Dips Modern Algebra Ring Theory Notes PDF

# Modern Algebra - Ring Theory ## Chapter 1: Basic Concepts and Definitions * 1.1 Basic Properties of Rings * 1.2 Definitions ## Chapter 2: Some Important Structures * 2.1 ($Z_{m}$, +<sub>m</sub>, *<sub>m</sub>) is a Commutative Ring with Unity * 2.2 V = {Set of All Functions from R to R} * 2.3 V = {Set of All Continuous Functions from R to R} * 2.4 Let's Combine Several Rings into One Large Product: Cartesian Product * 2.5 Boolean Ring * 2.6 Group Rings * 2.7 Matrix Ring * 2.8 C[0, 1] = {Set of all Continuous Functions from [0,1] to R } ## Chapter 3: Subring and Ideals * 3.1 Subring * 3.1.1 Subring Test * 3.2 Subfield * 3.2.1 Subfield Test * 3.3 Left Ideal * 3.3.1 Right Ideal * 3.3.2 Ideal * 3.3.3 Ideal Test * 3.3.4 Ideal generated by a set * 3.3.5 Co-maximal Ideals * 3.4 Simple Ring * 3.4.1 Maximal Ideal * 3.4.2 Prime Ideal ## Chapter 4: Ring of Polynomials * 4.1 Polynomial Ring * 4.2 Degree of a Polynomial * 4.3 Division Algorithm * 4.3.1 The Remainder Theorem * 4.3.2 The Factor Theorem * 4.3.3 Greatest Common Divisor (GCD) * 4.3.4 Least Common Multiple (LCM) * 4.4 Irreducible Polynomial and Reducible Polymial * 4.5 Irreducibility Tests * 4.6 Some Important Theorem * 4.7 Construction of Such Fields ## Chapter 5: ED, PID, UFD * 5.1 Square Free Number * 5.1.1 Quadratic Field * 5.2 Principal Ideal * 5.2.1 Principal Ideal Ring * 5.2.2 Principal Ideal Domain * 5.2.3 Norm on an Integral Domain * 5.2.4 Euclidean Domain * 5.2.5 Unique Factorization Domain (U.F.D.) * 5.3 Observations Over ED, PID, UFD * 5.4 Content of a Polynomial * 5.4.1 Primitive Polynomial ## Chapter 6: Ring Homomorphism * 6.1 Definition * 6.2 Kernel of Homomorphism * 6.3 Isomorphism * 6.3.1 Isomorphism Rings * 6.4 Quotient Rings * 6.5 Some Important Theorems * 6.6 Applications of Ring Homomorphism * 6.7 Embedding of Ring * 6.8 Prime Field * 6.9 Field of Quotients ## Assignment Sheet 1 * 1. Let C([0,1]) be the ring of all real valued continuous functions on [0,1]. Which of the following statements are true? * 2. We denote the characteristic of R by char(R). In the following, let R and S be nonzero commutative rings with unity. Then * 3. Let R be the ring obtained by taking the quotient of (Z/6Z)[X] by the principal ideal I =< 2X+4>. Then * 4. For which of the following values of n, does the finite field F_5^n with 5^n elements contain a non-trivial 93rd root of unity? * 5. Let F be a finite field such that for every a ∈ F the equation x^2=a has a solution in F. Then * 6. Let R be a commutative ring. Let I and J be ideals of R. * 7. In ring of Gaussian integers Z[i] * 8. Let I_1, I_2 be two ideals of a comparative ring R with identity. Which one of the following is true? * 9. Let R be the ring of all 2×2 matrices, E the ring of all even integers, T the ring of integers (mod 10) and S the ring of all multiples of 6. Then * 10. Let I be any ideal in the ring Z of integers. Then * 11. Let F be a finite field. If f: F → F, given by f(x) = x^3 is a ring homomorphism, then * 12. Consider Z_5 as field modulo 5 and let f(x) = x^4 + 4x^3 + 4x^2 + 4x + x + 1. Then the zero of f(x) and over Z_5 are 1 and 3, with respective multiplicity * 13. Let R be the polynomial ring Z_2 [x] and write the elements of Z_2 as {0, 1}. Let (f(x)) denote the ideal generated by the element f(x) ∈ R. If f(x) = x^2 + x + 1, then the quotient ring R/(f(x)) is * 14. Let R = Z_2×Z_2×Z and I = Z_2×Z_2×{0}. Then which of the following statement is correct? * 15. Which one of the following ideals of the ring Z[i] of Gaussian integers is NOT maximal? * 16. The number of maximal ideals in Z_27 is * 17. Let R be the ring of all real valued continuous functions on [0,1]. I = {f ∈ R: f(0) = 0} . Then * 18. Let M_3(R) be the ring of all 3×3 real matrices. If I,J⊆M_3(R) are defined as * 19. Let the set denote the ring of integers modulo n under addition and multiplication modulo n. Then is not a sub ring of * 20. Let S = { * 21. Set of multiples of 4 forms an ideal in Z, the ring of integers under usual addition and multiplication. This ideal is * 22. Which one of the following is TRUE? * 23. For n∈N, let nZ = {nk:k∈Z}. Then the number of units of Z/11Z and