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MTH 302: Modules Semester 2, 2013-2014

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MTH 302: Modules Semester 2, 2013-2014 MTH 302: Modules Semester 2, 2013-2014 Dr. Prahlad Vaidyanathan Contents Finite Abelian Groups...............................3 I. Rings......................................3 1. Definition and Examples.......................3 2. Ideals and Quotient Rings......................5 3. Prime and Maximal Ideals......................6 4. The Chinese Remainder Theorem..................7 5. Unique Factorization Domains....................8 6. Principal Ideal Domains and Euclidean Domains..........8 II. Modules....................................9 1. Definition and Examples.......................9 2. Homomorphisms and Quotient Modules.............. 10 3. Direct Sums of Modules....................... 11 4. Finitely Generated Modules..................... 11 III. Finitely Generated Modules over a PID................... 14 1. Free Modules over a PID....................... 14 2. Torsion Modules over a PID - I................... 15 3. Torsion Modules over a PID - II................... 17 4. Finite Abelian Groups........................ 18 5. Rational Canonical Form....................... 18 6. Cayley-Hamilton Theorem...................... 21 7. Jordan Canonical Form........................ 22 IV. Introduction to Commutative Algebra.................... 24 1. Hom and Direct Sums........................ 24 2. Exact Sequences........................... 25 3. Projective Modules.......................... 27 4. Noetherian Rings........................... 28 5. Noetherian Modules......................... 29 6. Artinian Modules........................... 30 7. Length of a module.......................... 30 V. Tensor Products................................ 32 1. Finite Dimensional Vector Spaces.................. 32 2. Tensor Product of Modules..................... 33 VI. Instructor Notes................................ 35 2 Finite Abelian Groups (See [Norman, x1]) 0.1. Fundamental Theorem of finite abelian groups 0.2. Examples : ∼ (i) If G = hgi is cyclic, then G = Z=mZ for some m 2 Z ∼ n (ii) If G = fg1; g2; : : : ; gng finite abelian, then G = Z =K for some subgroup K < Zn 0.3. Definition : (i) Z-basis for Zn. (ii) Notation : K = hv1; v2; : : : ; vti (iii) An invertible matrix P 2 Mn(Z) 0.4. Examples : 2 ∼ (i) If K = h(2; 0); (0; 4i, then G = Z =K = Z2 × Z4 4 6 (ii) If K0 = h(4; 6); (8; 10)i, then write A = , then A is similar to D = 8 10 2 0 , and so 2=K0 ∼= × as well. 0 4 Z Z2 Z4 0.5. Summary : ∼ n (i) Want to analyse a finite abelian group G, then write G = Z =K for some subgroup K < Zn (ii) Find a Z-basis fv1; v2; : : : ; vtg for K (Fact: Such a basis always exists with t ≤ n) 0 1 v1 B v2 C (iii) Write A = B C, then find invertible matrices P; T such that P AT −1 is @:::A vt diagonal (Fact: Such matrices P and T always exist) (iv) Use the diagonal matrix to express G as a product of cyclic groups. The diagonal matrix is called the Smith Normal form of A. (End of Day 1) I. Rings 1. Definition and Examples 1.1. Definition of a ring 1.2. Examples : 3 (i) Z; Q; R; C. N is not a ring. (ii)2 Z (iii) Zn = Z=nZ (iv) If R is a ring, then Mn(R) is a ring (v) Z[i] (vi) C[0; 1] (vii) If R is a ring, then R[x] is a ring. 1.3. Definition : (i) Multiplicative identity 1 = 1R. Note: If R has a multiplicative identity, then it is unique. (ii) Commutative ring (iii) Division ring (iv) Field (v) Zero divisor (vi) Integral domain 1.4. Examples : (i) Q; R; C are fields. Z is not. (ii) Zn is a field iff n is prime (See MTH 301.V.1.4) (iii) M2(R) has zero divisors. (iv) Any finite integral domain is a field. (v) If R is an integral domain, then R[x] is an integral domain. 1.5. Theorem: Let R be a ring, a; b 2 R, then (i)0 · a = a · 0 = 0 (ii)( −a)b = a(−b) = −(ab) (iii)( −a)(−b) = ab (iv)( na)b = a(nb) = n(ab) 8n 2 Z 1.6. Definition of subring. 1.7. Definition of ring homomomorphism, isomorphism. 1.8. Examples : (i) The quotient map Z ! Zn (ii) z 7! z from C to C (iii) f 7! f(0) from C[0; 1] to C (iv) x 7! 2x from Z to Z is not a homomorphism. 