Modules I: Basic Definitions and Constructions
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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. -
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. -
Arxiv:Math/0310146V1 [Math.AT] 10 Oct 2003 Usinis: Question Fr Most Aut the the of Algebra”
MORITA THEORY IN ABELIAN, DERIVED AND STABLE MODEL CATEGORIES STEFAN SCHWEDE These notes are based on lectures given at the Workshop on Structured ring spectra and their applications. This workshop took place January 21-25, 2002, at the University of Glasgow and was organized by Andy Baker and Birgit Richter. Contents 1. Introduction 1 2. Morita theory in abelian categories 2 3. Morita theory in derived categories 6 3.1. The derived category 6 3.2. Derived equivalences after Rickard and Keller 14 3.3. Examples 19 4. Morita theory in stable model categories 21 4.1. Stable model categories 22 4.2. Symmetric ring and module spectra 25 4.3. Characterizing module categories over ring spectra 32 4.4. Morita context for ring spectra 35 4.5. Examples 38 References 42 1. Introduction The paper [Mo58] by Kiiti Morita seems to be the first systematic study of equivalences between module categories. Morita treats both contravariant equivalences (which he calls arXiv:math/0310146v1 [math.AT] 10 Oct 2003 dualities of module categories) and covariant equivalences (which he calls isomorphisms of module categories) and shows that they always arise from suitable bimodules, either via contravariant hom functors (for ‘dualities’) or via covariant hom functors and tensor products (for ‘isomorphisms’). The term ‘Morita theory’ is now used for results concerning equivalences of various kinds of module categories. The authors of the obituary article [AGH] consider Morita’s theorem “probably one of the most frequently used single results in modern algebra”. In this survey article, we focus on the covariant form of Morita theory, so our basic question is: When do two ‘rings’ have ‘equivalent’ module categories ? We discuss this question in different contexts: • (Classical) When are the module categories of two rings equivalent as categories ? Date: February 1, 2008. -
On Quotient Modules – the Case of Arbitrary Multiplicity
On quotient modules – the case of arbitrary multiplicity Ronald G. Douglas∗ Texas A & M University College Station Texas 77843 Gadadhar Misra†and Cherian Varughese Indian Statistical Institute R. V. College Post Bangalore 560 059 Abstract: Let M be a Hilbert module of holomorphic functions defined m on a bounded domain Ω ⊆ C . Let M0 be the submodule of functions vanishing to order k on a hypersurface Z in Ω. In this paper, we describe the quotient module Mq. Introduction If M is a Hilbert module over a function algebra A and M0 ⊆ M is a submodule, then determining the quotient module Mq is an interesting problem, particularly if the function algebra consists of holomorphic functions on a domain Ω in Cm and M is a functional Hilbert space. It would be very desirable to describe the quotient module Mq in terms of the last two terms in the short exact sequence X 0 ←− Mq ←− M ←− M0 ←− 0 where X is the inclusion map. For certain modules over the disc algebra, this is related to the model theory of Sz.-Nagy and Foias. In a previous paper [6], the quotient module was described assuming that the sub- module M0 is the maximal set of functions in M vanishing on Z, where Z is an analytic ∗The research for this paper was partly carried out during a visit to ISI - Bangalore in January 1996 with financial support from the Indian Statistical Institute and the International Mathematical Union. †The research for this paper was partly carried out during a visit to Texas A&M University in August, 1997 with financial support from Texas A&M University. -
On the Derived Quotient Moduleo