Z/12Z, respectively, are * 24. Let R be the ring of all functions from R to R under point-wise addition and multiplication. Let I = {f: R → R | f(0) = 0} . Then * 25. Let R be the ring of all 2x2 matrices with integer entries. Which of the following subsets of R is an integral domain? * 26. A ring R has maximal ideals * 27. The power set P(X) of a set X with the binary operations symmetric difference A and intersection form a ring (the symmetric difference is the addition and the intersection is the multiplication) called the power set ring of the set X. If the set X has at least 3 elements, then in the power set ring (P(X), Δ, ∩) of X, every elements is * 28. The set {0, 2, 4} under addition and multiplication modulo 6 is * 29. Suppose a and b are elements in R, a commutative ring with unity. Then the equation ax = b * 30. Let C[0, 1] be the ring of continuous real-valued functions on [0, 1], with addition and multiplication defined pointwise. For any subset S of [0, 1] let Z(S) = {f∈ C[0, 1] | f(x) = 0 for all xe S}. Then which of the following statements are true? * 31. Pick out the true statements: * 32. Let f(x) ∈ Z_5[x] be a polynomial such that Z_5[x]/(f(x)) is a field, where (f(x)) denotes the ideal generated by f(x). Then one of the choices for f(x) is * 33. Consider Z_5 and Z_20 as ring modulo 5 and 20, respectively. Then the number of homomorphism φ: Z_5 → Z_20 is * 34. The polynomial ring Z[x] is * 35. The polynomial f(x) = x^2+5 is * 36. The number of non-trivial ring homomorphisms from Z_12 to Z_28 is * 37. Let I denote the ideal generated by x^2 + x^3 + x^2 +x+ 1 in Z_2 [x] and F = Z_3[x] /I. Then * 38. The polynomial x^3 - 7x^2 + 15x - 9 is * 39. The polynomial f(X) = X^2 + aX + 1 in Z_3[X] is * 40. Let f(x) = x^3 + 2x^2 +1 and g(x) = 2x^2 + x + 2. Then over Z_3, ## Assignment Sheet 2 * 1. Consider the following statements: * 2. Let R = Z_4[x] / (((x^2 + x + 1)(x^2 + x + 1)) is Z_2 [x] / (x^2 + x^2 + 1) is * a. a field having 8 elements * b. a field having 9 elements * c - an infinite field * d. NOT a field * 3. Let p(x) = 9x^5 + 10x^3 + 5x + 15 and q(x) = x^4 - x^2 - x - 2 be two polynomials in Q[x]. Then, over Q, * 4. Pick out the cases where the given ideal is a maximal ideal: * 5. Which of the following statements is false? * 6. Consider the polynomial ring Q[x]. The ideal of Q[x] generated by x^2-3 is * 7. Let R[X] be the ring of real polynomials in the variable X. The number of ideals in the quotient ring R[X]/( x^2-3X+2) is * 8. Let R = Q[x]. Let I be the principal ideal (x^2+1) and J be the principal ideal (x^2). Then * 9. Let F_125 be the field of 125 elements. The number of non-zero elements a∈F_125 such that a^5 = a is * 10. Let Z_n be the ring of integers modulo n, where n is an integer ≥ 2. Then Choose the correct in the following: * 11. Let F be a finite field. If f: F → F, given by f(x) = x^3 is a ring homomorphism, then * 12. Consider Z_5 as field modulo 5 and let f(x) = x^4 + 4x^3 + 4x^2 + 4x + x + 1. Then the zero of f(x) and over Z_5 are 1 and 3, with respective multiplicity * 13. Let R be the polynomial ring Z_2 [x] and write the elements of Z_2 as {0, 1}. Let (f(x)) denote the ideal generated by the element f(x) ∈ R. If f(x) = x^2 + x + 1, then the quotient ring R/(f(x)) is * 14. Consider the algebraic extension E = Q(√2, √3, √5) of the field Q of rational numbers. Then [E:Q] the degree of E over Q, is * 15. Let w be a complex number such that w^3 = 1 and w≠1. Suppose L is the field Q(√2, w) generated by √2 and w over the field of rational numbers. Then the number of subfields K of L such that K⊆L is * 16. Find the degree of the field extension Q(√2, √2, √2) over Q * 17. For a positive integer n let f_n(x) = x^n - 1 + x^(n - 2) + … + x + 1. Then * 18. Which of the following is true * 19. For a positive integer m, let a_m denote the number of distinct prime ideals of the ring Q[x] / (x^m-1) . Then * 20. Let Z[i] denote the ring of Gaussian integers. For which of the following values of n is the quotient ring Z[i] / nZ[i] as integer domain? * 21. Let F = F_3[X] / ((x^2 + 2x - 1)), where F_3 is the field with 3 elements. Which of the following statements are true? * 22. Let G be the Galois group of