Commutative algebra - Department of Mathematical Sciences - old ...

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14 1. A DICTIONARY ON RINGS AND IDEALS 1.3.6. Example. An ideal in Z is a prime ideal if it is generated by 0 or a prime number. Any nonzero prime ideal is a maximal ideal. 1.3.7. Definition. For an ideal I in a ring R the radical is √ I = {a ∈ R|a n ∈ I for some n} a is nilpotent is a n = 0 for some positive integer n. A ring is reduced if the nilradical √ 0 = 0, that is if 0 is the only nilpotent element. 1.3.8. Proposition. (1) The radical of an ideal is an ideal. (2) The nilradical is contained in any prime ideal. (3) A domain is reduced. Proof. (1) If am , bn ∈ I then by the binomial formula (a + b) m+n m+n m + n = a k m+n−k b k ∈ I k=0 and the radical is an ideal. (2) (3) Clearly a nilpotent element is contained in any prime ideal. 1.3.9. Exercise. (1) Show that the characteristic of a domain is either 0 or a prime number. (2) Let m, n be a natural numbers. Show that n + (m) ∈ Z/(m) is a unit if and only if m, n are relative prime. (3) Let m be a natural number. Show that Z/(m) is reduced if m is square free. (4) Show that a product of reduced rings is reduced. (5) Let a be nilpotent. Show that 1 − a is a unit. (6) Let I, J be ideals. Show that √ IJ = √ I ∩ J = √ I ∩ √ J. (7) Assume an ideal I is contained in a prime ideal P . Show that √ I ⊂ P . 1.4. Chinese remainders 1.4.1. Definition. Ideals I, J ⊂ R are comaximal ideals if I + J = R. 1.4.2. Proposition (Chinese remainder theorem). Let I1, . . . , Ik be pairwise comaximal ideals in a ring R. Then (1) For a1, . . . , ak ∈ R there is a a ∈ R, such that a−am ∈ Im for m = 1, . . . , k (2) (3) The product of projections is an isomorphism. Proof. (1) For each m I1 · · · Ik = I1 ∩ · · · ∩ Ik R/I1 · · · Ik → R/I1 × · · · × R/Ik R = (Im + In) = Im + n=m n=m So choose um ∈ Im and vm ∈ n=m In with um + vm = 1. Put a = a1v1 + · · · + akvk. Then a − am = · · · + amum + · · · ∈ Im. (2) For a in the intersection assume by induction that a ∈ I2 · · · Ik. From the proof of (1) a = u1a + av1 ∈ I1 · · · Ik. (3) Subjectivity follows from (1). The kernel is the intersection which by (2) is the product. 1.2.9 gives the isomorphism. In
1.5. UNIQUE FACTORIZATION 15 1.4.3. Corollary. Let P1, . . . , Pk be pairwise different maximal ideals in a ring R. Then and P n1 1 · · · P nk n1 nk k = P1 ∩ · · · ∩ Pk R/P n1 1 · · · P nk n1 nk k → R/P1 × · · · × R/Pk is an isomorphism. 1.4.4. Definition. An element e in a ring R is idempotent if e = e 2 . A nontrivial idempotent is an idempotent e = 0, 1. 1.4.5. Proposition. A ring R is a product of two nonzero rings if and only if it contains a nontrivial idempotent e. Proof. Use that the ideals Re and R(1 − e) are proper and comaximal. 1.4.6. Exercise. (1) Show that for a prime number p the rings Z/(p2 ) and Z/(p) × Z/(p) are not isomorphic. (2) Let n = p n1 1 be a factorization into different primes. Show that . . . pnk k Z/(p n1 1 · · · pnk k ) → Z/(pn1 1 ) × · · · × Z/(pnk k ) is an isomorphism. (3) Let elements e1 + e2 = 1 with e1e2 = 0 be given in a ring R. Show that R R/(e1) × R/(e2). (4) Let I, J be ideals such that √ I, √ J are comaximal. Show that I, J are comaximal. 1.5. Unique factorization 1.5.1. Lemma. Let R be a domain and (a) = (b) principal ideals. Then there is a unit u ∈ R such that b = ua. Proof. b = ua and a = vb giving b = uvb. If b = 0 then uv = 1 showing that u is a unit. 1.5.2. Definition. Let R be a domain and P the set of principal ideals different from (0) and R. By 1.5.1 multiplication of generators gives a well defined multiplication of principal ideals on P. An element in P maximal for inclusion is an irreducible principal ideal. A generator of an irreducible element is an irreducible element in R. 1.5.3. Definition. A domain R is a unique factorization domain if (1) Every irreducible element in P is a prime ideal. (2) Every element in P is a product of irreducible elements. 