Universal Localizations of Certain Noncommutative Rings
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Research Statement Robert Won
Research Statement Robert Won My research interests are in noncommutative algebra, specifically noncommutative ring the- ory and noncommutative projective algebraic geometry. Throughout, let k denote an alge- braically closed field of characteristic 0. All rings will be k-algebras and all categories and equivalences will be k-linear. 1. Introduction Noncommutative rings arise naturally in many contexts. Given a commutative ring R and a nonabelian group G, the group ring R[G] is a noncommutative ring. The n × n matrices with entries in C, or more generally, the linear transformations of a vector space under composition form a noncommutative ring. Noncommutative rings also arise as differential operators. The ring of differential operators on k[t] generated by multiplication by t and differentiation by t is isomorphic to the first Weyl algebra, A1 := khx; yi=(xy − yx − 1), a celebrated noncommutative ring. From the perspective of physics, the Weyl algebra captures the fact that in quantum mechanics, the position and momentum operators do not commute. As noncommutative rings are a larger class of rings than commutative ones, not all of the same tools and techniques are available in the noncommutative setting. For example, localization is only well-behaved at certain subsets of noncommutative rings called Ore sets. Care must also be taken when working with other ring-theoretic properties. The left ideals of a ring need not be (two-sided) ideals. One must be careful in distinguishing between left ideals and right ideals, as well as left and right noetherianness or artianness. In commutative algebra, there is a rich interplay between ring theory and algebraic geom- etry. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
Noncommutative Localization in Algebra and Topology
Noncommutative localization in algebra and topology ICMS Edinburgh 2002 Edited by Andrew Ranicki Electronic version of London Mathematical Society Lecture Note Series 330 Cambridge University Press (2006) Contents Dedication . vii Preface . ix Historical Perspective . x Conference Participants . xi Conference Photo . .xii Conference Timetable . xiii On atness and the Ore condition J. A. Beachy ......................................................1 Localization in general rings, a historical survey P. M. Cohn .......................................................5 Noncommutative localization in homotopy theory W. G. Dwyer . 24 Noncommutative localization in group rings P. A. Linnell . 40 A non-commutative generalisation of Thomason's localisation theorem A. Neeman . 60 Noncommutative localization in topology A. A. Ranicki . 81 v L2-Betti numbers, Isomorphism Conjectures and Noncommutative Lo- calization H. Reich . 103 Invariants of boundary link cobordism II. The Blanch¯eld-Duval form D. Sheiham . 143 Noncommutative localization in noncommutative geometry Z. Skoda· ........................................................220 vi Dedicated to the memory of Desmond Sheiham (13th November 1974 ¡ 25th March 2005) ² Cambridge University (Trinity College), 1993{1997 B.A. Hons. Mathematics 1st Class, 1996 Part III Mathematics, Passed with Distinction, 1997 ² University of Edinburgh, 1997{2001 Ph.D. Invariants of Boundary Link Cobordism, 2001 ² Visiting Assistant Professor, Mathematics Department, University of California at Riverside, 2001{2003 ² Research Instructor, International University Bremen (IUB), 2003{2005 vii Publications: 1. Non-commutative Characteristic Polynomials and Cohn Localization Journal of the London Mathematical Society (2) Vol. 64, 13{28 (2001) http://arXiv.org/abs/math.RA/0104158 2. Invariants of Boundary Link Cobordism Memoirs of the American Mathematical Society, Vol. 165 (2003) http://arXiv.org/abs/math.AT/0110249 3. Whitehead Groups of Localizations and the Endomorphism Class Group Journal of Algebra, Vol. -
Semicorings and Semicomodules Will Open the Door for Many New Applications in the Future (See [Wor2012] for Recent Applications to Automata Theory)
Semicorings and Semicomodules Jawad Y. Abuhlail∗ Department of Mathematics and Statistics Box # 5046, KFUPM, 31261 Dhahran, KSA [email protected] July 7, 2018 Abstract In this paper, we introduce and investigate semicorings over associative semir- ings and their categories of semicomodules. Our results generalize old and recent results on corings over rings and their categories of comodules. The generalization is not straightforward and even subtle at some places due to the nature of the base category of commutative monoids which is neither Abelian (not even additive) nor homological, and has no non-zero injective objects. To overcome these and other dif- ficulties, a combination of methods and techniques from categorical, homological and universal algebra is used including a new notion of exact sequences of semimodules over semirings. Introduction Coalgebraic structures in general, and categories of comodules for comonads in par- arXiv:1303.3924v1 [math.RA] 15 Mar 2013 ticular, are gaining recently increasing interest [Wis2012]. Although comonads can be defined in arbitrary categories, nice properties are usually obtained in case the comonad under consideration is isomorphic to −• C (or C • −) for some comonoid C [Por2006] in a monoidal category (V, •, I) acting on some category X in a nice way [MW]. However, it can be noticed that the most extensively studied concrete examples are – so far – the categories of coacts (usually called coalgebras) of an endo-functor F on the cate- gory Set of sets (motivated by applications in theoretical computer science [Gum1999] and universal (co)algebra [AP2003]) and categories of comodules for a coring over an associate algebra [BW2003], [Brz2009], i.e. -
