DOCSLIB.ORG

Math 100C: Homomorphism Rings of G-Modules Supplement

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 100C: Homomorphism Rings of G-Modules Supplement Math 100C: Homomorphism Rings of G-modules Supplement 1. Homomorphism rings Let V and W be G-modules (finite dimensional, over field C of complex numbers, where we assume the group G is finite). The aim of this blurb is to `determine' the C-vector space HomG(V; W ) of G-module endomorphisms ' : V ! W . Our basic tool is Schur's Lemma: Lemma 1.1. (Schur's Lemma) Let V be an irreducible G-module and let ' : V ! V be a G-module endomorphism. There exists α 2 C such that '(v) = αv for all v 2 V . Furthermore, if V and W are nonisomorphic irreducibles and if ' : V ! V is a G-module endomorphism, then ' ≡ 0. If V and W are both irreducible, Schur's Lemma gives us our answer { we have ( V ∼= W Hom (V; W ) ∼= C G G 0 otherwise. But what if V and W are not irreducible? The first step is to apply Maschke's Theorem to write V and W as direct sums of irreducible modules: ∼ ⊕n1 ⊕nr V =G (U1) ⊕ · · · ⊕ (Ur) ∼ ⊕m1 ⊕mr W =G (U1) ⊕ · · · ⊕ (Ur) ; where U1;:::;Ur are pairwise nonisomorphic irreducible G-modules and n1; : : : ; nr and m1; : : : ; mr are nonnegative integers (by allowing some of the ni; mi to be zero, we can always write down such a decomposition). We know ( i = j Hom (U ;U ) ∼= C G i j 0 i 6= j: The idea is to boost this result to get a reasonable picture of HomG(V; W ). To transfer between V; W; and the Ui modules, we will make use of inclusion and projection maps. For 1 ≤ i ≤ r and 1 ≤ j ≤ ni, let ⊕n1 ⊕nr ιi;j : Ui ,! (U1) ⊕ · · · ⊕ (Ur) = V th ⊕ni be the inclusion of Ui into the j factor in the direct sum Ui . The map ιi;j is an injection of G-modules. Similarly, for 1 ≤ i ≤ r and 1 ≤ j ≤ mi, let ⊕m1 ⊕mr πi;j : W = (U1) ⊕ · · · ⊕ (Ur) Ui th ⊕mi be projection of W onto the j factor in the direct sum Ui . 1 2 How do the ιi;j and πi;j maps help us? Well, if ' : V ! W is any G-module homomorphism, then for any i; i0; j; j0 we get a G-module homomorphism πi0;j0 ◦ ' ◦ ιi;j : Ui ! Ui0 between the irreducible G-modules Ui and Ui0 . These `little maps' characterize ': Problem 1.2. Let ' : V ! W and '0 : V ! W be two G-module homomor- 0 0 phisms. Prove that ' = ' if and only if πi0;j0 ◦ ' ◦ ιi;j = πi0;j0 ◦ ' ◦ ιi;j for all 0 0 1 ≤ i; i ≤ r and 1 ≤ j ≤ ni and 1 ≤ j ≤ mi. 0 Schur's Lemma tells us that πi0;j0 ◦ ' ◦ ιi;j ≡ 0 unless i = i , in which case there exists α 2 C such that πi;j ◦ ' ◦ ιi;j scales by α. Encoding the numbers 0 0 α = αi;j;i0;j0 for the various choices of i; j; i ; j gives rise to an injective C-linear map Φ : HomG(V; W ) ! Matm1×n1 (C) ⊕ · · · ⊕ Matmr×nr (C): Problem 1.3. Prove that Φ is a surjection. (Hint: Suppose you have an m × n complex matrix A. If U is an irreducible G-module, how can you build a G-module homomorphism U ⊕n ! U ⊕m using the matrix A?) By the above problem, the map Φ is an isomorphism of C-vector spaces, so that dimC HomG(V; W ) = m1n1 + ··· + mrnr: Problem 1.4. Let V be a G-module. Prove that the decomposition of V into irreducibles is unique. That is, suppose we have ∼ ⊕n1 ⊕nr V =G (U1) ⊕ · · · ⊕ (Ur) ; ∼ ⊕m1 ⊕mr V =G (U1) ⊕ · · · ⊕ (Ur) : for irreducible modules