic topics in Intn(D) · Closure operations in Zariski-Riemann spaces of valuation domains · Stability of domains · Non-Noetherian grade · Homotopy in integer-valued polynomials · Localizations of global properties of rings · Topics in integral closure · Monoids and submonoids of domains The book includes twenty articles written by many of the most prominent researchers in the field. Most contributions are authored by attendees of the conference in commutative algebra held at the Graz University of Technology in December 2012. There is also a small collection of invited articles authored by those who did not attend the conference. Following the model of the Graz conference, the volume contains a number of comprehensive survey articles along with related research articles featuring recent results that have not yet been published elsewhere.
| Author | : Rüdiger Göbel |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 450 |
| Release | : 1994 |
| Genre | : Mathematics |
| ISBN | : 0821851780 |
This volume contains the proceedings of a conference on abelian groups held in August 1993 at Oberwolfach. The conference brought together forty-seven participants from all over the world and from a range of mathematical areas. Experts from model theory, set theory, noncommutative groups, module theory, and computer science discussed problems in their fields that relate to abelian group theory. This book provides a window on the frontier of this active area of research.
| Author | : William Heinzer |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 426 |
| Release | : 2021-10-08 |
| Genre | : Education |
| ISBN | : 1470466422 |
Power series provide a technique for constructing examples of commutative rings. In this book, the authors describe this technique and use it to analyse properties of commutative rings and their spectra. This book presents results obtained using this approach. The authors put these results in perspective; often the proofs of properties of classical examples are simplified. The book will serve as a helpful resource for researchers working in commutative algebra.
| Author | : Niels Schwartz |
| Publisher | : Springer |
| Total Pages | : 276 |
| Release | : 2006-11-13 |
| Genre | : Mathematics |
| ISBN | : 3540482849 |
The book lays algebraic foundations for real geometry through a systematic investigation of partially ordered rings of semi-algebraic functions. Real spectra serve as primary geometric objects, the maps between them are determined by rings of functions associated with the spectra. The many different possible choices for these rings of functions are studied via reflections of partially ordered rings. Readers should feel comfortable using basic algebraic and categorical concepts. As motivational background some familiarity with real geometry will be helpful. The book aims at researchers and graduate students with an interest in real algebra and geometry, ordered algebraic structures, topology and rings of continuous functions.
