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Exercises and Solutions in Groups Rings and Fields
EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS Mahmut Kuzucuo˘glu Middle East Technical University [email protected] Ankara, TURKEY April 18, 2012 ii iii TABLE OF CONTENTS CHAPTERS 0. PREFACE . v 1. SETS, INTEGERS, FUNCTIONS . 1 2. GROUPS . 4 3. RINGS . .55 4. FIELDS . 77 5. INDEX . 100 iv v Preface These notes are prepared in 1991 when we gave the abstract al- gebra course. Our intention was to help the students by giving them some exercises and get them familiar with some solutions. Some of the solutions here are very short and in the form of a hint. I would like to thank B¨ulent B¨uy¨ukbozkırlı for his help during the preparation of these notes. I would like to thank also Prof. Ismail_ S¸. G¨ulo˘glufor checking some of the solutions. Of course the remaining errors belongs to me. If you find any errors, I should be grateful to hear from you. Finally I would like to thank Aynur Bora and G¨uldaneG¨um¨u¸sfor their typing the manuscript in LATEX. Mahmut Kuzucuo˘glu I would like to thank our graduate students Tu˘gbaAslan, B¨u¸sra C¸ınar, Fuat Erdem and Irfan_ Kadık¨oyl¨ufor reading the old version and pointing out some misprints. With their encouragement I have made the changes in the shape, namely I put the answers right after the questions. 20, December 2011 vi M. Kuzucuo˘glu 1. SETS, INTEGERS, FUNCTIONS 1.1. If A is a finite set having n elements, prove that A has exactly 2n distinct subsets. -
On Field Γ-Semiring and Complemented Γ-Semiring with Identity
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 8(2018), 189-202 DOI: 10.7251/BIMVI1801189RA Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) ON FIELD Γ-SEMIRING AND COMPLEMENTED Γ-SEMIRING WITH IDENTITY Marapureddy Murali Krishna Rao Abstract. In this paper we study the properties of structures of the semi- group (M; +) and the Γ−semigroup M of field Γ−semiring M, totally ordered Γ−semiring M and totally ordered field Γ−semiring M satisfying the identity a + aαb = a for all a; b 2 M; α 2 Γand we also introduce the notion of com- plemented Γ−semiring and totally ordered complemented Γ−semiring. We prove that, if semigroup (M; +) is positively ordered of totally ordered field Γ−semiring satisfying the identity a + aαb = a for all a; b 2 M; α 2 Γ, then Γ-semigroup M is positively ordered and study their properties. 1. Introduction In 1995, Murali Krishna Rao [5, 6, 7] introduced the notion of a Γ-semiring as a generalization of Γ-ring, ring, ternary semiring and semiring. The set of all negative integers Z− is not a semiring with respect to usual addition and multiplication but Z− forms a Γ-semiring where Γ = Z: Historically semirings first appear implicitly in Dedekind and later in Macaulay, Neither and Lorenzen in connection with the study of a ring. However semirings first appear explicitly in Vandiver, also in connection with the axiomatization of Arithmetic of natural numbers. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
1. Rings of Fractions
1. Rings of Fractions 1.0. Rings and algebras (1.0.1) All the rings considered in this work possess a unit element; all the modules over such a ring are assumed unital; homomorphisms of rings are assumed to take unit element to unit element; and unless explicitly mentioned otherwise, all subrings of a ring A are assumed to contain the unit element of A. We generally consider commutative rings, and when we say ring without specifying, we understand this to mean a commutative ring. If A is a not necessarily commutative ring, we consider every A module to be a left A- module, unless we expressly say otherwise. (1.0.2) Let A and B be not necessarily commutative rings, φ : A ! B a homomorphism. Every left (respectively right) B-module M inherits the structure of a left (resp. right) A-module by the formula a:m = φ(a):m (resp m:a = m.φ(a)). When it is necessary to distinguish the A-module structure from the B-module structure on M, we use M[φ] to denote the left (resp. right) A-module so defined. If L is an A module, a homomorphism u : L ! M[φ] is therefore a homomorphism of commutative groups such that u(a:x) = φ(a):u(x) for a 2 A, x 2 L; one also calls this a φ-homomorphism L ! M, and the pair (u,φ) (or, by abuse of language, u) is a di-homomorphism from (A, L) to (B, M). The pair (A,L) made up of a ring A and an A module L therefore form the objects of a category where the morphisms are di-homomorphisms. -