1.9. Lemma: Let ' : R ! R0 be a ring homomorphism, then 4 (i) '(0R) = OR0 (ii) '(−a) = −'(a) 1.10. Definition : (i) Kernel of a homomorphism (ii) Image of a homomorphism 1.11. Proposition : ' is injective iff ker(') = f0g 2. Ideals and Quotient Rings 2.1. Definition of ideal 2.2. Examples : (i) nZ C Z (ii) ff 2 C[0; 1] : f(0) = 0g C C[0; 1] (iii) If I C R, then Mn(I) C Mn(R). Proof that the converse is true in a commu- tative ring with 1. 0 (iv) If ' : R ! R is a homomorphism, then ker(') C R (End of Day 2) 2.3. Theorem: Let R be a ring, and I C R, then R=I := fa + I : a 2 Rg is a ring under the operations (a + I) + (b + I) := (a + b) + I and (a + I)(b + I) := (ab) + I Furthermore, the function π : R ! R=I given by π(a) := a + I is a surjective ring homomorphism and ker(π) = I 2.4. (First Isomorphism Theorem): Let ' : R ! R0 be a ring homomorphism, then (i) ker(') C R (ii) R= ker(') ∼= Im(') In particular, if ' is surjective, then R= ker(') ∼= R0 2.5. Examples: ∼ (i) If R = C[0; 1] and I = ff 2 R : f(0) = 0g, then R=I = C ∼ (ii) If I C R, then Mn(R)=Mn(I) = Mn(R=I) ∼ (iii) If R is any ring, and I = (x) := fxf(x): f(x) 2 R[x]g C R[x], then R[x]=I = R. 5 2.6. (Correspondence theorem): If R is a ring, and I C R, then there is a 1-1 correspon- dence between ideals in R=I and ideals in R containing I 2.7. Example : Ideals in Zn $ f divisors of ng 2.8. Lemma : Let R be a ring. Let fSλ : λ 2 Λg be a (possibly uncountable) collection of ideals in R. Then \ I := Sλ λ2Λ is an ideal in R 2.9. Definition : Let R be a ring, and X ⊂ R, then \ (X) := fI C R : X ⊂ Ig is called the ideal generated by X. If X = fag, we write (a) = (fag), and we call (a) the principal ideal generated by a. 2.10. Lemma: Let R be a commutative ring with 1 2 R, and let X ⊂ R, then 1 k [ X (X) = f rixi : ri 2 R; xi 2 Xg k=1 i=1 ie. (X) consists of all finite sums of elements of the form rx where r 2 R; x 2 X. In particular, (a) = fra : r 2 Rg 2.11. Example : If m; n 2 Z, then (fm; ng) = (gcd(m; n)) =: (m; n) (End of Day 3) 3. Prime and Maximal Ideals Let R be a commutative ring with 1 2 R 3.1. Definition of Maximal ideal 3.2. Theorem: R is a field iff R has no non-trivial ideals 3.3. Theorem: I C R is a maximal ideal iff R=I is a field 3.4. Examples: (i) If n 2 Z is prime iff nZ C Z is a maximal ideal. (ii) I := ff 2 C[0; 1] : f(0) = 0g C C[0; 1] is a maximal ideal. (iii)( x) C F [x] is a maximal ideal if F is a field. (iv)( x) C Z[x] is not a maximal ideal. 3.5. Definition of prime ideal 3.6. Theorem: I C R is a prime ideal iff R=I is an integral domain. 6 3.7. Example : nZ C Z is prime iff n is a prime number. 3.8. Corollaries/Examples : (i)( x) C Z[x] is a prime ideal. (ii) If I C R is a maximal ideal, then I is a prime ideal (3.3+3.7) 0 (iii) Let ' : R ! R be a surjective homomorphism, then ker(') C R is prime (maximal) if R0 is an integral domain (field) 4. The Chinese Remainder Theorem Let R be a commutative ring with 1 2 R 4.1. Definition : (i) Sum of two ideals (ii) Product of two ideals (iii) Comaximal ideals 4.2. Lemma: Let I;J C R be comaximal ideals, then IJ = I \ J 4.3. (Chinese Remainder Theorem): Let I;J C R and define ' : R ! R=I × R=J; given by a 7! (a + I; a + J) (i) ' is a homomorphism and ker(') = I \ J (ii) If I + J = R, then ' is surjective, and ker(') = IJ. In that case, R=IJ ∼= R=I × R=J (End of Day 4) 4.4. Corollary: If I1;I2;:::;In C R are such that for all i 6= j, one has Ii + Ij = R, then ∼ R=(I1I2 :::In) = R=I1 × R=I2 × ::: × R=In 4.5. Example : ∼ (i) mZ; nZ C Z are comaximal iff (m; n) = 1. In that case, Znm = Zm × Zn k1 k2 kt (ii) In particular, if n = p1 p2 : : : pt , then ∼ Zn = Z k1 × Z k2 × ::: × Z kt p1 p2 pt 4.6. Definition : (i) Unit of a ring R (ii) The group of units R∗ ∗ ∼ ∗ ∗ 4.7. Lemma: (R1 × R2) = R1 × R2 k1 k2 kt 4.8. Corollary: If n = p1 p2 : : : pt , then ∗ ∼ ∗ ∗ ∗ Zn = Z k1 × Z k2 × ::: × Z kt p1 p2 pt Hence k1 k1−1 k2 k2−1 kt kt−1 '(n) = (p1 − p1 )(p2 − p2 ) ::: (pt − pt ) 7 5. Unique Factorization Domains Let R be an integral domain with 1 2 R 5.1. Definition : Let a; b 2 R (i) a j b (a divides b) (ii) a ∼ b (a and b are associates) 5.2.
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