On Ring Extensions for Completely Primary Noncommutativerings
ON RING EXTENSIONS FOR COMPLETELY PRIMARY NONCOMMUTATIVERINGS BY E. H. FELLER AND E. W. SWOKOWSKI 0. Introduction. It is the authors' purpose in this paper to initiate the study of ring extensions for completely N primary noncommutative rings which satisfy the ascending chain condition for right ideals (A.C.C.). We begin here by showing that every completely N primary ring R with A.C.C. is properly contained in just such a ring. This is accomplished by first showing that R[x~\, x an indeterminate where ax = xa for all aeR,isN primary and then constructing the right quotient ring QCR[x]). The details of these results appear in §§1,7 and 8. The correspond- ing results for the commutative case are given by E. Snapper in [7] and [8]. If Re: A, where A is completely JVprimary with A.C.C. then,from the discussion in the preceding paragraph, it would seem natural to examine the structure of R(o) when oeA and ao = era for all aER in the cases where a is algebraic or transcendental over R. These structures are determined in §§ 6 and 8 of the present paper. The definitions and notations given in [2] will be used throughout this paper. As in [2], for a ring R, JV or N(R) denotes the union of nilpotent ideals^) of R, P or P(R) denotes the set of nilpotent elements of R and J or J(R) the Jacobson radical of R. The letter H is used for the natural homomorphism from R to R/N = R. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
A Brief History of Ring Theory
A Brief History of Ring Theory by Kristen Pollock Abstract Algebra II, Math 442 Loyola College, Spring 2005 A Brief History of Ring Theory Kristen Pollock 2 1. Introduction In order to fully define and examine an abstract ring, this essay will follow a procedure that is unlike a typical algebra textbook. That is, rather than initially offering just definitions, relevant examples will first be supplied so that the origins of a ring and its components can be better understood. Of course, this is the path that history has taken so what better way to proceed? First, it is important to understand that the abstract ring concept emerged from not one, but two theories: commutative ring theory and noncommutative ring the- ory. These two theories originated in different problems, were developed by different people and flourished in different directions. Still, these theories have much in com- mon and together form the foundation of today's ring theory. Specifically, modern commutative ring theory has its roots in problems of algebraic number theory and algebraic geometry. On the other hand, noncommutative ring theory originated from an attempt to expand the complex numbers to a variety of hypercomplex number systems. 2. Noncommutative Rings We will begin with noncommutative ring theory and its main originating ex- ample: the quaternions. According to Israel Kleiner's article \The Genesis of the Abstract Ring Concept," [2]. these numbers, created by Hamilton in 1843, are of the form a + bi + cj + dk (a; b; c; d 2 R) where addition is through its components 2 2 2 and multiplication is subject to the relations i =pj = k = ijk = −1. -
Noncommutative Localization in Noncommutative Geometry
Noncommutative localization in noncommutative geometry Zoran Skodaˇ Abstract The aim of these notes is to collect and motivate the basic localiza- tion toolbox for the geometric study of “spaces” locally described by noncommutative rings and their categories of modules. We present the basics of Ore localization of rings and modules in great detail. Common practical techniques are studied as well. We also describe a counterexample to a folklore test principle for Ore sets. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of localization of differential calculi. To aid the geometrical viewpoint, localization is studied with em- phasis on descent formalism, flatness, abelian categories of quasicoher- ent sheaves and generalizations, and natural pairs of adjoint functors for sheaf and module categories. The key motivational theorems from the seminal works of Gabriel on localization, abelian categories and schemes are quoted without proof, as well as the related statements of Popescu, Eilenberg-Watts, Deligne and Rosenberg. The Cohn universal localization does not have good flatness prop- erties, but it is determined by the localization map already at the ring level, like the perfect localizations are. Cohn localization is here related to the quasideterminants of Gelfand and Retakh; and this may help arXiv:math/0403276v2 [math.QA] 1 Mar 2005 the understanding of both subjects. Contents 1 Introduction 2 2 Noncommutative geometry 6 3 Abstract localization 12 1 2 Noncommutative localization in noncommutative geometry 4 Ore localization for monoids 15 5 Ore localization for rings 22 6 Practical criteria for Ore sets 25 7 Ore localization for modules 30 8 Monads, comonads and gluing 33 9 Distributive laws and compatibility 40 10 Commutative localization 45 11 Ring maps vs. -
A Generalization of the Field of Fractions of an Integral Domain
