Everand Logo

Welcome to Everand!

  • Discover millions of ebooks, audiobooks, and so much more with a free trial

    From $11.99/month after trial. Cancel anytime.

    Groups, Rings, Modules
    Groups, Rings, Modules
    Groups, Rings, Modules
    Ebook908 pages10 hours

    Groups, Rings, Modules

    By Maurice Auslander and David Buchsbaum

    ()

    Read preview

    About this ebook

    This classic monograph is geared toward advanced undergraduates and graduate students. The treatment presupposes some familiarity with sets, groups, rings, and vector spaces.
    The four-part approach begins with examinations of sets and maps, monoids and groups, categories, and rings. The second part explores unique factorization domains, general module theory, semisimple rings and modules, and Artinian rings. Part three's topics include localization and tensor products, principal ideal domains, and applications of fundamental theorem. The fourth and final part covers algebraic field extensions and Dedekind domains. Exercises are provided at the end of each chapter.
    LanguageEnglish
    PublisherDover Publications
    Release dateJun 1, 2014
    ISBN9780486795423
    Groups, Rings, Modules

    Related to Groups, Rings, Modules

    Titles in the series (100)

    View More
    View More

    Related ebooks

    Mathematics For You

    View More
    View More

    Related podcast episodes

    Related articles

    • UNLIMITED

      Why Mathematicians Should Stop Naming Things After Each Other

      Sep 2, 2020

      Any student of modern math must know what it feels like to drown in a well of telescoping terminology. For a high-profile example, let’s take the Calabi-Yau manifold, made famous by string theory. A Calabi-Yau manifold is a compact, complex Kähler ma

      6 min read

    • UNLIMITED

      The Math That Takes Newton Into the Quantum World

      Feb 28, 2019

      In my 50s, too old to become a real expert, I have finally fallen in love with algebraic geometry. As the name suggests, this is the study of geometry using algebra. Around 1637, René Descartes laid the groundwork for this subject by taking a plane,

      9 min read

    • UNLIMITED

      Toolkit Lets Chemists ‘Mix-and-match’ For New Nanoparticles

      May 6, 2018

      Chemists have developed a designer’s toolkit that lets them build various levels of complexity into nanoparticles using a simple, mix-and-match process. “Researchers in areas as diverse as medicine, energy, and electronics often design complex nanosc

      2 min read

    • UNLIMITED

      Semiconductors for Moving Holograms Twist Like Fusilli

      Apr 11, 2017

      Scientists have discovered a way to mass-produce spiral semiconductors necessary for smartphone displays with moving, holographic 3D images. Researchers at the University of Michigan unveiled some of the first holographic images in 1962, which they m

      2 min read

    • UNLIMITED

      The Relativity Drive

      Feb 22, 2024

      The Relativity Drive
      3 min read

    • UNLIMITED

      Fanning The Flame

      Sep 13, 2023

      Fanning The Flame
      3 min read

    • UNLIMITED

      Lesser-known Relative Of The Laser Could Leave The Lab Soon

      Feb 9, 2018

      Researchers may have found a way to solve the weakness of a type of light source similar to lasers. The alternative light source could lead to smaller, lower-cost, and more efficient sources of light pulses. “Sometimes you completely reshape your und

      3 min read

    • UNLIMITED

      Introduction To Shading Theory

      Nov 22, 2022

      Artist Charlie Pickard LOCATION: England Charlie Pickard is a classically trained fine artist and illustrator. Recently awarded the Philip de László Award for Excellence, Charlie continues to work, exhibit and teach out of his studio based in Londo

      5 min read

    • UNLIMITED

      Cheaper Materials Get The Toxic Stuff Out Of Solar Cells

      Mar 15, 2018

      New research could lead to the replacement of toxic materials that work so well in solar cells. Any substitute for the lead-containing perovskites used in some solar cells would have to really perform. The perovskite semiconductors have been so promi

      2 min read

    • UNLIMITED

      Meet The Element: Crazy Carbon – The Good, The Bad And The Beautiful

      Feb 1, 2021

      Carbon is such an important element that it divided the entire discipline of chemistry into two fields: the study of compounds with carbon, called organic chemistry and compounds without carbon, called inorganic chemistry. There is carbon in the soil

      1 min read

    • UNLIMITED

      Your First Smart Scope Targets

      Nov 14, 2024

      A classic target within the Sword of Orion, the Orion Nebula is large, detailed and bright. The Andromeda Galaxy is a popular target, but smart scopes with smaller sky coverage will make you think about framing. A great introduction for those who wan

      1 min read

    • UNLIMITED

      Heavyweight Particle

      Jun 12, 2022

      Heavyweight Particle
      13 min read

    • UNLIMITED

      Seeing Through The Dust

      Jun 3, 2020

      Seeing Through The Dust
      5 min read

    • UNLIMITED

      Interactive

      Dec 17, 2020

      Email us at [email protected] Karen Hall @kazcustard • Nov 22 Things to do in a howling gale part 2: hide in front of the garage to try & get a non shaken picture of tonight’s #AuroraBorealis #Mirriedancers #northernlights #Shetland I teac

      4 min read

    • UNLIMITED

      Book Reviews

      Mar 5, 2020

      Book Reviews
      3 min read

    Related categories

    Reviews for Groups, Rings, Modules

    0 ratings

    0 ratings0 reviews

    What did you think?