U1;:::;Ur and nonnegative integers n1; : : : ; nr and m1; : : : ; mr. Prove that ni = mi for all i. (Hint: Consider the C-vector space Hom(Ui;V ) { what is its dimension? This problem could also be solved by appeal to the Jordan-H¨olderTheorem.) As an example, suppose that V; U; and W are irreducible G-modules. Con- sider the direct sums V ⊕3 ⊕U ⊕2 and V ⊕4 ⊕U ⊕3 ⊕W ⊕3; what does the C-vector ⊕3 ⊕2 ⊕4 ⊕3 ⊕3 space HomG(V ⊕ U ;V ⊕ U ⊕ W ) look like? Answer: It looks like a direct sum of matrix vector spaces, namely Mat3×4(C) ⊕ Mat2×3(C) ⊕ Mat0×3(C); where we identify the space of `0 × 3' matrices Mat0×3(C) with the zero space 0. In particular, the dimension of this space is 3 × 4 + 2 × 3 + 0 × 3 = 12 + 6 + 0 = 18: 3 2. Endomorphism rings What if V = W in the above discussion? The homomorphism vector space HomG(V; V ) gets a new name: the endomorphism algebra EndG(V ) := HomG(V; V ) = fall G-module homomorphisms ' : V ! V g: As the name suggests, EndG(V ) is not just a C-vector space, but also a C- algebra { the multiplication operation is the composition of homomorphisms: ('1 · '2)(v) := '1('2(v)): If V 6= W , there is no natural algebra structure on the vector space HomG(V; W ) { try not to confuse these situations. What does the endomorphism ring EndG(V ) look like as a C-algebra? Let us decompose V as in Maschke's Theorem: ∼ ⊕n1 ⊕nr V =G (U1) ⊕ · · · ⊕ (Ur) ; for some nonisomorphic irreducibles U1;:::;Ur and some positive integers n1; : : : ; nr. In the last section we constructed an isomorphism of C-vector spaces: Φ : EndG(V ) ! Matn1 (C) ⊕ · · · ⊕ Matnr (C): The right hand side may be regarded as a C-algebra (where the multiplication is matrix multiplication in each direct summand). Here is a boring problem. Problem 2.1. Prove that Φ is an isomorphism of C-algebras (i.e. it sends composition of endomorphisms to multiplication of matrices). To make this theory more interesting, let's consider an explicit example. Let n > 1 be an integer and consider the permutation action of the symmetric n n group Sn on V = C . In the standard basis of C , this corresponds to the group homomorphism P : Sn ! GLn(C); where for any σ 2 Sn the matrix Pσ is the permutation matrix of σ: ( 1 σ(i) = j (P ) = σ i;j 0 otherwise. Since n > 1, the module V is not irreducible { rather, it breaks up into ∼ two irreducible modules V =Sn U ⊕ W , where U is the trivial 1-dimensional n representation of Sn and W = f(α1; : : : ; αn) 2 C : α1 + ··· + αn = 0g. Since U and W are not isomorphic (why?) we know that ∼ ∼ 2 EndSn (V ) = Mat1(C) ⊕ Mat1(C) = C : 4 Let us see the results of the above paragraph, explicitly, using matrices. We can identify Sn-module endomorphisms V ! V with n × n complex matrices T 2 Matn(C) such that TPσ = PσT for all σ 2 Sn. Here is a more interesting problem. Problem 2.2. Let T 2 Matn(C). Prove that TPσ = PσT for all permutation matrices Pσ if and only if T has the form 0α β ··· β1 Bβ α ··· βC T = B . C @ . A β β ··· α for some complex numbers α; β 2 C. Use this fact to deduce an isomorphism of C-algebras ∼ 2 EndSn (V ) = C : (Hint: What happens when you multiply two matrices in the above form to- gether? Try to write down an isomorphism that respects this multiplication!).
