Rings with Internal Cancellation
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Comultiplication on Monoids
Internat. J. Math. & Math. Sci. 803 VOL. 20 NO. 4 (1997) 803-812 COMULTIPLICATION ON MONOIDS MARTIN ARKOWITZ Mathematics Department Dartmouth College Hanover, NH 03755 E-mail address: [email protected] MAURIClO GUTIERREZ Mathematics Department Tufts University Medford, MA 02155 E-mail address: [email protected] (Received April 14, 1997) ABSTRACT. A comultiplication on a monoid S is a homomorphism rn: S S S (the free product of S with itself) whose composition with each projection is the identity homomorphism. We investigate how the existence of a comultiplication on S restricts the structure of S. We show that a monoid which satisfies the inverse property and has a comultiplication is cancellative and equidivisible. Our main result is that a monoid S which satisfies the inverse property admits a comultiplication if and only if S is the free product of a free monoid and a free group. We call these monoids semi-free and we study different comultiplications on them. KEY WORDS AND PHRASES: Monoid, free product, comultiplication, inverse property. cancellative monoid, equidivisibility, semi-free monoid. 1991 AMS SUBJECT CLASSIFICATION CODES: Primary: 20M05, 20M10 Secondary: 20M99, 20N99. 1. INTRODUCTION Comultiplications can be defined for objects in any category with coproducts and zero mor- phisms. Given such an object X, a comultiplication is a morphism m X X II X (the coproduct of X with itself), such that the composition of rn with either projection X t X X is the identity morphism. The general theory has been surveyed in [1], and in [3] it is specialized to algebraic systems. In [2], [5] and [6], comultiplications on groups have been studied. -
Comultiplication on Monoids
Internat. J. Math. & Math. Sci. 803 VOL. 20 NO. 4 (1997) 803-812 COMULTIPLICATION ON MONOIDS MARTIN ARKOWITZ Mathematics Department Dartmouth College Hanover, NH 03755 E-mail address: [email protected] MAURIClO GUTIERREZ Mathematics Department Tufts University Medford, MA 02155 E-mail address: [email protected] (Received April 14, 1997) ABSTRACT. A comultiplication on a monoid S is a homomorphism rn: S S S (the free product of S with itself) whose composition with each projection is the identity homomorphism. We investigate how the existence of a comultiplication on S restricts the structure of S. We show that a monoid which satisfies the inverse property and has a comultiplication is cancellative and equidivisible. Our main result is that a monoid S which satisfies the inverse property admits a comultiplication if and only if S is the free product of a free monoid and a free group. We call these monoids semi-free and we study different comultiplications on them. KEY WORDS AND PHRASES: Monoid, free product, comultiplication, inverse property. cancellative monoid, equidivisibility, semi-free monoid. 1991 AMS SUBJECT CLASSIFICATION CODES: Primary: 20M05, 20M10 Secondary: 20M99, 20N99. 1. INTRODUCTION Comultiplications can be defined for objects in any category with coproducts and zero mor- phisms. Given such an object X, a comultiplication is a morphism m X X II X (the coproduct of X with itself), such that the composition of rn with either projection X t X X is the identity morphism. The general theory has been surveyed in [1], and in [3] it is specialized to algebraic systems. In [2], [5] and [6], comultiplications on groups have been studied. -
Algebraic Systems, Spring 2014, January, 2014 Edition Gabriel Kerr
Algebraic Systems, Spring 2014, January, 2014 Edition Gabriel Kerr Contents Chapter 0. Peano Axioms for Natural Numbers - An Introduction to Proofs 5 0.1. Sets and Logic 5 Exercises 7 0.2. Peano Axioms 8 Exercises 15 0.3. Relations 15 Exercises 19 Chapter 1. Basic Arithmetic 21 1.1. The set Z 21 Exercises 23 1.2. The ring Z 23 Exercises 27 1.3. Factoring integers, part I 27 Exercises 30 1.4. Factoring integers, part II 30 Exercises 32 1.5. Modular Arithmetic 32 Exercises 34 1.6. The fields Q and R 34 Exercises 39 1.7. The field C 39 Exercises 40 3 CHAPTER 0 Peano Axioms for Natural Numbers - An Introduction to Proofs We begin this course with the construction of the natural number system. The Peano Axioms form the heart of much of mathematics and lay the foundation for algebra and analysis. Every attempt will be made to stay away from unnecessary abstraction. This should be a guiding principle when working out the exercises as well! 0.1. Sets and Logic This section should serve as a very short introduction to 20th century mathe- matics. That is, it is an introduction to proofs involving sets. Set theory is in fact a subject within itself that was initiated and studied by many in the early 20th century. This was in response to several paradoxes that had come up. For us, a set is a collection of things, called elements. For example, Atoms could be the set of all atoms in the universe while Adams could be the set of all people named Adam in your family. -
