DOCSLIB.ORG

Vector Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Vector Spaces Vector spaces We fix a field F. A vector space, V, over the field F, is a set V, equipped with: • An additive operation + : V × V ! V, denoted (v; w) ! v + w, for any v and w 2 V, which makes V into an abelian group with additive identity denoted 0 2 V and additive inverse of v 2 V denoted −v, for each v 2 V. • An operation of scalar multiplication F × V ! V, denoted (λ, v) ! λv = vλ, for each λ 2 F and each v 2 V, such that, for any v and w in V and any λ and µ in F, we have the relations: 1v = v; 0v = 0; (−1)v = −v; λ(v + w) = λv + λw; (λ + µ)v = λv + µv; λ(µv) = (λµ)v = µ(λ)v: Examples of vector spaces • The trivial vector space over a field F is a set with one element, denoted 0, with the operations 0 + 0 = 0 and λ0 = 0, for each λ 2 F. • The field F is a vector space over itself, with its usual operations. • Let S be a set. Then the space FS of all maps from S to F has the natural structure of a vector space, via the formulas, valid for each f : S ! F, g : S ! F and for each λ and µ in F: (λf + µg)(s) = λf(s) + µg(s); for each s 2 S: FS is called the free vector space over S. 1 Constructions of vector spaces Subspaces Given a vector space V, and a subset X of V, if X is invariant under the operations of V, we may restrict these operations to X and then X becomes a vector space in its own right, called a subspace of V. To verify that X ⊂ V qualifies as a subspace, one need only show that λx + µy 2 X, for any x and y in X and any λ and µ in F. In general a linear combination of vectors of X is a vector x of the form k x = Σi=1λixi, for some xi 2 X; i = 1; 2; : : : ; k, where λi 2 F; i = 1; 2; : : : ; k, for some positive integer k. Then X is a subspace if and only it contains all linear combinations of vectors of X. The zero element of any vector space constitutes a subspace, called the trivial subspace. The intersection of any family of subspaces of a vector space V is a sub- space. If a set X is a subset of a vector space V, then the intersection of all subspaces of V containing X is called [X], the subspace spanned by X. Then [[X]] = [X] and X = [X] if and only if X is a subspace. The space [X] consists of all possible linear combinations of the vectors of X. Every vector space is spanned by itself. The dimension of a non-trivial vector space V is the small- est cardinality of a set that spans it. The trivial vector space is assigned dimension zero. A space V is called finite dimensional if it is spanned by a finite set: it then has dimension zero if and only if it is trivial and otherwise its dimension is a positive integer. If X = fx1; x2; : : : xng ⊂ V, then [X] is written [x1; x2; : : : ; xn]. Then [X] has dimension at most n and has dimension n if and only if the vectors xj; j = 1; 2; : : : ; n are linearly independent: n Σj=1λjxj = 0 with λj 2 F, for each j = 1; 2; : : : ; n, if and only if λj = 0, for j = 1; 2; : : : ; n: The vectors are linearly dependent if and only if they are not linearly inde- n pendent, if and only if there is a relation 0 = Σj=1λjxj = 0, with all λi 2 F and at least one λk non-zero, if and only if the dimension of [X] is less than n. 2 If a collection of vectors fx1; x2; : : : ; xng ⊂ V spans a subspace X, then either X = V, or one can enlarge this collection to a list fx1; x2; : : : ; xn; xn+1g ⊂ V, such that X ⊂ [x1; x2; : : : ; xn+1] 6= X: the new vector xn+1 2 V just has to be chosen to lie in the complement of X in V. Then if the fx1; x2; : : : ; xng are linearly independent, so span a space of di- mension n, so are fx1; x2; : : : ; xn+1g and they span a space of dimension n+1. The collection of all n-dimensional subspaces of V is called the Grassmannian of all n-planes in V, Gr(n; V). If V has dimension m, then Gr(n; V) is a space of dimension n(m − n). Each one-dimensional subspace of V is a set [x] of the form [x] = fλx : λ 2 Fg for some 0 6= x 2 V. We have [x] = [y], for 0 6= y 2 V if and only if x and y are linearly dependent. The space of all one-dimensional subspaces of V is called P(V) = Gr(1; V), the projective space of V. Each two-dimensional subspace of V is a set [x; y] of the form [x; y] = fλx + µy : λ 2 F; µ 2 Fg for some linearly independent vectors x and y in V. a b We have [x; y] = [p; q] if and only if there is an invertible matrix, , c d with coefficients in F, such that p = ax + by; q = cx + dy, if and only if each x and y are linearly independent, as are p and q, but any three of the four vectors x; y; p and q are linearly dependent. A subset X of V is a subspace if and only if [x; y] ⊂ X, for any x and y in X. 3 The direct sum of vector spaces Let S and X be sets equipped with a surjective map π : X ! S, such that −1 Xs = π (fsg) ⊂ S has the structure of a vector space over F, for each s 2 S. A section of π is a map f : S ! X, such that π ◦ f = idS. Then the space Γ of all sections of π becomes a vector space by fibrewise addition and scalar multiplication: for any f and g in Γ, any λ 2 F and each s 2 S: (f + g)(s) = f(s) + g(s); (λf)(s) = λf(s): We write Γ = ⊕s2SXs, called the direct sum of the family of vector spaces Xs. An element f of Γ is written f = ⊕s2Sf(s). The support of a section f 2 Γ is the set of all s 2 S such that f(s) = 0. The collection of all elements of Γ with finite support is a subspace of Γ, called the restricted direct sum of the family of spaces fXs : s 2 Sg. These two notions of direct sum coincide when S is finite. S In particular if Xs = F for each s 2 S, then Γ = F . Homomorphisms Let V and W be vector spaces. Then a homomorphism or linear map, T from V to W is a set map T from V to W, respecting the vector space structure: T (λx + µy) = λT (x) + µT (y); for any x and y in V and any λ and µ in F. Then the image T (X) of a subspace X of V under T is a subspace of W, since if p and q lie in T (X) and if λ and µ are in F, then x and y in X exist, with T (x) = p and T (y) = q, so then z = λx + µy lies in X, since X is a subspace, so T (z) = λT (x) + µT (y) = λp + µq 2 T (X), so T (X) satisfies the subspace criterion. In particular the image of V itself is a subspace. • T is a monomorphism if and only if T is injective. • T is a epimorphism if and only if T is surjective. • T is an isomorphism if and only if T is invertible, if and only if T is both an epimorphism and a monomorphism. In this case the inverse map T −1 lies in Hom(W; V). 4 • T is called an endomorphism in the case that V = W and an automor- phism if T is an invertible endomorphism. The automorphisms of V form a group, under composition, called GL(V). The collection of all homomorphisms, T : V ! W itself naturally forms a vector space denoted Hom(V; W). If V; W and X are vector spaces, then the composition map Hom(V; W) × Hom(W; X), (f; g) ! g ◦f, for any f 2 Hom(V; W) and any g 2 Hom(W; X), is bilinear (i.e. linear in each argument). In the particular case that W = F, we put V∗ = Hom(V; F). Then V∗ is called the dual space of V. There is a natural homomorphism J from V ! (V∗)∗ given by the formula, valid for any v 2 V and any α 2 V∗: (Jv)(α) = α(v): If V is finite dimensional, then J is an isomorphism. A vector space V over F is finite dimensional if and only if any one of the following three conditions is met: • There is an epimorphism from FP to V, for some finite set P • There is a monomorphism from V to FQ, for some finite set Q. • There is an isomorphism from FS to V for some finite set S.
Load more

Recommended publications
  • Group Homomorphisms
    1-17-2018 Group Homomorphisms Here are the operation tables for two groups of order 4: · 1 a a2 + 0 1 2 1 1 a a2 0 0 1 2 a a a2 1 1 1 2 0 a2 a2 1 a 2 2 0 1 There is an obvious sense in which these two groups are “the same”: You can get the second table from the first by replacing 0 with 1, 1 with a, and 2 with a2. When are two groups the same? You might think of saying that two groups are the same if you can get one group’s table from the other by substitution, as above. However, there are problems with this. In the first place, it might be very difficult to check — imagine having to write down a multiplication table for a group of order 256! In the second place, it’s not clear what a “multiplication table” is if a group is infinite. One way to implement a substitution is to use a function. In a sense, a function is a thing which “substitutes” its output for its input. I’ll define what it means for two groups to be “the same” by using certain kinds of functions between groups. These functions are called group homomorphisms; a special kind of homomorphism, called an isomorphism, will be used to define “sameness” for groups. Definition. Let G and H be groups. A homomorphism from G to H is a function f : G → H such that f(x · y)= f(x) · f(y) forall x,y ∈ G.
