Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

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78 ADVANCED ABSTRACT ALGEBRA An element In such that ab = 0 for some is a divisor of zero. in R, is called a divisor of zero. In are divisors of zero. 2. Let IR be the set of real numbers, and is acommutative ring wth identity. + : defined by (Addition and multiplication are defined pointwise). Ibg= x ∀x ∈1 R, I is identity of the ring R. Note that (R, +, .) is a commutative ring with identity and also with divisors of zero. If R bg= S f x and g x x < T 0 , 0 1 , x ≥ 0 R bg= S x < T 1 , 0 0 , x ≥ 0 b gbg bgbg fbgbg x I x x IR x x IR fbg 1 x x IR b. gbg bgbg bg bg 0 bg 0 then f . g x = f x g x = 0 ∀x ∈1R. In above example, (f.I) (x) = ∀ ∈ = ∀ ∈ = ∀ ∈ If f g x = f x g x = I x = 1 ∀ x ∈IR and f x ≠ , g x ≠ ∀ x∈ R, then f has a multiplicative inverse ⇔ bg= + f x . Hence for example bg= 2 sin x has a multiplicative inverse, but g x Sin x does not. Definition: Integral Domain: If bR +,. gis a commutative ring with identity such that for all Examples: 1. Z 6 , + , . gis not an integral domain. 2. b Integral domain. Definition: the ring of real valued functions, the example given on page 5 is not an A non-commutative ring with identity is a skew field (or Division ring) if every non-zero element has its inverse in it. Zb b ba a, Rf f≠ bf
UNIT-IV 79 Example: (D, +, .) is a division ring, where + , . are defined as j i k b g b g where i D = m 2 = j 2 = k 2 = i j R = -1, ij = -ji = k, + : αα 0 0+ α 1i1 i+ α 2 j + α 3 k α+ 0, + βα o 1+ , + βα1 i2+ , + βα 2 3j ∈+ 1β R3r k jk = -kj = i, ki = -ik = j. = bα 0 + βog+ bα 1 + β1gi + bα 2 + β2gj + bα 3 + β3gk , Let x = α0 + α1i + α2 j + α3 k ≠ o in D. ∴ bα 0 + α1i + α2 j + α3kg• bβ o + β1i + β2 j + β3kg = bα 0βo − α1β1 − α2β2 − α3βkg+ bα 0β1 1β0 + 2β3 − α3β2gi + Then b ∃ y = α0 α0β2 − α2β0 − α3β1 − α1β 3g − α1 j + b i − α2 j − α3 k ∈D . β β β β α0β3 + α3β0 + α1β2 − α2β1gk, such that where Definition: x. y = 1, 2 0 2 1 2 2 2 3 β = α + α + α + α ≠o A commutative ring with identity is a field if its every non-zero element has inverse in it. Example: bIR, + ,. g , the ring of real numbers is a field. Theorem. Every field is without zero divisor. Proof. Let F be a field and x,y ∈ F, x ≠ 0. Then xy = ⇒ x− 1 (xy) = x− 1 = 0 ⇒ (x− 1 x)y = 0 ⇒ y = 0 Similarly, if y ≠ 0, then
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1 Advanced Abstract Algebra M.A./M.
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Contents 3 Unit I 5 Unit II 49 Unit
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UNIT-I 5 Unit-I Definition Group A
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UNIT-I 7 2. In example 1, a 1 1 0 1
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UNIT-I 9 Solutions: 1. CGbg≠ a φ
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UNIT-I 11 ∴ Ha ⊆ H. To show let
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UNIT-I 13 Corollary 4: (Feremat's L
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UNIT-I 15 Definition: Group Homomor
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UNIT-I 17 Proof: Consider the diagr
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UNIT-I 19 ( G ) N G ( H ) ≅ H N D
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UNIT-1 21 Theorem 8. Let H be a sub
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UNIT-1 23 From (i) and (ii), we hav
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UNIT-I 25 1 Note that αβ = 2 2 1
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UNIT-1 27 In this case the length o
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UNIT-1 29 | n Theorem 18. The set o
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UNIT-I b = φ ( φ φ ) ( ) g h t x
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UNIT-I 33 This is impossible. So no
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UNIT-I 35 integers x and y. Note th
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UNIT-I 37 O(7) = 4, as 7 1 = 7, 7 2
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UNIT-I 39 Put a x mt , = b = x Then
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UNIT-I n iii. N = ∈Gs, show that
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UNIT-I 43 abab------ab = a n b n
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UNIT-I Q.20. Q.21. Q.22. Q.23. Q.24
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UNIT-I 47 Remark : If G is abelian,
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UNIT-IV 81 Definition. Let S be a s
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UNIT-IV 83 (r+s) − (r 1 +s 1 ) =
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UNIT-IV 85 and = ψ(r+K) + ψ(s+K)
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UNIT-IV 87 M ⊂ M′ ⊂ R M′ =
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UNIT-IV 89 We define addition and m
Page 59 and 60: UNIT-IV 91 be the set of the ordere
Page 61 and 62: UNIT-IV 93 (iii) a + b − a b
Page 63 and 64: UNIT-IV 95 and Z/A ~ Im(f) = f (z)
Page 65 and 66: UNIT-IV 97 If a n = 0 for all n ≥
Page 67 and 68: UNIT-IV 99 x 3 = (0, 0, 0, 1,…) .
Page 69 and 70: UNIT-IV 101 Then either (i) or n i
Page 71 and 72: UNIT-IV 103 deg f(x) = 0 and deg g(
Page 73 and 74: UNIT-IV 105 If x, y ∈ A , then th
Page 75 and 76: UNIT-IV 107 The possibility φ(r) <
Page 77 and 78: UNIT-IV 109 Lemma 4. If f(x) is an
Page 79 and 80: UNIT-IV 111 Proof. Let, if possible
Page 81 and 82: UNIT-II 49 Unit-II Definition: Comp
Page 83 and 84: UNIT-II 51 If r = 1, then G is simp
Page 85 and 86: UNIT-II 53 group of the Commutator
Page 87 and 88: UNIT-II di + 1 idi + 1 i di + 1 idi
Page 89 and 90: UNIT-II 57 we write the refinement
Page 91 and 92: UNIT-II 59 of length e in which eac
Page 93 and 94: UNIT-II 61 of nilpotency of G) , so
Page 95 and 96: UNIT-III 63 n.v = R i.e. abelian gr
Page 97 and 98: UNIT-III 65 Theorem 1 Let M, N be R
Page 99 and 100: UNIT-III 67 F ( BA ad 1'= = , 1a d(
Page 101 and 102: UNIT-III 69 Vector space over F. Le
Page 103 and 104: UNIT-III 71 1. J = only one linearl
Page 105 and 106: UNIT-III Hence J = −L NM 4 0 0 0
Page 107 and 108: UNIT-III 75 Example 5 The direct su
Page 109: UNIT-IV 77 A. B ≠ B. A. If IR, th
Page 113 and 114: UNIT-IV 81 Examples 1. The polynomi
Page 115 and 116: UNIT-V 129 (Characteristic of a rin
Page 117 and 118: UNIT-V 131 As F = p dΘ F ≅ Z p i
Page 119 and 120: UNIT-V 133 In this case is called t
Page 121 and 122: + c c+ 0 0 1 c c+ 1 1 1 0 1+ c c c
Page 123 and 124: UNIT-V 137 element of defines a per
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Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

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