Free Ideal Rings and Localization in General Rings
This page intentionally left blank Free Ideal Rings and Localization in General Rings Proving that a polynomial ring in one variable over a field is a principal ideal domain can be done by means of the Euclidean algorithm, but this does not extend to more variables. However, if the variables are not allowed to commute, giving a free associative algebra, then there is a generalization, the weak algorithm, which can be used to prove that all one-sided ideals are free. This book presents the theory of free ideal rings (firs) in detail. Particular emphasis is placed on rings with a weak algorithm, exemplified by free associative algebras. There is also a full account of localization, which is treated for general rings, but the features arising in firs are given special attention. Each section has a number of exercises, including some open problems, and each chapter ends in a historical note. paul cohn is Emeritus Professor of Mathematics at the University of London and Honorary Research Fellow at University College London. NEW MATHEMATICAL MONOGRAPHS Editorial Board B´ela Bollob´as William Fulton Frances Kirwan Peter Sarnak Barry Simon For information about Cambridge University Press mathematics publications visit http://publishing.cambridge.org/stm/mathematics Already published in New Mathematical Monographs: Representation Theory of Finite Reductive Groups Marc Cabanes, Michel Enguehard Harmonic Measure John B. Garnett, Donald E. Marshall Heights in Diophantine Geometry Enrico Bombieri, Walter Gubler Free Ideal Rings and Localization in General Rings P. M. COHN Department of Mathematics University College London Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge ,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521853378 © Cambridge University Press 2006 This publication is in copyright. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title One-sided prime ideals in noncommutative algebra Permalink https://escholarship.org/uc/item/7ts6k0b7 Author Reyes, Manuel Lionel Publication Date 2010 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Tsit Yuen Lam, Chair Professor George Bergman Professor Koushik Sen Spring 2010 One-sided prime ideals in noncommutative algebra Copyright 2010 by Manuel Lionel Reyes 1 Abstract One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes Doctor of Philosophy in Mathematics University of California, Berkeley Professor Tsit Yuen Lam, Chair The goal of this dissertation is to provide noncommutative generalizations of the following theorems from commutative algebra: (Cohen's Theorem) every ideal of a commutative ring R is finitely generated if and only if every prime ideal of R is finitely generated, and (Kaplan- sky's Theorems) every ideal of R is principal if and only if every prime ideal of R is principal, if and only if R is noetherian and every maximal ideal of R is principal. We approach this problem by introducing certain families of right ideals in noncommutative rings, called right Oka families, generalizing previous work on commutative rings by T. Y. Lam and the author. As in the commutative case, we prove that the right Oka families in a ring R correspond bi- jectively to the classes of cyclic right R-modules that are closed under extensions. -
TWO RESULTS on PROJECTIVE MODULES We Wish to Present Two Results. the First Is from the Paper Pavel Prıhoda, Projective Modules
TWO RESULTS ON PROJECTIVE MODULES ANDREW HUBERY We wish to present two results. The first is from the paper Pavel Pˇr´ıhoda, Projective modules are determined by their radical factors, J. Pure Applied Algebra 210 (2007) 827{835. Theorem A. Let Λ be a ring, with Jacobson radical J, and let P and Q be two projective (right) Λ-modules. Then we can lift any isomorphism f¯: P=P J −!∼ Q=QJ to an isomorphism f : P −!∼ Q. In particular, projective modules are determined by their tops. Special cases have been known for a long time. For example, we say that Λ is right perfect if every right Λ-module admits a projective cover. Then P P=P J is a projective cover, so the result follows (Bass 1960). In general, though, P ! P=P J will not be a projective cover. Beck (1972) showed that if P=P J is a free Λ=J-module, then P is free. Note that if J = 0 then the theorem tells us nothing! The second result is from the appendix to the paper Michael Butler and Alastair King, Minimal resolutions of algebras, J. Algebra 212 (1999) 323{362. Theorem B. Let Λ = TA(M) be an hereditary tensor algebra, where A is semisim- ple and M is an A-bimodule. Then every projective module is isomorphic to an induced module, so one of the form X ⊗A Λ; equivalently, every projective module is gradable. This is trivial if there are no oriented cycles; that is, if the graded radical Λ+ is nilpotent, so Λ is semiprimary and J(Λ) = Λ+. -
