A Flavour of Noncommutative Algebra (Part 1)
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Math 601: Algebra Problem Set 2 Due: September 20, 2017 Emily Riehl Exercise 1. the Group Z Is the Free Group on One Generator
Math 601: Algebra Problem Set 2 due: September 20, 2017 Emily Riehl Exercise 1. The group Z is the free group on one generator. What is its universal property?1 Exercise 2. Prove that there are no non-zero group homomorphisms between the Klein four group and Z=7. Exercise 3. • Define an element of order d in Sn for any d < n. • For which n is Sn abelian? Give a proof or supply a counterexample for each n ≥ 1. Exercise 4. Is the product of cyclic groups cyclic? If so, give a proof. If not, find a counterexample. Exercise 5. Prove that if A and B are abelian groups, then A × B satisfies the universal property of the coproduct in the category Ab of abelian groups. Explain why the commutativity hypothesis is necessary. Exercise 6. Prove that the function g 7! g−1 defines a homomorphism G ! G if and only if G is abelian.2 Exercise 7. (i) Fix an element g in a group G. Prove that the conjugation function x 7! −1 gxg defines a homomorphism γg : G ! G. (ii) Prove that the function g 7! γg defines a homomorphism γ : G ! Aut(G). The image of this function is the subgroup of inner automorphisms of G. (iii) Prove that γ is the zero homomorphism if and only if G is abelian. Exercise 8. We have seen that the set of homomorphisms hom(B; A) between two abelian groups is an abelian group with addition defined pointwise in A. In particular End(A) := hom(A; A) is an abelian group under pointwise addition. -
Artinian Rings Having a Nilpotent Group of Units
JOURNAL OF ALGEBRA 121, 253-262 (1989) Artinian Rings Having a Nilpotent Group of Units GHIOCEL GROZA Department of Mathematics, Civil Engineering Institute, Lacul Tei 124, Sec. 2, R-72307, Bucharest, Romania Communicated by Walter Feit Received July 1, 1983 Throughout this paper the symbol R will be used to denote an associative ring with a multiplicative identity. R,[R”] denotes the subring of R which is generated over R, by R”, where R, is the prime subring of R and R” is the group of units of R. We use J(R) to denote the Jacobson radical of R. In Section 1 we prove that if R is a finite ring and at most one simple component of the semi-simple ring R/J(R) is a field of order 2, then R” is a nilpotent group iff R is a direct sum of two-sided ideals that are homomorphic images of group algebras of type SP, where S is a particular commutative finite ring, P is a finite p-group, and p is a prime number. In Section 2 we study the artinian rings R (i.e., rings with minimum con- dition) with RO a finite ring, R finitely generated over its center and so that every block of R is as in Lemma 1.1. We note that every finite-dimensional algebra over a field F of characteristic p > 0, so that F is not the finite field of order 2 is of this type. We prove that R is of this type and R” is a nilpotent group iff R is a direct sum of two-sided ideals that are particular homomorphic images of crossed products. -
Representations of Finite Rings
Pacific Journal of Mathematics REPRESENTATIONS OF FINITE RINGS ROBERT S. WILSON Vol. 53, No. 2 April 1974 PACIFIC JOURNAL OF MATHEMATICS Vol. 53, No. 2, 1974 REPRESENTATIONS OF FINITE RINGS ROBERT S. WILSON In this paper we extend the concept of the Szele repre- sentation of finite rings from the case where the coefficient ring is a cyclic ring* to the case where it is a Galois ring. We then characterize completely primary and nilpotent finite rings as those rings whose Szele representations satisfy certain conditions. 1* Preliminaries* We first note that any finite ring is a direct sum of rings of prime power order. This follows from noticing that when one decomposes the additive group of a finite ring into its pime power components, the component subgroups are, in fact, ideals. So without loss of generality, up to direct sum formation, one needs only to consider rings of prime power order. For the remainder of this paper p will denote an arbitrary, fixed prime and all rings will be of order pn for some positive integer n. Of the two classes of rings that will be studied in this paper, completely primary finite rings are always of prime power order, so for the completely primary case, there is no loss of generality at all. However, nilpotent finite rings do not need to have prime power order, but we need only classify finite nilpotent rings of prime power order, the general case following from direct sum formation. If B is finite ring (of order pn) then the characteristic of B will be pk for some positive integer k. -
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. -
AN INTRODUCTION to CATEGORY THEORY and the YONEDA LEMMA Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transfo