    Tap to rate

    Review must be at least 10 words

      Book preview

      Groups, Rings, Modules - Maurice Auslander

      Index

      PREFACE

      The main thrust of this book is easily described. It is to introduce the reader who already has some familiarity with the basic notions of sets, groups, rings, and vector spaces to the study of rings by means of their module theory. This program is carried out in a systematic way for the classically important semisimple rings, principal ideal domains, and Dedekind domains. The proofs of the well-known basic properties of these traditionally important rings have been designed to emphasize general concepts and techniques. Hopefully this will give the reader a good introduction to the unifying methods currently being developed in ring theory.

      Part I is a potpourri of background material, much of which is undoubtedly familiar to the reader, some of which is probably new. In addition to the usual notions of sets, monoids, and groups, heavy emphasis is put on maps and morphisms of monoids and groups. This naturally leads to the notion of a category, which is briefly discussed in Chapter 3. In Chapter 4, the notions already developed for sets, monoids, and groups are applied to a preliminary discussion of the category of rings. Chapter 5 is far less formal. It is devoted to the study of unique factorization in arbitrary commutative domains. Here the principal novelties are the heavy use of localization in commutative domains and the introduction of chain conditions for ideals.

      Part Two begins with a lengthy discussion of modules over general rings. Starting from the notion of a basis for vector spaces, we develop free modules as well as the general notion of sums and products in the category of modules over a ring. Among other things, it is shown that a ring R is a division ring if and only if every R-module is free. This is the first step of our general program of studying rings by means of their modules. Although Chapter 6 is too long to describe in further detail, we caution the reader that familiarity with the contents of this chapter is essential to the understanding of the rest of the book.

      The remainder of Part Two is devoted to the next step of our program of studying rings by means of their modules. Namely, it is shown that a ring is semisimple if and only if its modules are semisimple. In this context, projective modules arise naturally. So, also, does the notion of the radical of a ring. Although this part of the book is devoted mainly to semisimple rings, some fundamental facts are developed for general artin rings in the text as well as in the exercises.

      The rest of the book is devoted almost exclusively to commutative rings. Since localization and tensor products play such important roles in this theory, Part Three starts with a discussion of these techniques. This is then followed by the study of principal ideal domains. These rings are characterized as commutative rings R with the property that submodules of free R-modules are free. Thus, they arise naturally as the next step in our program of studying rings by means of their modules. In describing the structure of finitely generated modules over principal ideal domains, injective modules are introduced. Part Three ends with applications of this structure theory to the study of endomorphisms of finite-dimensional vector spaces. Included are such standard items as canonical forms of matrices and determinants.

      The final part of the book is devoted to algebraic extensions of fields and the study of integral extensions of noetherian domains. The major aim of Chapter 12 is to develop finite galois theory of fields. This theory is used to study integral extensions of noetherian domains which leads to the theory of Dedekind domains. As part of our general module theoretic point of view, we characterize Dedekind domains as those integral domains having the property that submodules of projective modules are projective. The book ends with a description of the ideals in Dedekind domains and the structure theorem for finitely generated modules over such domains.

      We recommend that the reader have pencil and paper close at hand when reading the text. Proofs for many assertions have been omitted. The reader will be able to supply the missing steps or proofs either by himself or after consulting outlines given in the exercises. In addition to exercises explaining the text, there are exercises dealing with related but supplementary material.

      The partitioning of the book was done on pedagogical as well as logical grounds. Part One can be used for a leisurely one-semester course on the fundamental structures of algebra. Parts Two and Three can serve as a one-semester introduction to general ring theory for more advanced students. For students familiar with Chapters 1, 2, and 4, the entire text should constitute a full year course in algebra.

      We thank our publishers, Harper & Row, for their patience during the preparation of the manuscript.

      M. A.

      D. A. B.

      PART

      ONE

      Chapter 1 SETS AND MAPS

      INTRODUCTION

      This chapter and the next are devoted to a review of the basic concepts of set and group theory. Because we are assuming the reader already has some familiarity with these topics, our exposition is neither systematic nor complete. Only a brief description of the basic concepts and results that are needed in the rest of this book is presented.

      This should serve to give the reader some idea of the mathematical background we are assuming as well as help fix conventions and notations for the rest of the book. Although few proofs are given, outlines of proofs of the less obvious results cited in the text are given in the exercises. It is hoped that the reader will find completing these outlines a useful way of familiarizing himself with any new concepts or results he may encounter in this or the next chapter.

      1. SETS AND SUBSETS

      We take a naive, nonaxiomatic view of set theory. We view a set as an actual collection of things called the elements of the set. We will often denote the fact that x is an element of the set X by writing x X. From this point of view it is obvious that two sets are the same if and only if they have the same elements. Or stated more precisely, two sets X and Y are the same if and only if both of the following statements are true:

      (a) If x X, then x Y.

      (b) If y Y, then y X.

      In this connection, we remind the reader that in mathematical usage, a statement of the form "If A, then B" is true unless A is true and B is false, in which case it is false. In particular, if A is false, then the statement "If A, then B" is true independent of whether B is true or false. To illustrate this point we show that there is only one empty set.

      We recall that a set X is said to be empty if X has no elements; or more precisely, if the statement "x X" is always false. Suppose now that the sets X and Y are empty. Then both of the statements "x X and y Y are always false. Hence, by our convention concerning sentences of the form If A, then B," both of the statements

      (a) If x X, then x Y;

      (b) If y Y, then y X;

      are true. This shows that if the sets X and Y are both empty, then X .

      An important set associated with a set X is the power set 2x of X which we will define once we have recalled the notion of a subset of a set.

      A set Y is said to be a subset of a set X if every element of Y is also an element of X, or equivalently, the set Y is a subset of the set X if and only if the statement "If y Y, then y X" is true. The fact that Y is a subset of X is often denoted by Y ⊂ X, which is sometimes also read as "Y is contained in X."