Littlewood–Richardson Coefficients and Birational Combinatorics
Littlewood{Richardson coefficients and birational combinatorics Darij Grinberg 28 August 2020 [corrected version] Algebraic and Combinatorial Perspectives in the Mathematical Sciences slides: http: //www.cip.ifi.lmu.de/~grinberg/algebra/acpms2020.pdf paper: arXiv:2008.06128 aka http: //www.cip.ifi.lmu.de/~grinberg/algebra/lrhspr.pdf 1 / 43 The proof is a nice example of birational combinatorics: the use of birational transformations in elementary combinatorics (specifically, here, in finding and proving a bijection). Manifest I shall review the Littlewood{Richardson coefficients and some of their classical properties. I will then state a \hidden symmetry" conjectured by Pelletier and Ressayre (arXiv:2005.09877) and outline how I proved it. 2 / 43 Manifest I shall review the Littlewood{Richardson coefficients and some of their classical properties. I will then state a \hidden symmetry" conjectured by Pelletier and Ressayre (arXiv:2005.09877) and outline how I proved it. The proof is a nice example of birational combinatorics: the use of birational transformations in elementary combinatorics (specifically, here, in finding and proving a bijection). 2 / 43 Manifest I shall review the Littlewood{Richardson coefficients and some of their classical properties. I will then state a \hidden symmetry" conjectured by Pelletier and Ressayre (arXiv:2005.09877) and outline how I proved it. The proof is a nice example of birational combinatorics: the use of birational transformations in elementary combinatorics (specifically, here, in finding and proving a bijection). 2 / 43 Chapter 1 Chapter 1 Littlewood{Richardson coefficients References (among many): Richard Stanley, Enumerative Combinatorics, vol. 2, Chapter 7. Darij Grinberg, Victor Reiner, Hopf Algebras in Combinatorics, arXiv:1409.8356. -
SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′. -
Algebraic Number Theory
Algebraic Number Theory William B. Hart Warwick Mathematics Institute Abstract. We give a short introduction to algebraic number theory. Algebraic number theory is the study of extension fields Q(α1; α2; : : : ; αn) of the rational numbers, known as algebraic number fields (sometimes number fields for short), in which each of the adjoined complex numbers αi is algebraic, i.e. the root of a polynomial with rational coefficients. Throughout this set of notes we use the notation Z[α1; α2; : : : ; αn] to denote the ring generated by the values αi. It is the smallest ring containing the integers Z and each of the αi. It can be described as the ring of all polynomial expressions in the αi with integer coefficients, i.e. the ring of all expressions built up from elements of Z and the complex numbers αi by finitely many applications of the arithmetic operations of addition and multiplication. The notation Q(α1; α2; : : : ; αn) denotes the field of all quotients of elements of Z[α1; α2; : : : ; αn] with nonzero denominator, i.e. the field of rational functions in the αi, with rational coefficients. It is the smallest field containing the rational numbers Q and all of the αi. It can be thought of as the field of all expressions built up from elements of Z and the numbers αi by finitely many applications of the arithmetic operations of addition, multiplication and division (excepting of course, divide by zero). 1 Algebraic numbers and integers A number α 2 C is called algebraic if it is the root of a monic polynomial n n−1 n−2 f(x) = x + an−1x + an−2x + ::: + a1x + a0 = 0 with rational coefficients ai. -
Kernel Methods for Knowledge Structures