transactions of the american mathematical society Volume 154, February 1971 ON THE DERIVED QUOTIENT MODULEO BY C. N. WINTON Abstract. With every Ä-module M associate the direct limit of Horn« (D, M) over the dense right ideals of R, the derived quotient module SHiM) of M. 3>(M) is a module over the complete ring of right quotients of R. Relationships between Si(M) and the torsion theory of Gentile-Jans are explored and functorial properties of S> are discussed. When M is torsion free, results are given concerning rational closure, rational completion, and injectivity. 1. Introduction. Let R be a ring with unity and let Q be its complete ring of right quotients. Following Lambek [8], a right ideal D of R is called dense provided that if an R-map f: E(RB) ->■E(RR), where E(RR) denotes the injective hull of RR, is such thatfi(D) = 0, then/=0. It is easy to see this is equivalent to xZ)^(O) for every xe E(RB). The collection of dense right ideals is closed under finite intersection; furthermore, any right ideal containing a dense right ideal is again dense. Hence, the collection becomes a directed set if we let Dy á D2 whenever D2 £ Dy for dense right ideals Dy and D2 of R. It follows that for any /{-module M we can form the direct limit £>(M) of {HomB (D, M) \ D is a dense right ideal of R} over the directed set of dense right ideals of R. We will call 3>(M) the derived quotient module of M. -
Arxiv:Math/0505004V2
CENTRALIZERS AND INVERSES TO INDUCTION AS EQUIVALENCE OF CATEGORIES LARS KADISON Abstract. Given a ring homomorphism B → A, consider its centralizer B R = A , bimodule endomorphism ring S = End BAB and sub-tensor-square B ring T = (A ⊗B A) . Nonassociative tensoring by the cyclic modules RT or S R leads to an equivalence of categories inverse to the functors of induction of restricted A-modules or restricted coinduction of B-modules in case A | B is separable, H-separable, split or left depth two (D2). If RT or S R are projec- tive, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is RT a progenerator, which replaces the key module AAe for an Azumaya algebra A. After establishing these char- acterizations, we characterize left D2 extensions in terms of the module TR, and ask whether a weak generator condition on RT might characterize left D2 extensions as well, possibly a problem in σ(M)-categories or its generaliza- tions. We also show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel and pre-braided commutative. For example, its normality yields a Hopf subalgebra analogue of a factoid for subgroups and their centralizers, and a special case of a conjecture that D2 Hopf subalgebras are normal. 1. Introduction and Preliminaries Given a ring homomorphism B → A, we pass to its induced bimodule BAB and B define its centralizer R = A , bimodule endomorphism ring S = End BAB and B sub-tensor-square ring T = (A ⊗B A) . -
Part IB — Groups, Rings and Modules
Part IB | Groups, Rings and Modules Based on lectures by O. Randal-Williams Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Groups Basic concepts of group theory recalled from Part IA Groups. Normal subgroups, quotient groups and isomorphism theorems. Permutation groups. Groups acting on sets, permutation representations. Conjugacy classes, centralizers and normalizers. The centre of a group. Elementary properties of finite p-groups. Examples of finite linear groups and groups arising from geometry. Simplicity of An. Sylow subgroups and Sylow theorems. Applications, groups of small order. [8] Rings Definition and examples of rings (commutative, with 1). Ideals, homomorphisms, quotient rings, isomorphism theorems. Prime and maximal ideals. Fields. The characteristic of a field. Field of fractions of an integral domain. Factorization in rings; units, primes and irreducibles. Unique factorization in principal ideal domains, and in polynomial rings. Gauss' Lemma and Eisenstein's irreducibility criterion. Rings Z[α] of algebraic integers as subsets of C and quotients of Z[x]. Examples of Euclidean domains and uniqueness and non-uniqueness of factorization. Factorization in the ring of Gaussian integers; representation of integers as sums of two squares. Ideals in polynomial rings. Hilbert basis theorem. [10] Modules Definitions, examples of vector spaces, abelian groups and vector spaces with an endomorphism. Sub-modules, homomorphisms, quotient modules and direct sums. Equivalence of matrices, canonical form. Structure of finitely generated modules over Euclidean domains, applications to abelian groups and Jordan normal form. -