      One easily verified consequence of this definition is that if X is a subset of X. For the statement "If x , then x X," is true for any set X because the statement "x " is always false. Also associated with an element x of X is the subset {x} of X consisting precisely of the element x of X. Further, the reader should have no difficulty verifying the following.

      Basic Properties 1.1

      Let X, Y, and Z be sets. Then:

      (a) X ⊂ X.

      (b) X = Y if and only if X ⊂ Y and Y ⊂ X.

      (c) If X ⊂ Y and Y ⊂ Z, then X ⊂ Z.

      We are now in a position to define the power set 2x of a set X. The set 2x is the set whose elements are precisely the subsets of X. Stated symbolically, the power set 2x of a set X is the set with the property that Y 2x if and only if Y ⊂ X.

      It is worth noting that 2x is never empty, even if X is always contained in X and is thus an element of 2x. Also, as we have already observed, there is associated with each element x of X the element {x} of 2x. Hence, 2x consists of a single element if and only if X is empty.

      We now recall the familiar notions of union and intersection of sets. Suppose X a subset of 2x. The intersection of the subsets of X of X consisting of all x in X such that the statement "If X' , then x Xof 2x = Xis empty, then the statement "If X' , then x X'" is true for all x in X since the statement "X' is false.

      The union of the subsets of X of X consisting of all x in X with the property that the statement "There is an X' such that x X'" is empty, then the statement "There is an X' such that x X'" is false for all x X since there are no X' in 2x satisfying the condition that X' .

      In practice, a particularly useful way of studying a set is to represent it as a union of some of its subsets. For this reason it is convenient to make the following definition.

      Definition

      Suppose X of 2x is called a covering of X if X .

      Although coverings of various types play an important role in all of mathematics, we will be particularly concerned with the type of coverings called partitions.

      Definition

      of a set X is said to be a partition of X provided:

      (a) If X' , then X' .

      (b) If X' and X" , then X' ∩ X" .

      of nonempty subsets of a set X is a partition of X if and only if each element in X is in one and only one subset of X is a partition of a set Xcontaining a particular element x of Xof X containing the element x of Xof a set X.

      Finally, we recall what is meant by the product X × Y of two sets X and Y. The set X × Y consists of all symbols (x, y) with x an element of X and y an element of Y. Hence, two elements (x, y) and (x', y’) in X × Y are the same if and only if x = X' and y = y’. Obviously, X × Y is empty if and only if either X or Y is empty.

      2. MAPS

      A map of sets consists of three things: a set X called the domain of the map, a set Y called the range of the map, and a subset f of X × Y having the property that if x is in X, then there is a unique y in Y such that the element (x, y) in X × Y is in f. These data X, Y, f will be denoted by f: X → Y which is to be read as "f is a map from X to Y." If x is in X, then the unique element y in Y such that (x, y) is in f is called the value of the map f at x and is denoted by f(x).

      It is important to observe that according to this definition two maps cannot be the same unless they have the same domain and range. Also, two maps f : X→Y and g : X→Y with the same domains and ranges are the same if and only if their values are the same for each x in X, that is, if and only if f(x) = g(x) for all x in X. Thus, once having specified the domain and range of a map, it only remains to describe its values for each x in X in order to completely determine the map. In the future, when defining particular maps from a set X to a set Y, we shall generally describe them by prescribing their values for each x in X rather than by writing down a subset of X × Yis a partition of a set X. Then we have already seen that for each x in X containing xas the subset of X × T in X with x in X.

      We now describe some important maps of sets.

      Example 2.1 Suppose f:X→Y is a map and X' is a subset of X. We define a map f|X' : X' →Y called the restriction of f to X' by (f|X')(X') = f(x') for all x' in X'

      Example 2.2 Associated with each subset X' of a set X is the inclusion map from X' to X which is denoted by inc : X'→X and is defined by inc(x) = x if x is an element of X which is in X'.

      Example 2.3 The inclusion map of a set X to itself is called the identity map and is usually denoted by idx for each set X.

      Example 2.4 is a subset of any set X→Xto X to a set X is called the empty map. In this connection, the reader should convince himself that there are no maps from a nonempty set to the empty set.

      Example 2.5 of a set X for each x in X where [x] is the unique subset of X containing the element xis called the canonical or natural map from the set X .

      Suppose X and Y are sets. Then each map with domain X and range Y is completely determined by a subset of X × Y and hence by an element of 2X×Y Thus, the collection of all maps from X to Y which we denote by (X, Y) is a set which is a subset of 2X×Y.

      Of fundamental importance in constructing and analyzing maps is the notion of the composition of maps. Given two maps f: X→Y and g: Y→Z with the range of f the same as the domain of g, we define their composition gf to be the map gf: X→Z given by gf(x)= g(f(x)) for each x in X. It follows immediately from this definition that if we are given three maps f : U→X, g : X→Y, and h: Y→Z, then the two maps h(gf):U→Z and (hg)f: U→Z are the same. This property of the composition of maps is referred to as the associativity of the composition of maps.

      As an example of the composition of maps we point out that if f : X → Y is a map of sets and X' is a subset of X, then f|X' : X' →0 Y, the restriction of f to X'where inc : X' → X is the inclusion map.

      3. ISOMORPHISMS OF SETS

      One of the most important problems in mathematics is deciding when two mathematical objects have the same or similar mathematical properties and can therefore be considered essentially the same. Since all the mathematical objects we will be considering in this book consist of an underlying set together with some additional structure, it is reasonable to first consider how sets are compared and the circumstances under which they are considered essentially the same.