    [Show full text]
  • The General Linear Group
    18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices.
    [Show full text]
  • The Fundamental Homomorphism Theorem
    Lecture 4.3: The fundamental homomorphism theorem Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.3: The fundamental homomorphism theorem Math 4120, Modern Algebra 1 / 10 Motivating example (from the previous lecture) Define the homomorphism φ : Q4 ! V4 via φ(i) = v and φ(j) = h. Since Q4 = hi; ji: φ(1) = e ; φ(−1) = φ(i 2) = φ(i)2 = v 2 = e ; φ(k) = φ(ij) = φ(i)φ(j) = vh = r ; φ(−k) = φ(ji) = φ(j)φ(i) = hv = r ; φ(−i) = φ(−1)φ(i) = ev = v ; φ(−j) = φ(−1)φ(j) = eh = h : Let's quotient out by Ker φ = {−1; 1g: 1 i K 1 i iK K iK −1 −i −1 −i Q4 Q4 Q4=K −j −k −j −k jK kK j k jK j k kK Q4 organized by the left cosets of K collapse cosets subgroup K = h−1i are near each other into single nodes Key observation Q4= Ker(φ) =∼ Im(φ). M. Macauley (Clemson) Lecture 4.3: The fundamental homomorphism theorem Math 4120, Modern Algebra 2 / 10 The Fundamental Homomorphism Theorem The following result is one of the central results in group theory. Fundamental homomorphism theorem (FHT) If φ: G ! H is a homomorphism, then Im(φ) =∼ G= Ker(φ). The FHT says that every homomorphism can be decomposed into two steps: (i) quotient out by the kernel, and then (ii) relabel the nodes via φ. G φ Im φ (Ker φ C G) any homomorphism q i quotient remaining isomorphism process (\relabeling") G Ker φ group of cosets M.
    [Show full text]
  • Homomorphisms and Isomorphisms
    Lecture 4.1: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 1 / 13 Motivation Throughout the course, we've said things like: \This group has the same structure as that group." \This group is isomorphic to that group." However, we've never really spelled out the details about what this means. We will study a special type of function between groups, called a homomorphism. An isomorphism is a special type of homomorphism. The Greek roots \homo" and \morph" together mean \same shape." There are two situations where homomorphisms arise: when one group is a subgroup of another; when one group is a quotient of another. The corresponding homomorphisms are called embeddings and quotient maps. Also in this chapter, we will completely classify all finite abelian groups, and get a taste of a few more advanced topics, such as the the four \isomorphism theorems," commutators subgroups, and automorphisms. M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 2 / 13 A motivating example Consider the statement: Z3 < D3. Here is a visual: 0 e 0 7! e f 1 7! r 2 2 1 2 7! r r2f rf r2 r The group D3 contains a size-3 cyclic subgroup hri, which is identical to Z3 in structure only. None of the elements of Z3 (namely 0, 1, 2) are actually in D3. When we say Z3 < D3, we really mean is that the structure of Z3 shows up in D3.
    [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]
  • Monoids: Theme and Variations (Functional Pearl)
    University of Pennsylvania ScholarlyCommons Departmental Papers (CIS) Department of Computer & Information Science 7-2012 Monoids: Theme and Variations (Functional Pearl) Brent A. Yorgey University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/cis_papers Part of the Programming Languages and Compilers Commons, Software Engineering Commons, and the Theory and Algorithms Commons Recommended Citation Brent A. Yorgey, "Monoids: Theme and Variations (Functional Pearl)", . July 2012. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_papers/762 For more information, please contact [email protected]. Monoids: Theme and Variations (Functional Pearl) Abstract The monoid is a humble algebraic structure, at first glance ve en downright boring. However, there’s much more to monoids than meets the eye. Using examples taken from the diagrams vector graphics framework as a case study, I demonstrate the power and beauty of monoids for library design. The paper begins with an extremely simple model of diagrams and proceeds through a series of incremental variations, all related somehow to the central theme of monoids. Along the way, I illustrate the power of compositional semantics; why you should also pay attention to the monoid’s even humbler cousin, the semigroup; monoid homomorphisms; and monoid actions. Keywords monoid, homomorphism, monoid action, EDSL Disciplines Programming Languages and Compilers | Software Engineering | Theory and Algorithms This conference paper is available at ScholarlyCommons: https://repository.upenn.edu/cis_papers/762 Monoids: Theme and Variations (Functional Pearl) Brent A. Yorgey University of Pennsylvania [email protected] Abstract The monoid is a humble algebraic structure, at first glance even downright boring.