Load more

Recommended publications
  • Modules and Vector Spaces
    Modules and Vector Spaces R. C. Daileda October 16, 2017 1 Modules Definition 1. A (left) R-module is a triple (R; M; ·) consisting of a ring R, an (additive) abelian group M and a binary operation · : R × M ! M (simply written as r · m = rm) that for all r; s 2 R and m; n 2 M satisfies • r(m + n) = rm + rn ; • (r + s)m = rm + sm ; • r(sm) = (rs)m. If R has unity we also require that 1m = m for all m 2 M. If R = F , a field, we call M a vector space (over F ). N Remark 1. One can show that as a consequence of this definition, the zeros of R and M both \act like zero" relative to the binary operation between R and M, i.e. 0Rm = 0M and r0M = 0M for all r 2 R and m 2 M. H Example 1. Let R be a ring. • R is an R-module using multiplication in R as the binary operation. • Every (additive) abelian group G is a Z-module via n · g = ng for n 2 Z and g 2 G. In fact, this is the only way to make G into a Z-module. Since we must have 1 · g = g for all g 2 G, one can show that n · g = ng for all n 2 Z. Thus there is only one possible Z-module structure on any abelian group. • Rn = R ⊕ R ⊕ · · · ⊕ R is an R-module via | {z } n times r(a1; a2; : : : ; an) = (ra1; ra2; : : : ; ran): • Mn(R) is an R-module via r(aij) = (raij): • R[x] is an R-module via X i X i r aix = raix : i i • Every ideal in R is an R-module.
    [Show full text]
  • Pub . Mat . UAB Vol . 27 N- 1 on the ENDOMORPHISM RING of A
    Pub . Mat . UAB Vol . 27 n- 1 ON THE ENDOMORPHISM RING OF A FREE MODULE Pere Menal Throughout, let R be an (associative) ring (with 1) . Let F be the free right R-module, over an infinite set C, with endomorphism ring H . In this note we first study those rings R such that H is left coh- erent .By comparison with Lenzing's characterization of those rings R such that H is right coherent [8, Satz 41, we obtain a large class of rings H which are right but not left coherent . Also we are concerned with the rings R such that H is either right (left) IF-ring or else right (left) self-FP-injective . In particular we pro- ve that H is right self-FP-injective if and only if R is quasi-Frobenius (QF) (this is an slight generalization of results of Faith and .Walker [31 which assure that R must be QF whenever H is right self-i .njective) moreover, this occurs if and only if H is a left IF-ring . On the other hand we shall see that if' R is .pseudo-Frobenius (PF), that is R is an . injective cogenera- tortin Mod-R, then H is left self-FP-injective . Hence any PF-ring, R, that is not QF is such that H is left but not right self FP-injective . A left R-module M is said to be FP-injective if every R-homomorphism N -. M, where N is a -finitély generated submodule of a free module F, may be extended to F .
    [Show full text]
  • Classification of Finite Abelian Groups
    Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k.
    [Show full text]
  • MA3203 Ring Theory Spring 2019 Norwegian University of Science and Technology Exercise Set 3 Department of Mathematical Sciences
    MA3203 Ring theory Spring 2019 Norwegian University of Science and Technology Exercise set 3 Department of Mathematical Sciences 1 Decompose the following representation into indecomposable representations: 1 1 1 1 1 1 1 1 2 2 2 k k k . α γ 2 Let Q be the quiver 1 2 3 and Λ = kQ. β δ a) Find the representations corresponding to the modules Λei, for i = 1; 2; 3. Let R be a ring and M an R-module. The endomorphism ring is defined as EndR(M) := ff : M ! M j f R-homomorphismg. b) Show that EndR(M) is indeed a ring. c) Show that an R-module M is decomposable if and only if its endomorphism ring EndR(M) contains a non-trivial idempotent, that is, an element f 6= 0; 1 such that f 2 = f. ∼ Hint: If M = M1 ⊕ M2 is decomposable, consider the projection morphism M ! M1. Conversely, any idempotent can be thought of as a projection mor- phism. d) Using c), show that the representation corresponding to Λe1 is indecomposable. ∼ Hint: Prove that EndΛ(Λe1) = k by showing that every Λ-homomorphism Λe1 ! Λe1 depends on a parameter l 2 k and that every l 2 k gives such a homomorphism. 3 Let A be a k-algebra, e 2 A be an idempotent and M be an A-module. a) Show that eAe is a k-algebra and that e is the identity element. For two A-modules N1, N2, we denote by HomA(N1;N2) the space of A-module morphism from N1 to N2.
