Noncommutative Determinant Is Hard: a Simple Proof Using an Extension of Barrington’S Theorem
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On Regular Matrix Semirings
Thai Journal of Mathematics Volume 7 (2009) Number 1 : 69{75 www.math.science.cmu.ac.th/thaijournal Online ISSN 1686-0209 On Regular Matrix Semirings S. Chaopraknoi, K. Savettaseranee and P. Lertwichitsilp Abstract : A ring R is called a (von Neumann) regular ring if for every x 2 R; x = xyx for some y 2 R. It is well-known that for any ring R and any positive integer n, the full matrix ring Mn(R) is regular if and only if R is a regular ring. This paper examines this property on any additively commutative semiring S with zero. The regularity of S is defined analogously. We show that for a positive integer n, if Mn(S) is a regular semiring, then S is a regular semiring but the converse need not be true for n = 2. And for n ≥ 3, Mn(S) is a regular semiring if and only if S is a regular ring. Keywords : Matrix ring; Matrix semiring; Regular ring; Regular semiring. 2000 Mathematics Subject Classification : 16Y60, 16S50. 1 Introduction A triple (S; +; ·) is called a semiring if (S; +) and (S; ·) are semigroups and · is distributive over +. An element 0 2 S is called a zero of the semiring (S; +; ·) if x + 0 = 0 + x = x and x · 0 = 0 · x = 0 for all x 2 S. Note that every semiring contains at most one zero. A semiring (S; +; ·) is called additively [multiplicatively] commutative if x + y = y + x [x · y = y · x] for all x; y 2 S: We say that (S; +; ·) is commutative if it is both additively and multiplicatively commutative. -
1 Jun 2020 Flat Semimodules & Von Neumann Regular Semirings
Flat Semimodules & von Neumann Regular Semirings* Jawad Abuhlail† Rangga Ganzar Noegraha‡ [email protected] [email protected] Department of Mathematics and Statistics Universitas Pertamina King Fahd University of Petroleum & Minerals Jl. Teuku Nyak Arief 31261Dhahran,KSA Jakarta12220,Indonesia June 2, 2020 Abstract Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of exact sequences of semimodules and its relationships with other notions of flatness for semimodules over semirings. We also prove that a subtractive semiring over which every right (left) semimodule is e-flat is a von Neumann regular semiring. Introduction Semirings are, roughly, rings not necessarily with subtraction. They generalize both rings and distributive bounded lattices and have, along with their semimodules many applications in Com- arXiv:1907.07047v2 [math.RA] 1 Jun 2020 puter Science and Mathematics (e.g., [1, 2, 3]). Some applications can be found in Golan’s book [4], which is our main reference on this topic. A systematic study of semimodules over semirings was carried out by M. Takahashi in a series of papers 1981-1990. However, he defined two main notions in a way that turned out to be not natural. Takahashi’s tensor products [5] did not satisfy the expected Universal Property. On *MSC2010: Primary 16Y60; Secondary 16D40, 16E50, 16S50 Key Words: Semirings; Semimodules; Flat Semimodules; Exact Sequences; Von Neumann Regular Semirings; Matrix Semirings The authors would like to acknowledge the support provided by the Deanship of Scientific Research (DSR) at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work through projects No. -
Subset Semirings
University of New Mexico UNM Digital Repository Faculty and Staff Publications Mathematics 2013 Subset Semirings Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy [email protected] Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebraic Geometry Commons, Analysis Commons, and the Other Mathematics Commons Recommended Citation W.B. Vasantha Kandasamy & F. Smarandache. Subset Semirings. Ohio: Educational Publishing, 2013. This Book is brought to you for free and open access by the Mathematics at UNM Digital Repository. It has been accepted for inclusion in Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Subset Semirings W. B. Vasantha Kandasamy Florentin Smarandache Educational Publisher Inc. Ohio 2013 This book can be ordered from: Education Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: 1-866-880-5373 Copyright 2013 by Educational Publisher Inc. and the Authors Peer reviewers: Marius Coman, researcher, Bucharest, Romania. Dr. Arsham Borumand Saeid, University of Kerman, Iran. Said Broumi, University of Hassan II Mohammedia, Casablanca, Morocco. Dr. Stefan Vladutescu, University of Craiova, Romania. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/eBooks-otherformats.htm ISBN-13: 978-1-59973-234-3 EAN: 9781599732343 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two SUBSET SEMIRINGS OF TYPE I 9 Chapter Three SUBSET SEMIRINGS OF TYPE II 107 Chapter Four NEW SUBSET SPECIAL TYPE OF TOPOLOGICAL SPACES 189 3 FURTHER READING 255 INDEX 258 ABOUT THE AUTHORS 260 4 PREFACE In this book authors study the new notion of the algebraic structure of the subset semirings using the subsets of rings or semirings. -
Arxiv:2006.00374V4 [Math.GT] 28 May 2021