      Because a map from a set X to a set Y associates with each element x in X an element y in Y, a map clearly can be viewed as a method for comparing the sets X and Y. If this is a reasonable idea, then we should be able to state in terms of maps what is probably the simplest comparison of sets we can make: the fact that a set is the same as itself. The reader should have no difficulty convincing himself that the identity map on a set does indeed express this fact. It is interesting to note that the identity map on a set can be completely described in terms of maps as is done in the following.

      Basic Property 3.1

      For a map f: X → X, the following statements are equivalent:

      (a) f = idx.

      (b) Given any map g : X → Y, then gf = g.

      (c) Given any map h : Y → X, then fh = h.

      Having decided that the identity map expresses the fact that a set is the same as itself, it is reasonable to ask what kind of maps between two sets X and Y must exist in order to conclude that X and Y resemble each other as much as possible. In view of our previous discussion, this amounts to asking when is a map f : X → Y close to being an identity map? A possible answer might be that there is a map g : Y → X such that the composition gf : X → X is the identity on X. But there is no reason to favor the set X over the set Y. Hence, we should also require that there be a map h : Y → X such that fh = idY. However, the associativity of the composition of maps implies that under these circumstances the two maps g and h are the same. Therefore, it seems reasonable to consider two sets X and Y as being essentially the same if there exists a pair of maps f : X → Y and g : Y → X such that gf = idX and fg = idY. In fact, this amounts to nothing more than the familiar notion of two sets being isomorphic, as we see in the following.

      Definition

      Let X and Y be sets. A map f : X → Y is said to be an isomorphism if and only if there is a map g : Y → X such that gf = idx and fg = idy. If f : X → Y is an isomorphism, then there is only one map g : Y → X with these properties, and this uniquely determined map from Y to X, which we denote by f−1 is called the inverse of f. Finally, the set X is said to be isomorphic to Y if there is a map f : X → Y which is an isomorphism.

      We remind the reader of the following.

      Basic Properties 3.2

      (a) All identity maps are isomorphisms which are their own inverses. Hence, all sets are isomorphic to themselves.

      (b) If f : X → Y is an isomorphism, then the inverse f−1 : Y → X is also an isomorphism whose inverse is f, that is, (f−1)−1 = f. Hence, if X is isomorphic to Y, then Y is isomorphic to X.

      (c) The composition gf of two isomorphisms g and f is also an isomorphism with inverse f−1 g−1. Thus, if X is isomorphic to Y and Y is isomorphic to Z, then X is isomorphic to Z.

      (d) If gf is an isomorphism, then g is an isomorphism if and only if f is an isomorphism.

      Experience has shown, roughly speaking, that a map f : X → Y is an isomorphism if and only if it gives a way of identifying the set X with the set Y. A precise formulation of this idea is given in the following familiar characterization of isomorphisms.

      Basic Property 3.3

      A map f : X → Y is an isomorphism if and only if it satisfies both of the following conditions:

      (a) If y Y, then there is an x in X such that f(x) = y.

      (b) If x1 and x2 are in X and f(x1) = f(x2), then x1 = x2.

      Although technically equivalent to the notion of an isomorphism, the conditions (a) and (b) of the above basic property are conceptually quite different from our original definition of an isomorphism since these conditions describe what the map does to the elements of the sets involved rather than how it is related to other maps. We will often refer to an isomorphism as a bijective map when we wish to emphasize this different approach to the concept of an isomorphism of sets.

      4. EPIMORPHISMS AND MONOMORPHISMS

      Yet another aspect of the notion of an isomorphism of sets is given in the following.

      Basic Property 4.1

      A map f: X → Y which is an isomorphism satisfies the following conditions:

      (a) If g1, g2: Y → Z are two maps such that g1f = g2f, then g1 = g2.

      (b) If h1, h2: U → C are two maps such that fh1 = fh2, then h1 = h2.

      It turns out that there are many important maps which satisfy one but not necessarily both of the above conditions. For this reason we make the following definitions.

      Definitions

      Let f : X → Y be a map.

      (a) f is called an epimorphism if given two maps g1, g2: Y → Z, we have g1 = g2 whenever g1f = g2f.

      (b) f is called a monomorphism if given two maps h1, h2: U → X, we have h1 = h2 whenever fh1 = fh2.

      Thus, if a map is an isomorphism, it is both an epimorphism and a monomorphism.

      We now list some easily verified properties of epimorphisms and mono-morphisms.

      Basic Properties 4.2

      Let f : X → Y and g : Y → Z be two maps.

      (a) If f and g are both epimorphisms (monomorphisms), then the composition gf : X → Z is an epimorphism (monomorphism).

      (b) If gf : X → Z is an epimorphism, then so is g.

      (c) If gf : X → Z is a monomorphism, then so is f.

      We have already seen how to describe in terms of what a map does to elements the fact that it is an isomorphism. The same can be done for the notions of epimorphisms and monomorphisms. In order to state this result, it is convenient to have the following.

      Definitions

      Let f : X → Y be a map.

      (a) f is said to be a surjection, or a surjective map, if for each y in Y there is an x in X such that f(x) = y.

      (b) f is said to be an injection, or an injective map, if given x1 and x2 in X with the property that f(x1) = f(x2), then x1 = x2.

      Basic Properties 4.3

      (a) A map is an epimorphism if and only if it is a surjective map.

      (b) A map is a monomorphism if and only if it is an injective map.

      (c) A map is an isomorphism if and only if it is an epimorphism and a monomorphism.

      As with isomorphisms, we will refer to an epimorphism (monomorphism) as a surjective map (injective map) whenever we wish to emphasize what the map does to the elements of the sets involved rather than its relation to other maps.