Geometries Over Finite Rings
Innovations in Incidence Geometry Volume 15 (2017), Pages 123–143 ISSN 1781-6475 50 years of Finite Geometry, the “Geometries over finite rings” part Dedicated to Prof. J.A. Thas on the occasion of his 70th birthday Dirk Keppens Abstract Whereas for a substantial part, “Finite Geometry” during the past 50 years has focussed on geometries over finite fields, geometries over finite rings that are not division rings have got less attention. Nevertheless, sev- eral important classes of finite rings give rise to interesting geometries. In this paper we bring together some results, scattered over the litera- ture, concerning finite rings and plane projective geometry over such rings. The paper does not contain new material, but by collecting information in one place, we hope to stimulate further research in this area for at least another 50 years of Finite Geometry. Keywords: ring geometry, finite geometry, finite ring, projective plane MSC 2010: 51C05,51E26,13M05,16P10,16Y30,16Y60 1 Introduction Geometries over rings that are not division rings have been studied for a long time. The first systematic study was done by Dan Barbilian [11], besides a mathematician also one of the greatest Romanian poets (with pseudonym Ion Barbu). He introduced plane projective geometries over a class of associative rings with unit, called Z-rings (abbreviation for Zweiseitig singulare¨ Ringe) which today are also known as Dedekind-finite rings. These are rings with the property that ab = 1 implies ba = 1 and they include of course all commutative rings but also all finite rings (even non-commutative). Wilhelm Klingenberg introduced in [52] projective planes and 3-spaces over local rings. -
Module Theory
Module Theory Claudia Menini 2014-2015 Contents Contents 2 1 Modules 4 1.1 Homomorphisms and Quotients ..................... 4 1.2 Quotient Module and Isomorphism Theorems . 10 1.3 Product and Direct Sum of a Family of Modules . 13 1.4 Sum and Direct Sum of Submodules. Cyclic Modules . 21 1.5 Exact sequences and split exact sequences . 27 1.6 HomR (M; N) ............................... 32 2 Free and projective modules 35 3 Injective Modules and Injective Envelopes 41 4 Generators and Cogenerators 59 5 2 × 2 Matrix Ring 65 6 Tensor Product and bimodules 68 6.1 Tensor Product 1 ............................. 68 6.2 Bimodules ................................. 78 6.3 Tensor Product 2 ............................. 82 7 Homology 94 7.1 Categories and Functors ......................... 94 7.2 Snake Lemma ...............................100 7.3 Chain Complexes .............................105 7.4 Homotopies ................................113 7.5 Projective resolutions . 116 7.6 Left Derived functors . 123 7.7 Cochain Complexes and Right Derived Functors . 139 8 Semisimple modules and Jacobson radical 145 9 Chain Conditions. 160 2 CONTENTS 3 10 Progenerators and Morita Equivalence 173 10.1 Progenerators ...............................173 10.2 Frobenius .................................199 Chapter 1 Modules 1.1 Homomorphisms and Quotients R Definition 1.1. Let R be a ring. A left R-module is a pair (M; µM ) where (M+; 0) is an abelian group and R µ = µM : R × M ! M is a map such that, setting a · x = µ ((a; x)) , the following properties are satisfied : M1 a · (x + y) = a · x + a · y; M2 (a + b) · x = a · x + b · x; M3 (a ·R b) x = a · (b · x); M4 1R · x = x for every a; b 2 R and every x; y 2 M. -
Symmetry on Rings of Differential Operators
