Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

More documents

Recommendations

Info

66 ADVANCED ABSTRACT ALGEBRA ( 0) – → R m n – → R – → M – → ( 0) where m < n . Fundamental Structure Theorem for finitely generated modules over a principal ideal domain : Theorem 2 Let R be a Principal Ideal Domain and M a finitely generated R-module. Then M is direct sum of cyclic modules : , where di di+ 1, i = 1,......, m − 1 (Recall that a module M over a using R is cyclic if M has an element x for which M = Rx . Thus a cyclic group is the same as a cyclic module over Z, the ring of integers. Every cyclic module is representable in the form of a quotient module of the free cyclic module, i.e., in the case of a ring of Principal ideals it has the form Proof ). Suppose M is generated by n elements, then M has a presentation φ ( 0) – → R m n – → R – → M – → ( 0) , m n Where m < n, where M = CoKernel if a homomorphism φ : R – → R , which is given by m × n matrix A. Now we Claim: invertible matrices P and Q of orders m, n respectively over R such that ∃(d (a M P Where di d for i+1 i = 1,.........., r − 1; more precisely PAQ = diag Two vector u,v are called right associated if ∃S ∈GL ( R) such that u = vS. We show here that any vector (a, b) is right associated to (h, o), where h is an HCF of a and b. Since R is a PID, a and b have an HCF h, a = h; a', b = hb', a', b' ∈R . Since h generates the ideal generated by a and b, we have h = ha'd' – hb' c', cancelling h we get or h = ha'd' - hb'c', Cancelling h we get 1 = a'd' - b'c', Hence ' −b' d i ( h, o) = ( a, b) −c' a' Which shows (a, b) is right associated to (h, o). Now we prove the general case, i.e. we find a matrix right associated to A which all entries of the 1st row 2
UNIT-III 67 F ( BA ad 1'= = , 1a d( SAT ,......., a) o ad , o ,....... a a o ) aA 11 b ij 1 1 2 HG I o 1K JF H G I o a2K J F = H G I R m 1 2 r o a2K J after the first one are zero. We continue in this way, we find a matrix right associated to A, such that bij = o for i
Page 1 and 2:
1 Advanced Abstract Algebra M.A./M.
Page 3 and 4:
Contents 3 Unit I 5 Unit II 49 Unit
Page 5 and 6:
UNIT-I 5 Unit-I Definition Group A
Page 7 and 8:
UNIT-I 7 2. In example 1, a 1 1 0 1
Page 9 and 10:
UNIT-I 9 Solutions: 1. CGbg≠ a φ
Page 11 and 12:
UNIT-I 11 ∴ Ha ⊆ H. To show let
Page 13 and 14:
UNIT-I 13 Corollary 4: (Feremat's L
Page 15 and 16:
UNIT-I 15 Definition: Group Homomor
Page 17 and 18:
UNIT-I 17 Proof: Consider the diagr
Page 19 and 20:
UNIT-I 19 ( G ) N G ( H ) ≅ H N D
Page 21 and 22:
UNIT-1 21 Theorem 8. Let H be a sub
Page 23 and 24:
UNIT-1 23 From (i) and (ii), we hav
Page 25 and 26:
UNIT-I 25 1 Note that αβ = 2 2 1
Page 27 and 28:
UNIT-1 27 In this case the length o
Page 29 and 30:
UNIT-1 29 | n Theorem 18. The set o
Page 31 and 32:
UNIT-I b = φ ( φ φ ) ( ) g h t x
Page 33 and 34:
UNIT-I 33 This is impossible. So no
Page 35 and 36:
UNIT-I 35 integers x and y. Note th
Page 37 and 38:
UNIT-I 37 O(7) = 4, as 7 1 = 7, 7 2
Page 39 and 40:
UNIT-I 39 Put a x mt , = b = x Then
Page 41 and 42:
UNIT-I n iii. N = ∈Gs, show that
Page 43 and 44:
UNIT-I 43 abab------ab = a n b n
Page 45 and 46:
UNIT-I Q.20. Q.21. Q.22. Q.23. Q.24
Page 47 and 48: UNIT-I 47 Remark : If G is abelian,
Page 49 and 50: UNIT-IV 81 Definition. Let S be a s
Page 51 and 52: UNIT-IV 83 (r+s) − (r 1 +s 1 ) =
Page 53 and 54: UNIT-IV 85 and = ψ(r+K) + ψ(s+K)
Page 55 and 56: UNIT-IV 87 M ⊂ M′ ⊂ R M′ =
Page 57 and 58: 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: UNIT-III 65 Theorem 1 Let M, N be R
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 and 110: UNIT-IV 77 A. B ≠ B. A. If IR, th
Page 111 and 112: UNIT-IV 79 Example: (D, +, .) is a
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
show all

Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

Create successful ePaper yourself

Delete template?

Save as template?