1.5.4. Proposition. In a unique factorization domain the factorization of elements in P into irreducibles is unique up to order. Proof. Proceed by induction on the shortest factorization of an element in P. Let (a1) . . . (am) = (b1) . . . (bn) be factorizations into irreducibles. (a1) is a prime ideal, so by 1.3.2 and reordering (b1) = (a1). By cancellation m − 1 = n − 1 and (ai) = (bi) after a reordering. 1.5.5. Definition. A domain R is a principal ideal domain if every ideal is principal. 1.5.6. Proposition. A principal ideal domain R is a unique factorization domain.
Page 1: ELEMENTARY COMMUTATIVE ALGEBRA LECT
Page 5 and 6: Contents Prerequisites 7 1. A dicti
Page 7: Prerequisites The basic notions fro
Page 10 and 11: 10 1. A DICTIONARY ON RINGS AND IDE
Page 22 and 23: 22 2. MODULES (5) If R is a field,
Page 24 and 25: 24 2. MODULES (1) IM = {a1y1 + ·
Page 26 and 27: 26 2. MODULES 2.3.6. Corollary. Let
Page 28 and 29: 28 2. MODULES (2) Give an example o
Page 30 and 31: 30 2. MODULES 2.4.11. Proposition.
Page 32 and 33: 32 2. MODULES Proof. By 2.5.3 the c
Page 34 and 35: 34 2. MODULES (3) The formation of
Page 36 and 37: 36 2. MODULES 2.6.12. Example. Let
Page 38 and 39: 38 2. MODULES (1) The change of rin
Page 40 and 41: 40 3. EXACT SEQUENCES OF MODULES 3.
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64 4. FRACTION CONSTRUCTIONS 4.6. T
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66 5. LOCALIZATION (2) For a prime
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68 5. LOCALIZATION (2) Any local ri
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70 5. LOCALIZATION 5.4. Exactness a
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72 5. LOCALIZATION (1) φ is flat.
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74 6. FINITE MODULES 6.1.7. Corolla
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76 6. FINITE MODULES (4) Let A be a
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78 6. FINITE MODULES Let the n × n
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80 6. FINITE MODULES 6.5. Finite Pr
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82 6. FINITE MODULES 6.5.9. Corolla
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84 6. FINITE MODULES (1) E is an in
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86 7. MODULES OF FINITE LENGTH 7.2.
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88 7. MODULES OF FINITE LENGTH Proo
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90 7. MODULES OF FINITE LENGTH (5)
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92 7. MODULES OF FINITE LENGTH 7.5.
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94 8. NOETHERIAN MODULES Proof. Use
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96 8. NOETHERIAN MODULES 8.3.2. Cor
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98 8. NOETHERIAN MODULES 8.5. Fract
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9 Primary decomposition 9.1. Suppor
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9.2. ASS OF MODULES 103 Proof. The
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9.2. ASS OF MODULES 105 9.2.12. Def
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9.4. DECOMPOSITION OF MODULES 107 9
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9.5. DECOMPOSITION OF IDEALS 109 9.
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10 Dedekind rings 10.1. Principal i
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10.3. DEDEKIND DOMAINS 113 10.2.4.
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10.3. DEDEKIND DOMAINS 115 10.3.10.
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118 BIBLIOGRAPHY
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120 INDEX finite type rings, 95 fin
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