    [Show full text]
  • BLECHER • All Numbers Below Are Integers, Usually Positive. • Read Up
    MATH 4389{ALGEBRA `CHEATSHEET'{BLECHER • All numbers below are integers, usually positive. • Read up on e.g. wikipedia on the Euclidean algorithm. • Diophantine equation ax + by = c has solutions iff gcd(a; b) doesn't divide c. One may find the solutions using the Euclidean algorithm (google this). • Euclid's lemma: prime pjab implies pja or pjb. • gcd(m; n) · lcm(m; n) = mn. • a ≡ b (mod m) means mj(a − b). An equivalence relation (reflexive, symmetric, transitive) on the integers. • ax ≡ b (mod m) has a solution iff gcd(m; a)jb. • Division Algorithm: there exist unique integers q and r with a = bq + r and 0 ≤ r < b; so a ≡ r (mod b). • Fermat's theorem: If p is prime then ap ≡ a (mod p) Also, ap−1 ≡ 1 (mod p) if p does not divide a. • For definitions for groups, rings, fields, subgroups, order of element, etc see Algebra fact sheet" below. • Zm = f0; 1; ··· ; m−1g with addition and multiplication mod m, is a group and commutative ring (= Z =m Z) • Order of n in Zm (i.e. smallest k s.t. kn ≡ 0 (mod m)) is m=gcd(m; n). • hai denotes the subgroup generated by a (the smallest subgroup containing a). For example h2i = f0; 2; 2+2 = 4; 2 + 2 + 2 = 6g in Z8. (In ring theory it may be the subring generated by a). • A cyclic group is a group that can be generated by a single element (the group generator). They are abelian. • A finite group is cyclic iff G = fg; g + g; ··· ; g + g + ··· + g = 0g; so G = hgi.
    [Show full text]
  • Idempotent Lifting and Ring Extensions
    IDEMPOTENT LIFTING AND RING EXTENSIONS ALEXANDER J. DIESL, SAMUEL J. DITTMER, AND PACE P. NIELSEN Abstract. We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring R for which idempotents lift modulo the Jacobson radical J(R), but idempotents do not lift modulo J(M2(R)). Thus, the property \idempotents lift modulo the Jacobson radical" is not a Morita invariant. We also prove that if I and J are ideals of R for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over I + J. On the positive side, if I and J are enabling ideals in R, then I + J is also an enabling ideal. We show that if I E R is (weakly) enabling in R, then I[t] is not necessarily (weakly) enabling in R[t] while I t is (weakly) enabling in R t . The latter result is a special case of a more general theorem about completions.J K Finally, we give examplesJ K showing that conjugate idempotents are not necessarily related by a string of perspectivities. 1. Introduction In ring theory it is useful to be able to lift properties of a factor ring of R back to R itself. This is often accomplished by restricting to a nice class of rings. Indeed, certain common classes of rings are defined precisely in terms of such lifting properties. For instance, semiperfect rings are those rings R for which R=J(R) is semisimple and idempotents lift modulo the Jacobson radical.
    [Show full text]
  • MAS4107 Linear Algebra 2 Linear Maps And
    Introduction Groups and Fields Vector Spaces Subspaces, Linear . Bases and Coordinates MAS4107 Linear Algebra 2 Linear Maps and . Change of Basis Peter Sin More on Linear Maps University of Florida Linear Endomorphisms email: [email protected]fl.edu Quotient Spaces Spaces of Linear . General Prerequisites Direct Sums Minimal polynomial Familiarity with the notion of mathematical proof and some experience in read- Bilinear Forms ing and writing proofs. Familiarity with standard mathematical notation such as Hermitian Forms summations and notations of set theory. Euclidean and . Self-Adjoint Linear . Linear Algebra Prerequisites Notation Familiarity with the notion of linear independence. Gaussian elimination (reduction by row operations) to solve systems of equations. This is the most important algorithm and it will be assumed and used freely in the classes, for example to find JJ J I II coordinate vectors with respect to basis and to compute the matrix of a linear map, to test for linear dependence, etc. The determinant of a square matrix by cofactors Back and also by row operations. Full Screen Close Quit Introduction 0. Introduction Groups and Fields Vector Spaces These notes include some topics from MAS4105, which you should have seen in one Subspaces, Linear . form or another, but probably presented in a totally different way. They have been Bases and Coordinates written in a terse style, so you should read very slowly and with patience. Please Linear Maps and . feel free to email me with any questions or comments. The notes are in electronic Change of Basis form so sections can be changed very easily to incorporate improvements.
