Topic
Commutative ring
About: Commutative ring is a research topic. Over the lifetime, 6507 publications have been published within this topic receiving 85692 citations.
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Book•
1 Jan 1988
TL;DR: In this article, the stable factorization approach is introduced to the synthesis of feedback controllers for linear control systems, where the controller is designed as a matrix over a fraction field associated with a commutative ring with identity, denoted by R, which also has no divisors of zero.
Abstract: This book introduces the so-called "stable factorization approach" to the synthesis of feedback controllers for linear control systems The key to this approach is to view the multi-input, multi-output (MIMO) plant for which one wishes to design a controller as a matrix over the fraction field F associated with a commutative ring with identity, denoted by R, which also has no divisors of zero In this setting, the set of single-input, single-output (SISO) stable control systems is precisely the ring R, while the set of stable MIMO control systems is the set of matrices whose elements all belong to R The set of unstable, meaning not necessarily stable, control systems is then taken to be the field of fractions F associated with R in the SISO case, and the set of matrices with elements in F in the MIMO case The central notion introduced in the book is that, in most situations of practical interest, every matrix P whose elements belong to F can be "factored" as a "ratio" of two matrices N,D whose elements belong to R, in such a way that N,D are coprime In the familiar case where the ring R corresponds to the set of bounded-input, bounded-output (BIBO)-stable rational transfer functions, coprimeness is equivalent to two functions not having any common zeros in the closed right half-plane including infinity However, the notion of coprimeness extends readily to discrete-time systems, distributed-parameter systems in both the continuous- as well as discrete-time domains, and to multi-dimensional systems Thus the stable factorization approach enables one to capture all these situations within a common framework The key result in the stable factorization approach is the parametrization of all controllers that stabilize a given plant It is shown that the set of all stabilizing controllers can be parametrized by a single parameter R, whose elements all belong to R Moreover, every transfer matrix in the closed-loop system is an affine function of the design parameter R Thus problems of reliable stabilization, disturbance rejection, robust stabilization etc can all be formulated in terms of choosing an appropriate R This is a reprint of the book Control System Synthesis: A Factorization Approach originally published by MIT Press in 1985 Table of Contents: Introduction / Proper Stable Rational Functions / Scalar Systems: An Introduction / Matrix Rings / Stabilization
2,375 citations
TL;DR: For each commutative ring R we associate a simple graph Γ(R) as discussed by the authors, and we investigate the interplay between the ring-theoretic properties of R and the graph-theory properties of Γ (R).
1,343 citations
1 Jan 2012
TL;DR: A compilation of two sets of notes at the University of Kansas was published in the Spring of 2002 by?? and the other in the spring of 2007 by Branden Stone.
Abstract: 1 A compilation of two sets of notes at the University of Kansas; one in the Spring of 2002 by ?? and the other in the Spring of 2007 by Branden Stone. These notes have been typed
1,289 citations
TL;DR: In this article, the authors present the idea of coloring of a commutative ring and show that the existence of an infinite clique implies that the clique R = co implies that there exists an infinitely many cliques.
1,183 citations
1 Jan 1990
TL;DR: In this article, a localization theorem for the K-theory of commutative rings and of schemes is presented, relating the k-groups of a scheme, of an open subscheme, and of those perfect complexes on the scheme which are acyclic on the open scheme.
Abstract: In this paper we prove a localization theorem for the K-theory of commutative rings and of schemes, Theorem 7.4, relating the K-groups of a scheme, of an open subscheme, and of the category of those perfect complexes on the scheme which are acyclic on the open subscheme. The localization theorem of Quillen [Q1] for K′- or G-theory is the main support of his many results on the G-theory of noetherian schemes. The previous lack of an adequate localization theorem for K-theory has obstructed development of this theory for the fifteen years since 1973. Hence our theorem unleashes a pack of new basic results hitherto known only under very restrictive hypotheses like regularity. These new results include the “Bass fundamental theorem” 6.6, the Zariski (Nisnevich) cohomolog-ical descent spectral sequence that reduces problems to the case of local (hensel local) rings 10.3 and 19.8, the Mayer-Vietoris theorem for open covers 8.1, invariance mod l under polynomial extensions 9.5, Vorst-van der Kallen theory for NK 9.12, Goodwillie and Ogle-Weibel theorems relating K-theory to cyclic cohomology 9.10, mod l Mayer-Vietoris for closed covers 9.8, and mod l comparison between algebraic and topological K-theory 11.5 and 11.9. Indeed most known results in K-theory can be improved by the methods of this paper, by removing now unnecessary regularity, affineness, and other hypotheses.
1,133 citations
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| 2025 | 66 |
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| 2020 | 386 |