Kernel Methods for Knowledge Structures Zur Erlangung des akademischen Grades eines Doktors der Wirtschaftswissenschaften (Dr. rer. pol.) von der Fakultät für Wirtschaftswissenschaften der Universität Karlsruhe (TH) genehmigte DISSERTATION von Dipl.-Inform.-Wirt. Stephan Bloehdorn Tag der mündlichen Prüfung: 10. Juli 2008 Referent: Prof. Dr. Rudi Studer Koreferent: Prof. Dr. Dr. Lars Schmidt-Thieme 2008 Karlsruhe Abstract Kernel methods constitute a new and popular field of research in the area of machine learning. Kernel-based machine learning algorithms abandon the explicit represen- tation of data items in the vector space in which the sought-after patterns are to be detected. Instead, they implicitly mimic the geometry of the feature space by means of the kernel function, a similarity function which maintains a geometric interpreta- tion as the inner product of two vectors. Knowledge structures and ontologies allow to formally model domain knowledge which can constitute valuable complementary information for pattern discovery. For kernel-based machine learning algorithms, a good way to make such prior knowledge about the problem domain available to a machine learning technique is to incorporate it into the kernel function. This thesis studies the design of such kernel functions. First, this thesis provides a theoretical analysis of popular similarity functions for entities in taxonomic knowledge structures in terms of their suitability as kernel func- tions. It shows that, in a general setting, many taxonomic similarity functions can not be guaranteed to yield valid kernel functions and discusses the alternatives. Secondly, the thesis addresses the design of expressive kernel functions for text mining applications. A first group of kernel functions, Semantic Smoothing Kernels (SSKs) retain the Vector Space Models (VSMs) representation of textual data as vec- tors of term weights but employ linguistic background knowledge resources to bias the original inner product in such a way that cross-term similarities adequately con- tribute to the kernel result. -
Abstract Algebra: Monoids, Groups, Rings
Notes on Abstract Algebra John Perry University of Southern Mississippi [email protected] http://www.math.usm.edu/perry/ Copyright 2009 John Perry www.math.usm.edu/perry/ Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States You are free: to Share—to copy, distribute and transmit the work • to Remix—to adapt the work Under• the following conditions: Attribution—You must attribute the work in the manner specified by the author or licen- • sor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial—You may not use this work for commercial purposes. • Share Alike—If you alter, transform, or build upon this work, you may distribute the • resulting work only under the same or similar license to this one. With the understanding that: Waiver—Any of the above conditions can be waived if you get permission from the copy- • right holder. Other Rights—In no way are any of the following rights affected by the license: • Your fair dealing or fair use rights; ◦ Apart from the remix rights granted under this license, the author’s moral rights; ◦ Rights other persons may have either in the work itself or in how the work is used, ◦ such as publicity or privacy rights. Notice—For any reuse or distribution, you must make clear to others the license terms of • this work. The best way to do this is with a link to this web page: http://creativecommons.org/licenses/by-nc-sa/3.0/us/legalcode Table of Contents Reference sheet for notation...........................................................iv A few acknowledgements..............................................................vi Preface ...............................................................................vii Overview ...........................................................................vii Three interesting problems ............................................................1 Part . -
Fields Besides the Real Numbers Math 130 Linear Algebra