      We conclude this section with the following useful property of surjective and injective maps.

      Proposition 4.4

      Suppose we are given a diagram of maps of sets

      satisfying:

      (a) ts = hg.

      (b) s is a surjective map and h is an injective map.

      Then there is one and only one map j: U → Z such that is js = g and hj = t.

      5. THE IMAGE ANALYSIS OF A MAP

      A map f : X → Y of sets not only serves as a way of comparing the sets X and Y, but also as a way of comparing subsets of X and subsets of Y. In the following definitions we point out some of these relationships. Others will be discussed later on.

      Definitions

      Suppose f : X → Y is a map of sets. If X' is a subset of X, then the subset of Y consisting of all elements f(x) in Y with x in X' is called the image of X' under f and is denoted by f(X'). The subset f(X) of Y is called the image of the map f and is usually denoted by Im f.

      Suppose we are given a map f : X → Y. It is clear that f is a surjective map, or equivalently an epimorphism, if and only if Im f = Y. However, regardless of whether the map f itself is surjective, the map f0: X → Im f, defined by f0(x) = f(x) for all x in X is always surjective. Hence, associated with each map f: X → Y is the surjective map f0: X → Im f.

      The importance of the map f0 lies in the fact that it completely determines the map f if we assume that we know the range of f. For it is easily checked that the map f is the composition

      where inc : Im f → Y is the inclusion map of the subset Im f of Y. This representation of a map f : X → Y as the composition inc fo is called the image analysis of the map f.

      Although we have already pointed out that the map f0 is always a surjective map, it is equally important to observe that all inclusion maps are injective maps, or equivalently monomorphisms. The image analysis of a map therefore shows that every map can be written as the composition of a surjective map followed by an injective map. Because the representation of a map as the composition of a surjective map followed by an injective map plays a critical role in analyzing maps, we make the following definition.

      Definition

      Let f: X → Y be a map. By an analysis of f we mean a set A together with a surjective map g : X → A and an injective map h: A → Y such that f = hg.

      We end this preliminary discussion of the analysis of a map by pointing out that all analyses of a map are essentially the same. Precisely, we have the following.

      Basic Property 5.1

      Suppose

      and

      are both analyses of the map f: X → Y. Then there is one and only one map j : A → A' such that jg = g' and h' j = h, and this uniquely determined map j: A → A' is an isomorphism.

      6. THE COIMAGE ANALYSIS OF A MAP

      Another important standard analysis of a map is the coimage analysis. Before describing this analysis, it is convenient to have the following.

      Definitions

      Suppose f : X → Y is a map. If Y' is a subset of Y, then the set of all x in X with the property f(x) is in Y' is called the preimage of Y' under f and is denoted by f→1 (Y'). If y is an element of Y, we write f−1(y) for f→1({y}).

      Suppose we are given a map f : X → Y. Then it is not difficult to show that the subset Coim f of 2x consisting of all subsets of X of the form f→1(y) with y in Im f is a partition of X which we call the coimage of f. a set X for all x in Xcontaining xis a surjective map. In particular, the map kCoim f :X → Coim f is surjective.

      The map kCoim f : X → Coim f has another important property: There is a unique map jf: Coim f → Y such that f = jfkCoim f. We first show that such a map exists. Suppose the subset X' of X is an element of Coim f. Then by definition there is a y in Im f such that f−1(y)= X'. Hence, f(X')= {y}. Define the map jf: Coim f → Y by jf(X') is the unique element y of Y such that f(X') = {yfor all x in X which shows that the map jf:Coim f → Y does indeed have the property f = jfkCoim f. That there is only one such map from Coim f → Y follows from the fact that KCoim f is surjective and hence an epimorphism. We leave it to the reader to verify that jf:Coim f → Y is also an injective map with Im jf = Im f and so (jf)0: Coim f → Im j is an isomorphism from Coim f to Im f. The map jf: Coim f → Y is called the map from Coim f to Y induced by the map f : X → Y.

      is an analysis of the map f. This analysis is called the coimage analysis of the map f. Further, because

      are both analyses of f, we know that there is a unique isomorphism g : Coim f → Im f such that gkCoim f = f0 and inc g = jf. The map g is easily seen to be the isomorphism (jf)0:Coim f → Im f. This isomorphism is called the canonical isomorphism from Coim f to Im f.

      7. DESCRIPTION OF SURJECTIVE MAPS

      Suppose f : X → Y of f has the property that f is a surjective map if and only if jf: Coim f → Y is a surjective map and hence an isomorphism. This certainly suggests that when f : X → Y is a surjective map, the maps kCoim f: X → Coim f and f : X → Y are intimately connected. It is precisely this connection that we describe in this section. We begin with the following.

      Definition

      be two partitions on a set Xis a refinement if given a subset X' of X , there is a subset X" of X such that X"⊃X'.

      , then given any subset X' of X , not only is there a subset X" of X containing X', but there is only one such subset of X , for each element X' containing the element X' is called the canonical map from , to .

      , such that the diagram

      of X . In summary, we have the following.

      Basic Properties 7.1

      be partitions of a set X. Then:

      .

      .

      We can now state the main results concerning the connections between arbitrary surjective maps f : X → Y and their associated surjective maps kcoim f: X → Coim f.

      Proposition 7.2

      Let f1 : X → Y1 and f2 : X → Y2 be two surjective maps.

      (a) The following statements are equivalent:

      (i) There is a map h: Y1 → Y2 such that hf1 = f2.

      (ii) There is a map g :Coim f1 Coim f2 such that gkCoim f1 = kCoim f2.

      (iii) Coim f1 ≥ Coim f2.

