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9.15 Quotient Rings 339 this, however, we need to create the notion of a quotient ring, which we will do in the next section. 9.15 Quotient Rings With a group and a normal subgroup we can build a quotient group. In an analogous way, given a ring and an ideal, we can build a quotient ring. Part of the work in building a quotient ring is exactly like building a quotient group, so the fact that a quotient ring has properties R1–R10 has already been done in part. As with groups, building the quotient structure begins by defining a form of equivalence. Theorem 9.15.1 Let R be a ring and I an ideal of R. For ring elements a and b, define a ≡I b if a − b ∈ I. Then ≡I is an equivalence relation on R. Since R is an abelian group with respect to its addition operation, I is a normal additive subgroup of R. Therefore, Theorem 9.15.1 is merely a restatement of Exercise 8.3.2 in its additive form and needs no additional proof. So we are ready to define the set R/I with its addition and multiplication operations, and show that it has all properties R1–R10. There are no surprises in the definitions of the binary operations on R/I. But considering the need for H to be a normal subgroup of G in showing the binary operation on G/H is well defined, little bells should be going off in your head as you consider the burden of showing that addition and multiplication on R/I are well defined. It should be no surprise that you will exploit the fact that I is an ideal of R in at least part of this demonstration. What you might not see at first is what kind of ideal I needs to be and where you will call on the fact that I is this kind of ideal. Since addition in R is commutative, I is normal as an additive subgroup of R. Thus addition is well defined by our work in Chapter 8. It’s a different story for multiplication, though. If multiplication is not commutative, then a left ideal might not be a right ideal, and vice versa. So we might wonder whether I needs merely to be either a left ideal or a right ideal, or perhaps both. It turns out that I needs to be a two-sided ideal, and you will see why when you provide the missing piece of the proof of the following. Theorem 9.15.2 Let R be a ring and I a two-sided ideal of R. Fora ∈ R write I + a =[a], where [a] is the equivalence class of a modulo I. Define addition ⊕ and multiplication ⊗ on R/I ={I + a : a ∈ R} by (I + a) ⊕ (I + b) = I + (a + b) and (I + a) ⊗ (I + b) = I + (ab) (9.70) Then R/I is a ring under the operations ⊕ and ⊗. We can talk our way through almost all properties R1–R10, so that your work in proving Theorem 9.15.2 will be minimal. Since R is an abelian additive group and I is a normal additive subgroup, R/I has properties R1–R6. Properties R8–R10 are immediate from the definitions of ⊕ and ⊗. The only property that takes any real work is R7, showing that ⊗ is well defined, and this is what you
340 Chapter 9 Rings will show in the next exercise. True to form, you suppose I + a = I + b and I + c = I + d and use this to show that I + ac = I + bd. That is, supposing a − b and c − d are in I should somehow allow you to show that ac − bd is in I also. EXERCISE 9.15.3 Finish the proof of Theorem 9.15.2 by showing that multiplication is well defined. With R/I defined and shown to be a ring, we should be able to bypass all the chatty exposition as in Chapter 8 and jump right to results analogous to Theorems 8.6.19 and 8.6.21. EXERCISE 9.15.4 (a) State a result analogous to Theorem 8.6.19 for a ring and the quotient ring created by “modding” out an ideal. (b) Explain how all except one component of the proof of this theorem follows from Theorem 8.6.19. (c) Provide the missing component to complete the proof of your theorem. EXERCISE 9.15.5 Suppose R and S are rings and φ : R → S is an epimorphism. Then S ∼ = R/ Ker(φ). Let’s return to the morphism in Exercise 9.14.9 defined by φ(n) = ne. If a ring has nonzero characteristic, then there is a smallest positive integer n for which ne = 0. By Exercise 9.14.19, Ker(φ) is an ideal in the integers, which is a PID. Thus Ker(φ) = (k) for some integer k. Since φ(k) = 0, and since n is the smallest positive integer for which ne = 0, it must be that k ≥ n. But since n ∈ Ker(φ), it must be that n is a multiple of k, so that k ≤ n. Thus k = n. By Exercise 9.15.5, the range of φ is isomorphic to Z/(n). That is, R contains a subring isomorphic to Zn. If we think of a ring with unity e as an abelian additive group and let S be the subgroup generated by e, the identity element of the other operation, then the form of S can be determined by looking at the additive form of Eq. (8.22). S ={ne : n ∈ Z} (9.71) which is precisely the range of φ. Thus as far as addition is concerned, S is the smallest additive subgroup of R that contains e. If we could show that S is closed under multiplication, we would have that S is the smallest subring of R that contains e. But closure is immediate by Exercise 9.3.20. In bits and pieces, we have proved the following. Theorem 9.15.6 Suppose R is a ring with unity. If R has characteristic zero, then it contains a subring isomorphic to the integers. If R has positive characteristic n, then it contains a subring isomorphic to Zn. In either case, such is the smallest subring of R that contains e.