manner, which are both commutative and asso- ciative, both have identity elements (the additive identity denoted 0 and the multiplicative identity denoted 1), addition has inverse elements (the ad- ditive inverse of x denoted −x as usual), multipli- cation has inverses of nonzero elements (the multi- Fields besides the Real Numbers 1 −1 plicative inverse of x denoted x or x ), multipli- Math 130 Linear Algebra cation distributes over addition, and 0 6= 1. D Joyce, Fall 2015 Of course, one example of a field is the field of Most of the time in linear algebra, our vectors real numbers R. What are some others? will have coordinates that are real numbers, that is to say, our scalar field is R, the real numbers. Example 2 (The field of rational numbers, Q). Another example is the field of rational numbers. But linear algebra works over other fields, too, A rational number is the quotient of two integers like C, the complex numbers. In fact, when we a=b where the denominator is not 0. The set of discuss eigenvalues and eigenvectors, we'll need to all rational numbers is denoted Q. We're familiar do linear algebra over C. Some of the applications with the fact that the sum, difference, product, and of linear algebra such as solving linear differential quotient (when the denominator is not zero) of ra- equations require C as as well. tional numbers is another rational number, so Q Some applications in computer science use linear has all the operations it needs to be a field, and algebra over a two-element field Z (described be- 2 since it's part of the field of the real numbers R, its low). -
Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference
Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference Sunday 30 May to Friday 4 June 2010 Index of Talks and Abstracts Main talks 1. Simeon Ball, On subsets of a finite vector space in which every subset of basis size is a basis 2. Simon Blackburn, Honeycomb arrays 3. G`abor Korchm`aros, Curves over finite fields, an approach from finite geometry 4. Cheryl Praeger, Basic pregeometries 5. Bernhard Schmidt, Finiteness of circulant weighing matrices of fixed weight 6. Douglas Stinson, Multicollision attacks on iterated hash functions Short talks 1. Mari´en Abreu, Adjacency matrices of polarity graphs and of other C4–free graphs of large size 2. Marco Buratti, Combinatorial designs via factorization of a group into subsets 3. Mike Burmester, Lightweight cryptographic mechanisms based on pseudorandom number generators 4. Philippe Cara, Loops, neardomains, nearfields and sets of permutations 5. Ilaria Cardinali, On the structure of Weyl modules for the symplectic group 6. Bill Cherowitzo, Parallelisms of quadrics 7. Jan De Beule, Large maximal partial ovoids of Q−(5, q) 8. Bart De Bruyn, On extensions of hyperplanes of dual polar spaces 1 9. Frank De Clerck, Intriguing sets of partial quadrangles 10. Alice Devillers, Symmetry properties of subdivision graphs 11. Dalibor Froncek, Decompositions of complete bipartite graphs into generalized prisms 12. Stelios Georgiou, Self-dual codes from circulant matrices 13. Robert Gilman, Cryptology of infinite groups 14. Otokar Groˇsek, The number of associative triples in a quasigroup 15. Christoph Hering, Latin squares, homologies and Euler’s conjecture 16. Leanne Holder, Bilinear star flocks of arbitrary cones 17. Robert Jajcay, On the geometry of cages 18. -
36 Rings of Fractions
36 Rings of fractions Recall. If R is a PID then R is a UFD. In particular • Z is a UFD • if F is a field then F[x] is a UFD. Goal. If R is a UFD then so is R[x]. Idea of proof. 1) Find an embedding R,! F where F is a field. 2) If p(x) 2 R[x] then p(x) 2 F[x] and since F[x] is a UFD thus p(x) has a unique factorization into irreducibles in F[x]. 3) Use the factorization in F[x] and the fact that R is a UFD to obtain a factorization of p(x) in R[x]. 36.1 Definition. If R is a ring then a subset S ⊆ R is a multiplicative subset if 1) 1 2 S 2) if a; b 2 S then ab 2 S 36.2 Example. Multiplicative subsets of Z: 1) S = Z 137 2) S0 = Z − f0g 3) Sp = fn 2 Z j p - ng where p is a prime number. Note: Sp = Z − hpi. 36.3 Proposition. If R is a ring, and I is a prime ideal of R then S = R − I is a multiplicative subset of R. Proof. Exercise. 36.4. Construction of a ring of fractions. Goal. For a ring R and a multiplicative subset S ⊆ R construct a ring S−1R such that every element of S becomes a unit in S−1R. Consider a relation on the set R × S: 0 0 0 0 (a; s) ∼ (a ; s ) if s0(as − a s) = 0 for some s0 2 S Check: ∼ is an equivalence relation.