CONTROLLED MATHER-THURSTON THEOREMS MICHAEL FREEDMAN ABSTRACT. Classical results of Milnor, Wood, Mather, and Thurston produce flat connections in surprising places. The Milnor-Wood inequality is for circle bundles over surfaces, whereas the Mather-Thurston Theorem is about cobording general manifold bundles to ones admitting a flat connection. The surprise comes from the close encounter with obstructions from Chern-Weil theory and other smooth obstructions such as the Bott classes and the Godbillion-Vey invariant. Contradic- tion is avoided because the structure groups for the positive results are larger than required for the obstructions, e.g. PSL(2,R) versus U(1) in the former case and C1 versus C2 in the latter. This paper adds two types of control strengthening the positive results: In many cases we are able to (1) refine the Mather-Thurston cobordism to a semi-s-cobordism (ssc) and (2) provide detail about how, and to what extent, transition functions must wander from an initial, small, structure group into a larger one. The motivation is to lay mathematical foundations for a physical program. The philosophy is that living in the IR we cannot expect to know, for a given bundle, if it has curvature or is flat, because we can’t resolve the fine scale topology which may be present in the base, introduced by a ssc, nor minute symmetry violating distortions of the fiber. Small scale, UV, “distortions” of the base topology and structure group allow flat connections to simulate curvature at larger scales. The goal is to find a duality under which curvature terms, such as Maxwell’s F F and Hilbert’s R dvol ∧ ∗ are replaced by an action which measures such “distortions.” In this view, curvature resultsR from renormalizing a discrete, group theoretic, structure. -
Torsion Homology and Cellular Approximation 11
TORSION HOMOLOGY AND CELLULAR APPROXIMATION RAMON´ FLORES AND FERNANDO MURO Abstract. In this note we describe the role of the Schur multiplier in the structure of the p-torsion of discrete groups. More concretely, we show how the knowledge of H2G allows to approximate many groups by colimits of copies of p-groups. Our examples include interesting families of non-commutative infinite groups, including Burnside groups, certain solvable groups and branch groups. We also provide a counterexample for a conjecture of E. Farjoun. 1. Introduction Since its introduction by Issai Schur in 1904 in the study of projective representations, the Schur multiplier H2(G, Z) has become one of the main invariants in the context of Group Theory. Its importance is apparent from the fact that its elements classify all the central extensions of the group, but also from his relation with other operations in the group (tensor product, exterior square, derived subgroup) or its role as the Baer invariant of the group with respect to the variety of abelian groups. This last property, in particular, gives rise to the well- known Hopf formula, which computes the multiplier using as input a presentation of the group, and hence has allowed to use it in an effective way in the field of computational group theory. A brief and concise account to the general concept can be found in [38], while a thorough treatment is the monograph by Karpilovsky [33]. In the last years, the close relationship between the Schur multiplier and the theory of co-localizations of groups, and more precisely with the cellular covers of a group, has been remarked. -
Derivations of a Matrix Ring Containing a Subring of Triangular Matrices S
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2011, Vol. 55, No. 11, pp. 18–26. c Allerton Press, Inc., 2011. Original Russian Text c S.G. Kolesnikov and N.V. Mal’tsev, 2011, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2011, No. 11, pp. 23–33. Derivations of a Matrix Ring Containing a Subring of Triangular Matrices S. G. Kolesnikov1* andN.V.Mal’tsev2** 1Institute of Mathematics, Siberian Federal University, pr. Svobodnyi 79, Krasnoyarsk, 660041 Russia 2Institute of Core Undergraduate Programs, Siberian Federal University, 26 ul. Akad. Kirenskogo 26, Krasnoyarsk, 660041 Russia Received October 22, 2010 Abstract—We describe the derivations (including Jordan and Lie ones) of finitary matrix rings, containing a subring of triangular matrices, over an arbitrary associative ring with identity element. DOI: 10.3103/S1066369X1111003X Keywords and phrases: finitary matrix, derivations of rings, Jordan and Lie derivations of rings. INTRODUCTION An endomorphism ϕ of an additive group of a ring K is called a derivation if ϕ(ab)=ϕ(a)b + aϕ(b) for any a, b ∈ K. Derivations of the Lie and Jordan rings associated with K are called, respectively, the Lie and Jordan derivations of the ring K; they are denoted, respectively, by Λ(K) and J(K). (We obtain them by replacing the multiplication in K with the Lie multiplication a ∗ b = ab − ba and the Jordan one a ◦ b = ab + ba, respectively.) In 1957 I. N. Herstein proved [1] that every Jordan derivation of a primary ring of characteristic =2 is a derivation. His theorem was generalized by several authors ([2–9] et al.) for other classes of rings and algebras. -