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A Transition to Abstract Mathematic
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A Transition to Abstract Mathematic
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For Topo my little mouse
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Contents Why Read This Book? xiii P
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3.10 Irrational Numbers 107 3.11 Re
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8.2 Subgroups 252 8.2.1 Subgroups D
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Why Read This Book? One of Euclid
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Preface A Transition to Abstract Ma
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Preface to the First Edition This t
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Preface to the First Edition xix ea
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Acknowledgments It takes an entire
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Notation and Assumptions 0 Suppose
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0.2 Assumptions about the Real Numb
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0.2 Assumptions about the Real Numb
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0.2.3 Other Assumptions 0.2 Assumpt
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P A R T I Foundations of Logic and
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Language and Mathematics 1 One main
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1.1 Introduction to Logic 13 Now im
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The compound statement we call “p
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1.1 Introduction to Logic 17 exampl
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1.2 If-Then Statements 19 Definitio
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1.2 If-Then Statements 21 (h) The o
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1.2 If-Then Statements 23 (g) If ei
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p : Meghan is at least 25 years old
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1.3 Universal and Existential Quant
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1.4 Negations of Statements 33 2. F
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1.4 Negations of Statements 35 want
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Solution 1.4 Negations of Statement
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Solution 1. Everyone in this class
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1.5 How We Write Proofs 41 Our purp
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1.5 How We Write Proofs 43 the nega
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Properties of Real Numbers 2 It’s
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2.1 Basic Algebraic Properties of R
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2.1.2 Properties of Multiplication
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2.2 Ordering Properties of the Real
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(c) For all real numbers a, a2 ≥
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2.3 Absolute Value 55 (⇐) Now sup
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2.4 The Division Algorithm 57 mn =
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2.5 Divisibility and Prime Numbers
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2.5 Divisibility and Prime Numbers
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Sets and Their Properties 3 All of
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U A B Figure 3.5 Venn diagram illus
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(f) For all sets A, B, and C,ifA
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3.2 Proving Basic Set Properties 69
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3.3 Families of Sets 71 Notice that
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Example 3.3.2 Consider the family o
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3.3 Families of Sets 75 x ∈ � A
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3.3 Families of Sets 77 and ... and
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Example 3.4.2 For any positive inte
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3.4 The Principle of Mathematical I
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Proof. To clean up the notation, we
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3.5 Variations of the PMI 85 EXERCI
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(g) (a −m ) −n = a mn (h) (ab)
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Strong Induction 3.5 Variations of
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3.6 Equivalence Relations 91 EXERCI
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3.6 Equivalence Relations 93 beginn
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3.6 Equivalence Relations 95 constr
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3.7 Equivalence Classes and Partiti
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3.8 Building the Rational Numbers 1
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3.8 Building the Rational Numbers 1
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3.10 Irrational Numbers 107 EXERCIS
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3.10 Irrational Numbers 109 To the
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EXERCISE 3.10.2 √ 2 is irrational
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(a) {(x, y) : x ≤ y} (b) {(x, y)
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3.11 Relations in General 115 (O3)
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3.11 Relations in General 117 at al
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Functions 4 Second only to sets, fu
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4.1 Definition and Examples 121 pro
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Solution We show that T satisfies p
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4.2 One-to-one and Onto Functions 4
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4.2 One-to-one and Onto Functions 1
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A 1 x A B f (A 1 ) Figure 4.4 The i
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4.4 Composition and Inverse Functio
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4.4 Composition and Inverse Functio
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4.5 Three Helpful Theorems 135 The
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4.6 Finite Sets 137 EXERCISE 4.5.3
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4.7 Infinite Sets 139 The next exer
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4.7 Infinite Sets 141 of A. This or
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4.7 Infinite Sets 143 EXERCISE 4.7.