      (b) If there is a map h :Y1 → Y2 such that hf1 = f2, then:

      (i) There is only one such map.

      (ii) There is only one map g : Coim f1 → Coim f

      (e) The following are equivalent:

      (i) There is an isomorphism h :Y1 → Y2 such that hf1 = f2.

      (ii) Coim f1 = Coim f2.

      (d) If Y is a partition of X and f: X → Y is the canonical map, then Coim f = Y and f = kCoim f.

      Roughly speaking, this proposition says that all surjective maps f : X → Y with a fixed domain X are essentially given by the canonical maps k : X of X. Hence, it is of considerable importance to know how to create partitions of a set X One of the most widely used devices for accomplishing this is known as an equivalence relation, a notion we discuss in the next section. Before doing this we point out the following generalization of some of our results to date concerning surjective maps.

      Proposition 7.3

      Let f : X → Y be a surjective map of sets. If g : X → Z is a map of sets, then there exists a map h : Y → Z such that hf = g if and only if Coim f ≥ g. If Coim f ≥ Coim g, then there is only one map h : Y → Z such that hf = g.

      8. EQUIVALENCE RELATIONS

      By definition a relation R on a set X is simply any subset of X × X. We usually denote the fact that an element(x1, x2) in X × X is in the relation R of X by writing x1 R x2. If R is a relation on a set X and X' is a subset of X, then we denote by R|X' the relation on X' given by R ∩ (X' × Xare in XR|XR. The relation R|X' on X' is called the relation on X' induced by R.

      Definition

      A relation R on a set X is called an equivalence relation if it satisfies the following conditions:

      (a) x R x holds for all x in X.

      (b) If x1 R x2 holds, then x2 R x1 also holds.

      (c) If x1 R x2 and x2 R x3 hold for x1, x2, and x3 in X, then x1 R x3 also holds.

      We now describe how to associate with each equivalence relation R on a set X a partition X/R of X. For each element x in X, denote by [x]R the subset of X consisting of all elements x' in X such that x R x' holds. Let X/R be the subset of 2x consisting of the subsets [x]R of X as x ranges through all elements of X. Then it is not difficult to show that X/R is a partition of X with the property [x]x/R = [x]R for each x in X. Hence, one way to create a partition on a set X is to start out with an equivalence relation R on the set X and construct the partition X/R of X.

      on X). Namely, for x1 and x2 in X define x)x2 if and only if there is a subset X' of X such that x1 and x2 are both in X) we just defined is actually an equivalence relation.

      Moreover, it is equally easy to see that if R is an equivalence relation on a set X, then R(Xis a partition on a set X, then X. This description of how to go back and forth between equivalence relations and partitions of a set shows that these are really interchangeable notions, a fact that we shall use freely from now on.

      To illustrate this point, the reader should check the validity of the following proposition.

      Proposition 8.1

      Let R and R' be two equivalence relations on a set X. Then X/R ≥ X/R' if and only if x1 R x2 implies x1 R' x2 for all x1 and x2 in X.

      This suggests the following definition.

      Definition

      Let R and R' be two relations on a set X. Then R ≤ R' if and only if x1 R x2 implies x1 R' x2; or, equivalently, R ⊂ R'.

      It should be noted that if R and R' are equivalence relations on a set X, then R ≤ R' if and only if X/R ≥ X/R'.

      As our final example of the correspondence between the partitions and equivalence relations on a set, we point out that if f : X → Y is a map, then the equivalence relation Ron X corresponding to the partition Coim f of X is given by x1 R x2 if and only if f(x1)= f(x2). This equivalence relation is called the equivalence relation associated with f and is sometimes denoted by R(f).

      9. CARDINALITY OF SETS

      One of the earliest and most important mathematical processes one learns is that of counting; and one of the basic problems in counting is to determine when two sets of things have the same number of objects. This is usually done by showing that the objects in one collection can be matched up with the objects in the other collection. But the matching up of the objects in a set X with the objects in a set Y is nothing more than a bijective map from X to Y. This leads us to say that an arbitrary set X has the same number of elements as a set Y if and only if there is a bijective map f : X → Y. Obviously, a set X has the same number of elements as a set Y if and only if the set Y has the same number of elements as X. The fact that two sets X and Y have the same number of elements is often denoted by card(X) = card(Y) where card(X) is read as the cardinality of X. Or stated slightly differently, two sets X and Y have the same cardinality (that is, the same number of elements) if and only if they are isomorphic sets.

      In addition to knowing when two sets have the same number of elements, it is also important to know when one set Y has at least as many elements as another set X. A little thought should convince the reader that in usual practice this simply means that there is injective map f : X → Y. This observation leads us to define the cardinality of an arbitrary set X as being less than or equal to the cardinality of a set Y; symbolically, card(X) ≤ card(Y) if and only if there is an injective map f : X → Y.

      If this definition of card(X) ≤ card(Y) really corresponds to the notion that the set X has at most as many elements as the set Y, then it should have the following properties:

      (a) card(X) ≤ card(X) for all sets X.

      (b) If card(X) ≤ card(Y) and card(Y) ≤ card(X), then card(X) = card(Y).

      (c) If card(X) ≤ card(Y) and card(Y): ≤ card(Z), then card(X) ≤ card(Z).

      It is trivial to verify that (a) and (c) are true. The fact that (b) is true is less obvious and is equivalent to the following well-known Bernstein-Schroeder Theorem.

      Theorem 9.1

      Suppose X and Y are two sets and f : X → Y and g : Y → X are injective maps. Then X and Y are isomorphic sets.