AN INTRODUCTION TO CATEGORY THEORY AND THE YONEDA LEMMA SHU-NAN JUSTIN CHANG Abstract. We begin this introduction to category theory with definitions of categories, functors, and natural transformations. We provide many examples of each construct and discuss interesting relations between them. We proceed to prove the Yoneda Lemma, a central concept in category theory, and motivate its significance. We conclude with some results and applications of the Yoneda Lemma. Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transformations 6 4. The Yoneda Lemma 9 5. Corollaries and Applications 10 Acknowledgments 12 References 13 Introduction Category theory is an interdisciplinary field of mathematics which takes on a new perspective to understanding mathematical phenomena. Unlike most other branches of mathematics, category theory is rather uninterested in the objects be- ing considered themselves. Instead, it focuses on the relations between objects of the same type and objects of different types. Its abstract and broad nature allows it to reach into and connect several different branches of mathematics: algebra, geometry, topology, analysis, etc. A central theme of category theory is abstraction, understanding objects by gen- eralizing rather than focusing on them individually. Similar to taxonomy, category theory offers a way for mathematical concepts to be abstracted and unified. What makes category theory more than just an organizational system, however, is its abil- ity to generate information about these abstract objects by studying their relations to each other. This ability comes from what Emily Riehl calls \arguably the most important result in category theory"[4], the Yoneda Lemma. The Yoneda Lemma allows us to formally define an object by its relations to other objects, which is central to the relation-oriented perspective taken by category theory. -
Categories, Functors, and Natural Transformations I∗
Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0. -
The Projective Line Over the Finite Quotient Ring GF(2)[X]/⟨X3 − X⟩ and Quantum Entanglement I
The Projective Line Over the Finite Quotient Ring GF(2)[x]/hx3 − xi and Quantum Entanglement I. Theoretical Background Metod Saniga, Michel Planat To cite this version: Metod Saniga, Michel Planat. The Projective Line Over the Finite Quotient Ring GF(2)[x]/hx3 − xi and Quantum Entanglement I. Theoretical Background. 2006. hal-00020182v1 HAL Id: hal-00020182 https://hal.archives-ouvertes.fr/hal-00020182v1 Preprint submitted on 7 Mar 2006 (v1), last revised 6 Jun 2006 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Projective Line Over the Finite Quotient Ring GF(2)[x]/hx3 − xi and Quantum Entanglement I. Theoretical Background Metod Saniga† and Michel Planat‡ †Astronomical Institute, Slovak Academy of Sciences SK-05960 Tatransk´aLomnica, Slovak Republic ([email protected]) and ‡Institut FEMTO-ST, CNRS, D´epartement LPMO, 32 Avenue de l’Observatoire F-25044 Besan¸con, France ([email protected]) Abstract 3 The paper deals with the projective line over the finite factor ring R♣ ≡ GF(2)[x]/hx −xi. The line is endowed with 18 points, spanning the neighbourhoods of three pairwise distant points. As R♣ is not a local ring, the neighbour (or parallel) relation is not an equivalence relation so that the sets of neighbour points to two distant points overlap. -
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. -
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. -
Math 250A: Groups, Rings, and Fields. H. W. Lenstra Jr. 1. Prerequisites
Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites This section consists of an enumeration of terms from elementary set theory and algebra. You are supposed to be familiar with their definitions and basic properties. Set theory. Sets, subsets, the empty set , operations on sets (union, intersection, ; product), maps, composition of maps, injective maps, surjective maps, bijective maps, the identity map 1X of a set X, inverses of maps. Relations, equivalence relations, equivalence classes, partial and total orderings, the cardinality #X of a set X. The principle of math- ematical induction. Zorn's lemma will be assumed in a number of exercises. Later in the course the terminology and a few basic results from point set topology may come in useful. Group theory. Groups, multiplicative and additive notation, the unit element 1 (or the zero element 0), abelian groups, cyclic groups, the order of a group or of an element, Fermat's little theorem, products of groups, subgroups, generators for subgroups, left cosets aH, right cosets, the coset spaces G=H and H G, the index (G : H), the theorem of n Lagrange, group homomorphisms, isomorphisms, automorphisms, normal subgroups, the factor group G=N and the canonical map G G=N, homomorphism theorems, the Jordan- ! H¨older theorem (see Exercise 1.4), the commutator subgroup [G; G], the center Z(G) (see Exercise 1.12), the group Aut G of automorphisms of G, inner automorphisms. Examples of groups: the group Sym X of permutations of a set X, the symmetric group S = Sym 1; 2; : : : ; n , cycles of permutations, even and odd permutations, the alternating n f g group A , the dihedral group D = (1 2 : : : n); (1 n 1)(2 n 2) : : : , the Klein four group n n h − − i V , the quaternion group Q = 1; i; j; ij (with ii = jj = 1, ji = ij) of order 4 8 { g − − 8, additive groups of rings, the group Gl(n; R) of invertible n n-matrices over a ring 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.