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EXERCISE 4.8.3 If A1,A2, and B are
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4.8 Cartesian Products and Cardinal
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4.8 Cartesian Products and Cardinal
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4.9 Combinations and Partitions 151
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4.10 The Binomial Theorem 157 EXERC
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EXERCISE 4.10.3 Determine the follo
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(c) The coefficient of ab 3 c 2 in
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P A R T I I Basic Principles of Ana
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The Real Numbers 5 Let’s look at
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5.1 The Least Upper Bound Axiom 167
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5.2 The Archimedean Property 169 EX
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5.2 The Archimedean Property 171 No
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5.3 Open and Closed Sets 173 Thus A
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5.4 Interior, Exterior, Boundary, a
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5.4 Interior, Exterior, Boundary, a
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5.5 Closure of Sets 179 The constru
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5.6 Compactness 181 EXERCISE 5.6.3
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5.6 Compactness 183 EXERCISE 5.6.13
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Sequences of Real Numbers 6.1 Seque
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6.1 Sequences Defined 187 Example 6
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EXERCISE 6.1.15 Consider the sequen
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L 1 e L L 2 e . . . . . . 1 2 3 4 N
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6.2 Convergence of Sequences 193 EX
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6.2 Convergence of Sequences 195 To
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6.3 The Nested Interval Property 19
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a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
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Example 6.4.7 The terms a0 = 1 a1 =
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Functions of a Real Variable 7 Func
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7.1 Bounded and Monotone Functions
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50 40 30 20 7 1 ε 7 7 2 ε 0 ( ( F
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7.2 Limits and Their Basic Properti
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7.2 Limits and Their Basic Properti
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7.3 More on Limits 217 values of 0
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7.4 Limits Involving Infinity 219 W
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7.4 Limits Involving Infinity 221 T
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(a) limx→a f(x) =−∞ (b) limx
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7.5 Continuity 225 If f is not cont
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1 2 3 4 5 6 Figure 7.6 Some example
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|a − x| |xa| = |x − a| |x||a| 7
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7.6 Implications of Continuity 231
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7.6 Implications of Continuity 233
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7.7 Uniform Continuity 235 we decla
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7.7 Uniform Continuity 237 Next, le
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7.7.2 Uniform Continuity and Compac
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P A R T I I I Basic Principles of A
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Groups 8 In its simplest terms, alg
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8.1 Introduction to Groups 245 are
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8.1 Introduction to Groups 247 Defi
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◦ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 0
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Show that C × is an abelian group
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8.2 Subgroups 253 G1 and G3 are aut
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8.2 Subgroups 255 Notice how proper
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8.2 Subgroups 257 Example 8.2.15 su
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EXERCISE 8.2.23 If G is a group and
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8.3 Quotient Groups 261 [0] ={...,
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8.3 Quotient Groups 263 is standard
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Step 2: Define an operation ∗H on
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(iii) (Z + 3.8) +Z (Z + 1.2) (iv)
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then for a given f ∈ S and any a
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Thus f 2 = f ◦ f = (132)(56) ◦
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2 3 1 3 1 Figure 8.1 Rigid square u
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EXERCISE 8.4.12 Complete Table 8.64
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8.5 Normal Subgroups 277 Theorem 8.
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8.5 Normal Subgroups 279 point to b
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8.6 Group Morphisms 281 EXERCISE 8.
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8.6 Group Morphisms 283 Notice that
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Ker � e x 2 x maps to x Ker � G
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Rings 9 We can create algebraic str
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Page 360 and 361: 9.14 Ring Morphisms 337 EXERCISE 9.
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Page 368 and 369: Index =,51 ɛ (epsilon), 167-168 ɛ
Page 370 and 371: Index 347 Cosets, 263, 264, 265 Lag
Page 372 and 373: Index 349 Gauchy sequences, 202-206
Page 374 and 375: Index 351 identity, 124 relations v
Page 376 and 377: Index 353 Quaternions, 248, 253, 25
Page 378 and 379: Index 355 Tower of Hanoi, 84 Transc
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