      An outline of a proof of this theorem is given in the exercises for the convenience of those readers not familiar with this result.

      There are two more properties of the cardinality of sets that one might expect to be true judging from one’s ordinary experience with counting. Namely, (1) given a set X there is a set Y such that Y has actually more elements than X, that is, card(X) ≤ card(Y) but card(X) ≠ card(Y); or, more simply, card(X) < card(Y) and (2) if X and Y are sets, then either card(X) ≤ card(Y) or card(Y) ≤ card(X).

      The fact that given any set X there is a set Y such that card(X) < card(Y) follows from the following proposition, a proof of which is outlined in the exercises.

      Proposition 9.2

      Let X be any set. Then there is no surjective map from X to 2x.

      Hence, X and 2x are never isomorphic sets, which means card(X)≠card(2x). On the other hand, the map f: X → 2x given by f(x)= {x} for each x in X is clearly an injective map. Therefore, card(X) ≤ card(2x) which implies that card(X) < card(2x) for each set X. Thus, given any set X there is a set Y such that card(X) < card(Y), which settles the first question raised.

      However, the second question, whether given two sets X and Y, either card(X) ≤ card(Y) or card(Y) ≤ card(X), is much more complicated. In fact, it cannot be settled except by the introduction of a notion of set theory which we have not discussed at all; namely, the axiom of choice. Therefore, we shall return to this second question in a later section after we have discussed this axiom of set theory.

      10. ORDERED SETS

      There are various equivalent forms of the axiom of choice. We shall be concerned with only three of them: the existence of choice functions, the well-ordering axiom, and Zorn’s lemma. Because all but the first of these forms of the axiom of choice use the notion of an ordered set in their formulation, we shall begin this discussion with the notion of an ordered set.

      Definition

      A relation R on a set X is said to be an order relation on X or an ordering of X, if it satisfies:

      (a) x R x for all x in X.

      (b) If x1 R x2 and x2 R x1, then x1 = x2.

      (c) If x1 R x2 and x2 R x3, then x1 R x3.

      An ordering R of X is called a total ordering of X if it also satisfies:

      (d) If x1 and x2 are in X, then either x1 R x2 or x2 R x1.

      Finally, a set X together with an ordering R (total ordering R) is called an ordered set (totally ordered set).

      The reader should observe that if the relation R on a set X is an order relation and X' is a subset of X, then the relation R|X' is an order relation on X' called the induced ordering on X'. Unless stated explicitly to the contrary, if X' is a subset of an ordered set X, we always consider X' an ordered set under the induced ordering. Obviously, if X is a totally ordered set so is X' for each subset X' of X.

      When there is no danger of confusion concerning which ordering we mean, we shall follow the usual practice of writing x1 ≤ x2 for x1 R x2 when R is an order relation on the set X.

      We now offer as examples certain ordered sets that will be occurring frequently in the rest of the book.

      Example 10.1 Suppose X is a set. It is easy to check that the relation R on 2x given by X' R X" if and only if X' ⊂ X" is an order relation. This is the only order relation we shall ever consider on the set 2x. Hence, when we consider 2x an ordered set it is always with respect to this ordering. The reader should observe that 2x is a totally ordered set if and only if X has at most one element.

      The next example, which is closely related to our first one, is extremely useful in constructing maps, as we shall see later on.

      Example 10.2 Suppose we are given two sets X and Y(X, Y) consist of all triples (X', Y', f) where X' and Y' are subsets of X and Y, respectively, and f is a map from X' to Y(X, Y) given by (X', Y', f') R (X", Y", f") if and only if X' ⊂ X", Y' ⊂ Y", and f"(x) = f'(x) for all x in X(X, Y(X, Y) as an ordered set, it is always with respect to this ordering.

      Finally, we have the following familiar ordered sets.

      Example 10.3 All of the following sets with their usual ordering are totally ordered sets:

      (a) The set N of all nonnegative integers, that is, all integers n ≥ 0.

      (b) Z, the set of all integers.

      (c) Q, the set of all rational numbers.

      (d) R, the set of all real numbers.

      Now that we have defined the notion of an ordered set we can start discussing the axiom of choice.

      11. AXIOM OF CHOICE

      That every set X has a choice function is perhaps the simplest and most appealing form of the axiom of choice. What this amounts to saying is that given any nonempty collection of nonempty subsets of a set X, it is possible to choose an element out of each one. Although this seems self-evident, it nonetheless cannot be proven on the basis of the types of manipulations of sets we have permitted ourselves until now. Formulated somewhat more precisely, this assertion becomes the following.

      Axiom of Choice 1

      Given any set X, there is a map c } → X } is the set of all nonempty subsets of X) such that c(XX' for all nonempty subsets X' of X. Such a map c is called a choice function on the set X.

      As an illustration of how this form of the axiom of choice is used, we prove the following proposition.

      Proposition 11.1

      Let f : X → Y be a surjective map of sets. Then there is a map g : Y → X such that fg = idY.

      PROOF: If Y , then X is empty and f is an isomorphism so there is nothing to prove.

      Suppose now that Yand c :2x} → X is a choice function on X. Then define g : Y → X by g(y) = c(f−1(y)) for each y in Y. Since c(f−1(y)) is in r(y), it follows that f(c(f−1(y)))= y for all y in Y. Therefore, the map g: Y → X has the property fg =idY.

      In order to state the next form of the axiom of choice that interests us, it is necessary to recall the definition of a well-ordered set.

      Definition

      An ordered set X is said to be well ordered if

      (a) X is totally ordered.

      (b) If X' is a nonempty subset of X, then there is an element xo in X', called the first element of X', having the property x0 ≤ x for all x in X'.