MICIIAUI), ROBERT EUGENE. A Generalization oi the Field of Fractions of an Integral Domain. (1970) Directed by: Dr. E. E. Posey pp. 21 I ii this paper the author deals with < he problem of construct- ing the field of fractions of an integral domain and a general i'. of one of the methods of construction used to construct a ring oi left quotients for an arbitrary ring. In this generalization thi author relies heavily upon the concept of a faithful complete Eilter and defines partial endomorphisms from the filter el erne into the ring. After partitioning these partial endomorphisms Lnti equivalence classes and after defining operations on the equlvalei classes the author then shows that the resultant structure Is I ring of left quotients. In addition to showing that a faithful complete filter assures the existence of a ring of left quotients the author shows that these rings of left quotients can be embedded in the Utumi ring oJ left quotients. The paper concludes with theorems showing necessary and Hiifflclfiil conditions for the classical ring oi left quotient! ,., fin}! i; to exist .in.l .i theorem establishing the uniqueness up to Isomorphism of the classical ring of left quotients oi a ri R. A GENERALIZATION OF THE FIELD OF FRACTIONS OF AN INTEGRAL DOMAIN by Robert Eugene Michaud A Thesis Submitted to the Faculty of the Graduate School at The University of North Carolina at Greensboro in Partial Fulfillment of the Requirements for the Degree Master of Arts Greensboro August, 1970 Approved by APPROVAL SHEET This thesis has been approved by the following committee of the Faculty of the Graduate School at The University of North Carolina at Greensboro. -
Semirings and Their Applications Semirings and Their Applications
Semirings and their Applications Semirings and their Applications by Jonathan S. Golan University ofHaifa. Haifa. Israel Springer-Science+Business Media, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5252-0 ISBN 978-94-015-9333-5 (eBook) DOI 10.1007/978-94-015-9333-5 Printed on acid-free paper This book is an updated and considerably expended version of the book "The Theory of Semirings, with Applications to Mathematics and Theoretical science", published by Longinov in 1992. All Rights Reserved © 1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999. Softcover reprint ofthe hardcover 1st edition 1999 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. In memory of my mother, Prof. Naomi Golan CONTENTS Preface IX 1. Hemirings and semirings: definitions and examples 1 2. Sets and relations with values in a semiring 19 3. Building new semirings from old 27 4. Some conditions on semirings . 43 5. Complemented elements in semirings 59 6. Ideals in semirings ... 65 7. Prime and semiprime ideals in semirings . .. 85 8. Factor semirings .. 95 9. Morphisms of semirings 105 10. Kernels of morphisms 121 11. Semirings of fractions 129 12. Euclidean semirings 135 13. Additively-regular semirings 143 14. Semimodules over semirings 149 15. Factor semimodules .. 163 16. Some constructions for semimodules .. -
The Real Spectrum of a Noncommutative Ring
JOURNAL OF ALGEBRA 198, 412]427Ž. 1997 ARTICLE JA977145 The Real Spectrum of a Noncommutative Ring Ka Hin LeungU National Uni¨ersity of Singapore, 119260, Singapore View metadata, citation and similar papers at core.ac.uk brought to you by CORE and provided by Elsevier - Publisher Connector Murray Marshall² and Yufei Zhang³ Uni¨ersity of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E6, Canada Communicated by Susan Montgomery Received September 30, 1996 Spaces of orderings were introduced by the second author in a series of paperswx 15]19 and various structure results were obtained generalizing results proved earlier for formally real fields by various people: E. Becker, L. Brocker,È R. Brown, T. Craven, A. Prestel, and N. Schwartz. Inwx 6, 26 Žalso seewx 20, 23, 25. Craven and Tschimmel prove that formally real skew fields give rise to spaces of orderings. Inwx 8Ž also seewx 9. Kalhoff extends this to ternary fieldsŽ. also called planar ternary rings . Inwx 10, 11Ž also see wx28. Kleinstein and Rosenberg and Knebusch, respectively, show how spaces of orderings arise in the study of realŽ. commutative semi-local rings. Orderings on commutative rings have been studied extensively since the introduction of the real spectrum by Coste and Roy in the early 1980s. * E-mail address: [email protected]. ² The research of the second author was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. E-mail address: [email protected] ask.ca. ³ E-mail address: [email protected]. 412 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. -
Noncommutative Algebra University of Manchester, Autumn 2018
MATH 42041/62041: NONCOMMUTATIVE ALGEBRA UNIVERSITY OF MANCHESTER, AUTUMN 2018 NOTES BY TOBY STAFFORD, SLIGHTLY MODIFIED BY MIKE PREST Contents 0. Introduction. 1 1. Preliminaries and examples. 4 2. Modules. 19 3. Chain conditions. 38 4. The nilradical and nilpotent ideals. 48 5. Artinian Rings 56 6. Modules over Principal Ideal Domains. 69 0. Introduction. Noncommutative algebra is a very rich subject with a great many different types of rings, so here I will give an informal introduction to a few of them, and use these examples to illustrate the main ideas of the course. Among commutative rings one has the ring of integers, fields like the rationals Q the reals R and the complexes C, and (factor rings of) polynomial rings like C[x, y ] or Z[x, y ]. Examples of noncommutative rings include the full ring of n n matrices Mn(k) over a field k; these have no ideals. (See the next chapter × for the formal definition, but in brief an ideal is an abelian subgroup closed under left or right multiplication by elements of the ring. We will also want to consider a left ideal which is an abelian subgroup closed under left multiplication by elements of the ring.) Here is an idea of the proof of this statement in the special case of M2(k). So, if P is a nonzero ideal, a b pick a nonzero element α = in P . For simplicity assume that a = 0 (the case a = 0 is left as an c d 6 a−1 0 1 0 1 0 exercise).