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. -
Problems in Abstract Algebra
STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Projective Representations of Groups
PROJECTIVE REPRESENTATIONS OF GROUPS EDUARDO MONTEIRO MENDONCA Abstract. We present an introduction to the basic concepts of projective representations of groups and representation groups, and discuss their relations with group cohomology. We conclude the text by discussing the projective representation theory of symmetric groups and its relation to Sergeev and Hecke-Clifford Superalgebras. Contents Introduction1 Acknowledgements2 1. Group cohomology2 1.1. Cohomology groups3 1.2. 2nd-Cohomology group4 2. Projective Representations6 2.1. Projective representation7 2.2. Schur multiplier and cohomology class9 2.3. Equivalent projective representations 10 3. Central Extensions 11 3.1. Central extension of a group 12 3.2. Central extensions and 2nd-cohomology group 13 4. Representation groups 18 4.1. Representation group 18 4.2. Representation groups and projective representations 21 4.3. Perfect groups 27 5. Symmetric group 30 5.1. Representation groups of symmetric groups 30 5.2. Digression on superalgebras 33 5.3. Sergeev and Hecke-Clifford superalgebras 36 References 37 Introduction The theory of group representations emerged as a tool for investigating the structure of a finite group and became one of the central areas of algebra, with important connections to several areas of study such as topology, Lie theory, and mathematical physics. Schur was Date: September 7, 2017. Key words and phrases. projective representation, group, symmetric group, central extension, group cohomology. 2 EDUARDO MONTEIRO MENDONCA the first to realize that, for many of these applications, a new kind of representation had to be introduced, namely, projective representations. The theory of projective representations involves homomorphisms into projective linear groups. Not only do such representations appear naturally in the study of representations of groups, their study showed to be of great importance in the study of quantum mechanics. -
Group Matrix Ring Codes and Constructions of Self-Dual Codes
Group Matrix Ring Codes and Constructions of Self-Dual Codes S. T. Dougherty University of Scranton Scranton, PA, 18518, USA Adrian Korban Department of Mathematical and Physical Sciences University of Chester Thornton Science Park, Pool Ln, Chester CH2 4NU, England Serap S¸ahinkaya Tarsus University, Faculty of Engineering Department of Natural and Mathematical Sciences Mersin, Turkey Deniz Ustun Tarsus University, Faculty of Engineering Department of Computer Engineering Mersin, Turkey February 2, 2021 arXiv:2102.00475v1 [cs.IT] 31 Jan 2021 Abstract In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring Mk(R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring Mk(R) are one sided ideals in the group matrix k ring Mk(R)G and the corresponding codes over the ring R are G - codes of length kn. Additionally, we give a generator matrix for self- dual codes, which consist of the mentioned above matrix construction. 1 We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes. 1 Introduction Self-dual codes are one of the most widely studied and interesting class of codes. They have been shown to have strong connections to unimodular lattices, invariant theory, and designs. -
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. -
Matrix Rings and Linear Groups Over a Field of Fractions of Enveloping Algebras and Group Rings, I
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 88, 1-37 (1984) Matrix Rings and Linear Groups over a Field of Fractions of Enveloping Algebras and Group Rings, I A. I. LICHTMAN Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel, and Department of Mathematics, The University of Texas at Austin, Austin, Texas * Communicated by I. N. Herstein Received September 20, 1982 1. INTRODUCTION 1.1. Let H be a Lie algebra over a field K, U(H) be its universal envelope. It is well known that U(H) is a domain (see [ 1, Section V.31) and the theorem of Cohn (see [2]) states that U(H) can be imbedded in a field.’ When H is locally finite U(H) has even a field of fractions (see [ 1, Theorem V.3.61). The class of Lie algebras whose universal envelope has a field of fractions is however .much wider than the class of locally finite Lie algebras; it follows from Proposition 4.1 of the article that it contains, for instance, all the locally soluble-by-locally finite algebras. This gives a negative answer of the question in [2], page 528. Let H satisfy any one of the following three conditions: (i) U(H) is an Ore ring and fi ,“=, Hm = 0. (ii) H is soluble of class 2. (iii) Z-I is finite dimensional over K. Let D be the field of fractions of U(H) and D, (n > 1) be the matrix ring over D.