Introduction to Groups, Rings and Fields
Introduction to Groups, Rings and Fields HT and TT 2011 H. A. Priestley 0. Familiar algebraic systems: review and a look ahead. GRF is an ALGEBRA course, and specifically a course about algebraic structures. This introduc- tory section revisits ideas met in the early part of Analysis I and in Linear Algebra I, to set the scene and provide motivation. 0.1 Familiar number systems Consider the traditional number systems N = 0, 1, 2,... the natural numbers { } Z = m n m, n N the integers { − | ∈ } Q = m/n m, n Z, n = 0 the rational numbers { | ∈ } R the real numbers C the complex numbers for which we have N Z Q R C. ⊂ ⊂ ⊂ ⊂ These come equipped with the familiar arithmetic operations of sum and product. The real numbers: Analysis I built on the real numbers. Right at the start of that course you were given a set of assumptions about R, falling under three headings: (1) Algebraic properties (laws of arithmetic), (2) order properties, (3) Completeness Axiom; summarised as saying the real numbers form a complete ordered field. (1) The algebraic properties of R You were told in Analysis I: Addition: for each pair of real numbers a and b there exists a unique real number a + b such that + is a commutative and associative operation; • there exists in R a zero, 0, for addition: a +0=0+ a = a for all a R; • ∈ for each a R there exists an additive inverse a R such that a+( a)=( a)+a = 0. • ∈ − ∈ − − Multiplication: for each pair of real numbers a and b there exists a unique real number a b such that · is a commutative and associative operation; • · there exists in R an identity, 1, for multiplication: a 1 = 1 a = a for all a R; • · · ∈ for each a R with a = 0 there exists an additive inverse a−1 R such that a a−1 = • a−1 a = 1.∈ ∈ · · Addition and multiplication together: forall a,b,c R, we have the distributive law a (b + c)= a b + a c. -
Homomorphisms of C*-Algebras and Their K-Theory 3
HOMOMORPHISMS OF C*-ALGEBRAS AND THEIR K-THEORY PARASTOO NADERI AND JAMAL ROOIN Abstract. Let A and B be C*-algebras and ϕ: A → B be a ∗-homomorphism. We discuss the properties of the kernel and (co-)image of the induced map K0(ϕ): K0(A) → K0(B) on the level of K-theory. In particular, we are inter- ested in the case that the co-image is torsion free, and show that it holds when A and B are commutative and unital, B has real rank zero, and ϕ is unital and injective. We also show that A is embeddable in B if K0(ϕ) is injective and A has stable rank one and real rank zero. 1. Introduction and Main Results K-theory is a powerful tool in operator algebras and their aplications. It has been used in the classification of certain C*-algebras, starting from AF algebras [6] and now extending to larger classes [13, 16]. Although, it has been used to study homomorphisms between C*-algebras [5], it seems that the main focus has been on the use of K-theory to understand the structure of the objects of the category of C*-algebras rather than the morphisms. Consider a ∗-homomorphism ϕ: A → B between C*-algebras A and B. A natural question is that can we determine properties of ϕ (such as injectivity, surjectivity, unitality, and so on) by looking at its lift K0(ϕ): K0(A) → K0(B), or vice-versa? (One may also consider K1(ϕ) and Cu(ϕ).) In this paper we investigate such a relation between properties of a morphism in the level of C*-algebras and those of its lift to the level of K-theory. -
Stably Free Modules Over Innite Group Algebras
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UCL Discovery Department of Mathematics University College London Stably free modules over infinite group algebras Pouya Kamali A thesis submitted for the degree of Doctor of Philosophy Supervisor: Professor F.E.A. Johnson June 2010 1 I, Pouya Kamali, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. ..................................................... 2 Abstract We study finitely generated stably-free modules over infinite integral group algebras by using the language of cyclic algebras and relating it to well-known results in K-theory. For G a free or free abelian group and Q8n, the quaternionic group of order 8n, we show that there exist infinitely many isomorphically distinct stably-free modules of rank 1 over the integral group algebra of the group Γ = Q8n × G whenever n admits an odd divisor. This result implies that the stable class of the augmentation ideal Ω1(Z) displays infinite splitting at minimal level whenever G is the free abelian group on at least 2 generators. This is of relevance to low dimensional topol- ogy, in particular when computing homotopy modules of a cell complex with fundamental group Γ. 3 Acknowledgements First and foremost, I would like to thank my supervisor, Professor F.E.A. Johnson, who accepted me as his student and introduced me to the fascinat- ing subject of algebraic K-theory. My mathematical personality has greatly been shaped by his generous attitude towards research, but also by the man- ner in which he communicated it. -