    [Show full text]
  • 1 Vector Spaces
    AM 106/206: Applied Algebra Prof. Salil Vadhan Lecture Notes 18 November 25, 2009 1 Vector Spaces • Reading: Gallian Ch. 19 • Today's main message: linear algebra (as in Math 21) can be done over any field, and most of the results you're familiar with from the case of R or C carry over. • Def of vector space. • Examples: { F n { F [x] { Any ring containing F { F [x]=hp(x)i { C a vector space over R • Def of linear (in)dependence, span, basis. • Examples in F n: { (1; 0; 0; ··· ; 0), (0; 1; 0;:::; 0), . , (0; 0; 0;:::; 1) is a basis for F n { (1; 1; 0), (1; 0; 1), (0; 1; 1) is a basis for F 3 iff F has characteristic 2 • Def: The dimension of a vector space V over F is the size of the largest set of linearly independent vectors in V . (different than Gallian, but we'll show it to be equivalent) { A measure of size that makes sense even for infinite sets. • Prop: every finite-dimensional vector space has a basis consisting of dim(V ) vectors. Later we'll see that all bases have exactly dim(V ) vectors. • Examples: { F n has dimension n { F [x] has infinite dimension (1; x; x2; x3;::: are linearly independent) { F [x]=hp(x)i has dimension deg(p) (basis is (the cosets of) 1; x; x2; : : : ; xdeg(p)−1). { C has dimension 2 over R 1 • Proof: Let v1; : : : ; vk be the largest set of linearly independent vectors in V (so k = dim(V )).
    [Show full text]
  • Homomorphism Learning Problems and Its Applications to Public-Key Cryptography
    Homomorphism learning problems and its applications to public-key cryptography Christopher Leonardi1, 2 and Luis Ruiz-Lopez1, 2 1University of Waterloo 2Isara Corporation May 23, 2019 Abstract We present a framework for the study of a learning problem over abstract groups, and introduce a new technique which allows for public-key encryption using generic groups. We proved, however, that in order to obtain a quantum resistant encryption scheme, commuta- tive groups cannot be used to instantiate this protocol. Keywords: Learning With Errors, isogenies, non-commutative cryptography 1 Introduction Lattice based cryptography is nowadays the most prominent among the candidate areas for quantum resistant cryptography. The great popularity of lattice based cryptography is, in great part, due to its versatility|several different primitives have been constructed based on lattice problems|and security guarantees such as average-case to worst-case reductions to problems that are presumably hard even for quantum algorithms. Particularly, the short integers solutions problem (SIS), used by Ajtai in his seminal paper [1] to construct a one-way function, and the learning with errors problem (LWE), introduced by Regev in [18], have served as the backbone for several cryptographic constructions. The importance of these two problems goes beyond their applications in cryptography, since their formulation was motivated by purely mathematical problems of a mixed geometric and algebraic character. For example, SIS can be thought as the problem of finding short elements in the kernel of a linear function. For its part, LWE is the problem of finding solutions to a system of noisy linear equations. With these statements of the problems it is possible to imagine several generalizations of them, since some elements in the statements may seem rather arbitrary.
    [Show full text]
  • Commutative Algebra
    Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions .
    [Show full text]
  • Isomorphisms, Automorphisms, Homomorphisms, Kernels
    Isomorphisms, Automorphisms, Homomorphisms, Kernels September 10, 2009 Definition 1 Let G; G0 be groups and let f : G ! G0 be a mapping (of the underlying sets). We say that f is a (group) homomorphism if f(x · y) = f(x) · f(y) for all x; y 2 G. The composition f G−!G0−!h G" of two homomorphisms (as displayed) is again a homomorphism. For any two groups G, G0 by the trivial homomorphism from G to G0 we mean the mapping G ! G0 sending all elements to the identity element in G0. Recall: Definition 2 A group homomorphism f : G ! G0 is called a group isomorphism|or, for short, an isomorphism|if f is bijective. An isomorphism from a group G to itself is called an automorphism. Exercise 1 Let f : G ! G0 be a homomorphism of groups. The image under the homomorphism 0 f of any subgroup of G is a subgroup of G ; if H ⊂ G is generated by elements fx1; x2; : : : ; xng then its image f(H) is generated by the images ff(x1); f(x2); : : : ; f(xn)g; the image of any cyclic subgroup is cyclic; If f is an isomorphism from G to G0 and x 2 G then the order of x (which|by definition—is the order of the cyclic subgroup generated by x) is equal to the order of f(x). For G a group, let Aut(G) be the set of all automorphisms of G, with `composition of automorphisms" as its `composition law' (denoted by a center-dot (·). As noted in class, Proposition 1 If G is a group then Aut(G) is also a group.
    [Show full text]
  • Oriented Steiner Quasigroups
    Oriented Steiner quasigroups Izabella Stuhl Abstract We introduce the notion of an oriented Steiner quasigroup and de- velop elements of a relevant algebraic apparatus. The approach is based upon (modified) Schreier-type f-extensions for quasigroups (cf. earlier works [14, 11, 10]) achieved through oriented Steiner triple sys- tems. This is done in a fashion similar to in [13] where an analogous construction was established for loops. As a justification of this con- cept briefly discuss an application of oriented Steiner triple systems in cryptography using oriented Steiner quasigroups. 1 Introduction A Steiner triple system S is an incidence structure consisting of points and blocks such that every two distinct points are contained in precisely one block and any block has precisely three points. A finite Steiner triple system with n points exists if and only if n ≡ 1 or 3 (mod 6). A possible way to enrich Steiner triple systems is to introduce in each block a cyclic order. These objects occur in the literature as oriented Steiner triple systems (S, T ) (see [16]) or as Mendelsohn triple systems (see [8] Section 28, p. 388). Steiner triple systems play an important role in applications, particulary in cryptography, coding theory, statistics and computer science. Cf. [3], [6] (where a number of chapters are dedicated to this concept), and also [5] and the references therein. A particular example of an application is given in [15]. The results of the present paper develop an algebraic background that arXiv:1402.4644v2 [math.GR] 22 Apr 2014 can be useful, e.g., in the context of [2].
    [Show full text]
  • Lecture 7.3: Ring Homomorphisms
    Lecture 7.3: Ring homomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 7.3: Ring homomorphisms Math 4120, Modern algebra 1 / 10 Motivation (spoilers!) Many of the big ideas from group homomorphisms carry over to ring homomorphisms. Group theory The quotient group G=N exists iff N is a normal subgroup. A homomorphism is a structure-preserving map: f (x ∗ y) = f (x) ∗ f (y). The kernel of a homomorphism is a normal subgroup: Ker φ E G. For every normal subgroup N E G, there is a natural quotient homomorphism φ: G ! G=N, φ(g) = gN. There are four standard isomorphism theorems for groups. Ring theory The quotient ring R=I exists iff I is a two-sided ideal. A homomorphism is a structure-preserving map: f (x + y) = f (x) + f (y) and f (xy) = f (x)f (y). The kernel of a homomorphism is a two-sided ideal: Ker φ E R. For every two-sided ideal I E R, there is a natural quotient homomorphism φ: R ! R=I , φ(r) = r + I . There are four standard isomorphism theorems for rings. M. Macauley (Clemson) Lecture 7.3: Ring homomorphisms Math 4120, Modern algebra 2 / 10 Ring homomorphisms Definition A ring homomorphism is a function f : R ! S satisfying f (x + y) = f (x) + f (y) and f (xy) = f (x)f (y) for all x; y 2 R: A ring isomorphism is a homomorphism that is bijective. The kernel f : R ! S is the set Ker f := fx 2 R : f (x) = 0g.
    [Show full text]