Symmetry on rings of differential operators Eamon Quinlan-Gallego ∗ September 17, 2021 Abstract If k is a field and R is a commutative k-algebra, we explore the question of when the ring DRSk of k-linear differential operators on R is isomorphic to its opposite ring. Under mild hypotheses, we prove this is the case whenever R Gorenstein local or when R is a ring of invariants. As a key step in the proof we show that in many cases of interest canonical modules admit right D-module structures. 1 Introduction Let k be a field. A construction of Grothendieck assigns to every k-algebra R its ring DRSk of k-linear differential operators, which is a noncommutative ring that consists of certain k-linear operators on R [Gro65, §16]. Suppose k has characteristic zero. If R ∶= k x1,...,xn is a polynomial ring over k then the ring DRSk is called the Weyl algebra over k and it has a particularly[ ] pleasant structure; for example, it is (left and right) Noetherian and its global dimension is n [Roo72]. These facts adapt readily to the case where R is an arbitrary regular k-algebra that is essentially of finite type [Bj¨o79, §3], and in this context the study of DRSk and its modules has numerous applications in singularity theory (e.g. Bernstein-Sato polynomials) and in commutative algebra (e.g. the study of local cohomology [Lyu93]). Since the ring DRSk is noncommutative, a priori its left and right modules could behave very differently. op However, a key feature of the Weyl algebra DRSk is that it is isomorphic to its opposite ring DRSk via an involutive isomorphism that fixes the subring R (see Remark 2.6). -
Chapter I: Groups 1 Semigroups and Monoids
Chapter I: Groups 1 Semigroups and Monoids 1.1 Definition Let S be a set. (a) A binary operation on S is a map b : S × S ! S. Usually, b(x; y) is abbreviated by xy, x · y, x ∗ y, x • y, x ◦ y, x + y, etc. (b) Let (x; y) 7! x ∗ y be a binary operation on S. (i) ∗ is called associative, if (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x; y; z 2 S. (ii) ∗ is called commutative, if x ∗ y = y ∗ x for all x; y 2 S. (iii) An element e 2 S is called a left (resp. right) identity, if e ∗ x = x (resp. x ∗ e = x) for all x 2 S. It is called an identity element if it is a left and right identity. (c) The set S together with a binary operation ∗ is called a semigroup if ∗ is associative. A semigroup (S; ∗) is called a monoid if it has an identity element. 1.2 Examples (a) Addition (resp. multiplication) on N0 = f0; 1; 2;:::g is a binary operation which is associative and commutative. The element 0 (resp. 1) is an identity element. Hence (N0; +) and (N0; ·) are commutative monoids. N := f1; 2;:::g together with addition is a commutative semigroup, but not a monoid. (N; ·) is a commutative monoid. (b) Let X be a set and denote by P(X) the set of its subsets (its power set). Then, (P(X); [) and (P(X); \) are commutative monoids with respective identities ; and X. (c) x∗y := (x+y)=2 defines a binary operation on Q which is commutative but not associative. -
There Is No Analog of the Transpose Map for Infinite Matrices
Publicacions Matem`atiques, Vol 41 (1997), 653–657. THERE IS NO ANALOG OF THE TRANSPOSE MAP FOR INFINITE MATRICES Juan Jacobo Simon´ ∗ Abstract In this note we show that there are no ring anti-isomorphism be- tween row finite matrix rings. As a consequence we show that rowfinite and column finite matrix rings cannot be either iso- morphic or Morita equivalent rings. We also showthat anti- isomorphisms between endomorphism rings of infinitely generated projective modules may exist. 1. Introduction and notation The motivation for this note was to find out the extent to which the transpose for infinite matrices behaves like that for finite matrices. It is well-known that for any commutative ring, R, the transpose map t : Mn(R) → Mn(R) is an anti-automorphism. For infinite matrix rings the analogous transpose map yields an anti-isomorphism t : RFMA(R) → CFMA(R), where RFMA(R) (resp. CFMA(R)) denotes the ring of row-finite (resp. column-finite) matrices, indexed by the set A, having entries in R. Unlike the finite case, this is obviously not an anti-AUTOmorphism of RFMA(R). Thus we are led to ask: for a commutative ring, R, does there ex- ist an anti-automorphism of RFMA(R)? We answer this question in the negative, with a vengeance. In fact, we show in Theorem 2.2 that (A) there are no anti-isomorphisms between rings of the form End(RR ) (B) and End(SS ) for any rings R and S (commutative or not), and any infinite sets A and B. As a corollary we show that for any two rings R and S the rings RFMA(R) and CFMB(S) cannot be either isomorphic or Morita equivalent rings.