M373K, Homework #8
M373K, Homework #8 April 3, 2013 From the book, please work out: x3.2 #1 (a + b)(c + d) = ac + ad + bc + bd. The point to this problem is that multiplication is non-commutative. So bc does not necessarily equal cb, for example. x3.2 #6 Let D be an integral domain with finite characteristic m. Recall that this means ma = 0 for every a 2 D (and m is a positive integer). Wlog we can assume that m is the smallest positive integer for which this holds. Suppose j; k are positive integers and jk = m. Then for any a; b 2 D,(ja)(kb) = jkab = 0 (by the previous problem). However, D is an integral domain, so it has no zero divisors. So either ja = 0 or kb = 0. In particular, by setting a = b we see that either ja = 0 for every a 2 R or ka = 0 for every a 2 R. So if m is not prime then we can choose j; k so that both j and k are smaller than m. But this contradicts our choice of m. So m must be prime. x3.2 #10 Let D be an integral domain and a; b; c 2 D with ab = ac and a 6= 0. Then ab − ac = 0 so a(b − c) = 0. Because a 6= 0 and D has no zero divisors, this implies b − c = 0 which implies b = c. On the other hand, suppose that D is a commutative ring which satisfies the cancellation property: for every a; b; c 2 D with a 6= 0 and ab = ac, we must have b = c. -
Notes on Ring Theory
Notes on Ring Theory S. F. Ellermeyer Department of Mathematics Kennesaw State University January 15, 2016 Abstract These notes contain an outline of essential definitions and theorems from Ring Theory but only contain a minimal number of examples. Many examples are presented in class and thus it is important to come to class. Also of utmost importance is that the student put alot of effort into doing the assigned homework exercises.The material in these notes is outlined in the order of Chapters 17—33 of Charles Pinter’s “A Book of Abstract Algebra”, which is the textbook we are using in the course. 1Rings A ring is a non—empty set, , along with two operations called addition (usually denoted by the symbol +) and multiplication (usually denoted by the symbol or by no symbol) such that · 1. is an abelian group under addition. (The additive identity element of is usually denoted by 0.) 2. The multiplication operation is associative. 3. Multiplication is distributive over addition, meaning that for all and in we have ( + )= + · · · 1 and ( + ) = + . · · · Letuslookatafewexamplesofrings. Example 1 The trivial ring is the one—element set = 0 with addition operation defined by 0+0 = 0 and multiplication operation de{fi}ned by 0 0=0. · Example 2 The set of integers, , with the standard operations of addition and multiplication is a ring. The set of real numbers, , with the standard operations of addition and multiplication is a ring. The set of rational num- bers, , with the standard operations of addition and multiplication is a ring. The set of complex numbers, , with the standard operations of addition and multiplication is a ring. -
The Varieties of One-Sided Loops of Bol-Moufang Type 1. Introduction
The Varieties of One-Sided Loops of Bol-Moufang Type Rachel R. Aldrich and Sarah J. Drummond Abstract. We show that there are 20 different varieties of one-sided loops of Bol-Moufang type. We also show all inclusions between these varieties and give all necessary counterexamples. This work extends the research done by Phillips and Vojt˘echovsk´ywhich found the relationships among 26 different varieties of quasigroups and 14 different varieties of loops of Bol-Moufang type. We then show which varieties of one-sided loops of Bol-Moufang type have left and right inverse properties. The proofs and counterexamples are aided by the automated theorem prover, Prover9, and the finite model builder, Mace4. 1. Introduction Bol-Moufang Identities Let ∗ be a binary operation on a set G. Let La : G ! G be defined by La(x) = a ∗ x and let Ra : G ! G be defined by Ra(x) = x ∗ a. Definition 1. (G; ·) is called a quasigroup if La and Ra are bijective. Quasigroups have the cancellation property. Let (G; ·) be a quasigroup with x; y; z 2 G. Then if x ∗ z = y ∗ z, we have x = y, and if z ∗ x = z ∗ y, we have x = y. Definition 2. A loop is a quasigroup (G; ·) for which ∗ has a two-sided identity element e, such that e ∗ x = x ∗ e = x. A left loop is a quasigroup (G; ·) for which ∗ has a left identity element, and a right loop is a quasigroup (G; ·) for which ∗ has a right identity element. An identity is said to be of Bol-Moufang type if there are three variables on each side, with one of the variables occurring twice, and the other two occurring once.