      It is important to note that if X is a well-ordered set, then the first element of a nonempty subset X' of X is uniquely determined. For if xare both first elements in X', then xx0 which means that x.

      Axiom of Choice 2

      If X is a set, then there is an ordering on X which makes X a well-ordered set.

      As stated earlier, these two forms of the axiom of choice that have been given are equivalent. Although it is certainly not trivial to show that the assumption that every set has a choice function implies that every set can be well ordered (see the exercises for a discussion of this point), the reverse implication is quite simple to establish as we now show.

      Proposition 11.2

      Let X be a well-ordered set. Then X has a choice function.

      PROOF: Since the first element of any nonempty subset X' of X is a uniquely determined element of X', we obtain a choice function c on X by defining the map c :2X } → X as follows: c(X') is the first element of X' for each nonempty subset X' of X.

      As a check on his understanding of well-ordered sets the reader should convince himself that while the set N of nonnegative integers is a well-ordered set, neither the integers, rational numbers, nor real numbers is a well-ordered set even though each of them is totally ordered.

      We now turn our attention to the third and final form of the axiom of choice which is of concern to us, namely, Zorn’s lemma. Although this form of the axiom of choice is much more technical and therefore has less intuitive appeal than the others, it has the advantage of being the easiest to apply in most situations of interest to us.

      Before stating Zorn’s lemma we review the notion of an inductive set.

      Definition

      An ordered set X is said to be an inductive set if every nonempty totally ordered subset X' of X has an upper bound in X. That is, for each nonempty totally ordered X' of X there is an element x in X such that x ≥ x' for all elements X' in X'

      To help clarify this definition we give some important examples of inductive sets.

      Example 11.3 If X is a set, then the ordered set 2x is an inductive set.

      PROOF: is any subset of 2x . Then it is obvious that Y is an element of 2x in the sense that Y ⊃ X' for all Xof 2x has an upper bound in 2x.

      Example 11.4 Let X and Y (X, Y) is an inductive set.

      PROOF: (X, Y. Let X

      is totally ordered there is a map f0 : X0 → Y0 such that (X0, Y0, f. For suppose x0 is in X0, then by the definition of Xsuch that xwith x, for otherwise the value f0(xis totally ordered. For we know that either

      for all x

      Hence, we have shown that there is a map f0 : X0 → Y0. It is not hard to show now that (X0, Y0, fsince (X0, Y0, f0) ≤ (X', Y', f') for all (X', Y', f.

      The form of the axiom of choice known as Zorn’s lemma is simply the following.

      Axiom of Choice 3

      If X is an inductive set, then there is an element x0 in X such that if x is in X and x → x0, then x = x0. Such an element x of X is called a maximal element of X.

      As an illustration of how Zorn’s lemma is used, we finally give the much delayed proof that if X and Y are two sets, then either card(X) ≤ card(Y) or card(Y) ≤ card(X).

      Proposition 11.5

      If X and Y are sets, then there is either an injective map from X to Y or from Y to X.

      PROOF: Let Inj(X, Y(X, Y) consisting of all triples (X', Y', f') with the property that f' : X' → Y' is an injective map. Using the same type of argument as in Example 11.4, it is not difficult to see that Inj(X, Y) is an inductive set.

      Because Inj(X, Y) is an inductive set we know by Zorn’s lemma that there is a maximal element (X', Y', f') in Inj(X, Y). This maximal element (X', Y', f') has the property that either X' = X or Y' = Y. For suppose there is an x0 in X but not in X' and an element y0 in Y but not in Y'. Then the map g : X' ∪ {x0} → Y' ∪ {y0} defined by g(x) = f'(x) for x in X' and g(x0) = y0 is injective. Therefore, (X' ∪ Y' ∪ {y0}, g) is an element of Inj(X, Y) with the property that (X', Y', f') < (X' ∪ {x0}, Y' ∪ {y0}, g). This contradicts the fact that (X', Y', f') is a maximal element of Inj(X, Y). Therefore, our contention that either X' = X or Y' = Y has been established.

      If X' = Xof injective maps is an injective map from X to Y.

      On the other hand, if Y' = Y, we can define the map g : Y → X by letting g(y) be the unique element in X such that f'(g(y)) = y. It is obvious that g is an injective map, and so in this case we obtain an injective map from Y to X. This completes the proof of the proposition.

      We finish this discussion of the axiom of choice by pointing out that in the exercises there is an outline of a proof of our repeated assertions that the various conditions Axiom of Choice 1, 2, and 3 are equivalent. From now on we will make free use of these forms of the axiom of choice, especially Zorn’s lemma.

      12. PRODUCTS AND SUMS OF SETS

      In discussing the product of sets, it is convenient to have the notion of an indexed family of subsets of a fixed set.

      Definition

      A family of subsets of a set X indexed by a set I is a map ψ : I → 2x. The set I is called the indexing set and the subset ψ(i) of X is usually denoted by Xi. In practice one denotes the map ψ by {Xi}i I

      In connection with this definition we observe that associated with an indexed family {Xi}i I of subsets of a set X of 2x consisting of all subsets X' of X such that X' = Xi for some i in I. The reader should construct examples of different indexed families of subsets of a fixed set X of 2x.

      Definition

      The product of an indexed family {Xi}i I of subsets of a set X is the set of all maps f : I → X such that f(iXi, for all i in IX.

      of an indexed set {Xi}i I of subsets of X. If f then the element f(i) in Xi is denoted by x1 and the element f is denoted by {xi}i I where x1 = f(i) for all i in I.

      If {Xi}i I is an

      Enjoying the preview?
      Page 1 of 1