    [Show full text]
  • Cohomology Theory of Lie Groups and Lie Algebras
    COHOMOLOGY THEORY OF LIE GROUPS AND LIE ALGEBRAS BY CLAUDE CHEVALLEY AND SAMUEL EILENBERG Introduction The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions con- cerning Lie algebras^). This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham's theorems(2) (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the con- sideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group. We study here the question not only of the topological nature of the whole group, but also of the manifolds on which the group operates. Chapter I is concerned essentially with step 2 of the list above (step 1 depending here, as in the case of the whole group, on de Rham's theorems). Besides consider- ing invariant forms, we also introduce "equivariant" forms, defined in terms of a suitable linear representation of the group; Theorem 2.2 states that, when this representation does not contain the trivial representation, equi- variant forms are of no use for topology; however, it states this negative result in the form of a positive property of equivariant forms which is of interest by itself, since it is the key to Levi's theorem (cf.
    [Show full text]
  • Lecture 1.3: Direct Products and Sums
    Lecture 1.3: Direct products and sums Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Advanced Linear Algebra M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 1 / 5 Overview In previous lectures, we learned about vectors spaces and subspaces. We learned about what it meant for a subset to span, to be linearly independent, and to be a basis. In this lecture, we will see how to create new vector spaces from old ones. We will see several ways to \multiply" vector spaces together, and will learn how to construct: the complement of a subspace the direct sum of two subspaces the direct product of two vector spaces M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 2 / 5 Complements and direct sums Theorem 1.5 (a) Every subspace Y of a finite-dimensional vector space X is finite-dimensional. (b) Every subspace Y has a complement in X : another subspace Z such that every vector x 2 X can be written uniquely as x = y + z; y 2 Y ; z 2 Z; dim X = dim Y + dim Z: Proof Definition X is the direct sum of subspaces Y and Z that are complements of each other. More generally, X is the direct sum of subspaces Y1;:::; Ym if every x 2 X can be expressed uniquely as x = y1 + ··· + ym; yi 2 Yi : We denote this as X = Y1 ⊕ · · · ⊕ Ym. M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 3 / 5 Direct products Definition The direct product of X1 and X2 is the vector space X1 × X2 := (x1; x2) j x1 2 X1; x2 2 X2 ; with addition and multiplication defined component-wise.
    [Show full text]
  • OBJ (Application/Pdf)
    GROUPOIDS WITH SEMIGROUP OPERATORS AND ADDITIVE ENDOMORPHISM A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS POR THE DEGREE OF MASTER OF SCIENCE BY FRED HENDRIX HUGHES DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA JUNE 1965 TABLE OF CONTENTS Chapter Page I' INTRODUCTION. 1 II GROUPOIDS WITH SEMIGROUP OPERATORS 6 - Ill GROUPOIDS WITH ADDITIVE ENDOMORPHISM 12 BIBLIOGRAPHY 17 ii 4 CHAPTER I INTRODUCTION A set is an undefined termj however, one can say a set is a collec¬ tion of objects according to our sight and perception. Several authors use this definition, a set is a collection of definite distinct objects called elements. This concept is the foundation for mathematics. How¬ ever, it was not until the latter part of the nineteenth century when the concept was formally introduced. From this concept of a set, mathematicians, by placing restrictions on a set, have developed the algebraic structures which we employ. The structures are closely related as the diagram below illustrates. Quasigroup Set The first structure is a groupoid which the writer will discuss the following properties: subgroupiod, antigroupoid, expansive set homor- phism of groupoids in semigroups, groupoid with semigroupoid operators and groupoids with additive endormorphism. Definition 1.1. — A set of elements G - f x,y,z } which is defined by a single-valued binary operation such that x o y ■ z é G (The only restriction is closure) is called a groupiod. Definition 1.2. — The binary operation will be a mapping of the set into itself (AA a direct product.) 1 2 Definition 1.3» — A non-void subset of a groupoid G is called a subgroupoid if and only if AA C A.
    [Show full text]
  • 6. Localization
    52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a field, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can define addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i.
    [Show full text]
  • 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.
    [Show full text]