the splitting field of x^5-2 over Q. Then, which of the following statements are true? * 23. Let R be the ring of all entire functions, i.e., R is the ring of functions f: C → C that are analytic at every point of C, with respect to pointwise addition and multiplication. Then * 24. Consider the polynomial f(x) = x^4 - x^3 + 14x^2 + 5x +16. Also for a prime number p, let F_p denote the field with p elements. Which of the following are always true? * 25. Which of the following are true: * 26. Let R be a commutative ring (with unity). Let I and J be ideal in R. Pick out the true statements: * 27. Let p and q be two distinct primes. Pick the correct statements from the following * 28. Let w = cos^(2π/10) + i sin^(2π/10) . Let K = Q(w^2) and let L = Q(w). Then * 29. Let R be a ring. If R[x] is a principal ideal domain, then R is necessarily a * 30. What is the degree of the following numbers over Q? * a. √2 + √3 * b. √2 - √3 * 31. Which of the following rings is a PID? * 32. Let f(x) ∈ Z_5[x] be a polynomial such that Z_5[x]/(f(x)) is a field, where (f(x)) denotes the ideal generated by f(x). Then one of the choices for f(x) is * 33. Consider Z_5 and Z_20 as ring modulo 5 and 20, respectively. Then the number of homomorphism φ: Z_5 → Z_20 is * 34. The polynomial ring Z[x] is * 35. The polynomial f(x) = x^2 + 5 is * 36. The number of non-trivial ring homomorphisms from Z_12 to Z_28 is * 37. Let I denote the ideal generated by x^2 + x^3 + x^2 + x + 1 in Z_2 [x] and F = Z_3[x] /I. Then * 38. The polynomial x^3 - 7x^2 + 15x - 9 is * 39. The polynomial f(X) = X^2 +aX + 1 in Z_3[X] is * 40. Let'f(x) = x^3 + 2x^2 +1 and g(x) = 2x^2 + x + 2. Then over Z_3, ## Assignment Sheet 3 * 1. For the rings L = * which one of the following is TRUE? * 2. In which of the following fields, the polynomial x^2-312312x+123123 is irreducible in F[x]? * 3. Pick out the rings which are integral domains: * 4. The degree of the extension Q(√2+√3) Over the field Q(√2) is * 5. Let Q denote the field of rational numbers. The ring Q[x]/(x^2+1) is isomorphic to * a - The field of complex numbers. * b - Q[x]/(x^2) * c - Z[x]/((x^2+2x+2)). * d - None of above. * 6. Pick out the correct statements from the following list: * 7. Let K be a field, L a finite extension of K and M a finite extension of L. Then * 8. Let K be an extension of the field of rational numbers * 9. An algebraic number is one which occurs as the root of a monic polynomial with rational coefficients. Which of the following numbers are algebraic? * a - 5+√3 * b - 7+2√3 * c - cos^(2π/n), when n∈N * d - None of these * 10. Suppose that R is unique factorization domain and that a, b∈R are distinct irreducible elements. Which of the following statements is TRUE? * 11. Which of the following is a field? * 12. Let G denote the group of all the automorphisms of the field F_3^100 that consists of 3^100 elements. Then the number of distinct subgroups of G is equal to * 13. If Z[i] is the ring of Gaussian integers, the quotient Z[i]/((3-i)) is isomorphic to * 14. Consider the algebraic extension E = Q(√2, √3, √5) of the field Q of rational numbers. Then [E:Q] the degree of E over Q, is * 15. Let w be a complex number such that w^3 = 1 and w≠1. Suppose L is the field Q(√2, w) generated by √2 and w over the field of rational numbers. Then the number of subfields K of L such that K⊆L is * 16. Find the degree of the field extension Q(√2, √2, √2) over Q * 17. For a positive integer n let f_n(x) = x^n - 1 + x^(n - 2) + … + x + 1. Then * 18. Which of the following is true * 19. For a positive integer m, let a_m denote the number of distinct prime ideals of the ring Q[x]/(x^m-1) . Then * 20. Let Z[i] denote the ring of Gaussian integers. For which of the following values of n is the quotient ring Z[i] / nZ[i] as integer domain? * 21. Let F = F_3[X] / ((x^2 + 2x - 1)), where F_3 is the field with 3 elements. Which of the following statements are true? * 22. Let G be the Galois group of the splitting field of x^5-2 over Q. Then, which of the following statements are true? * 23. Let R be the ring of all entire functions, i.e., R is the ring of functions f: C → C that are analytic at every point of C, with respect to pointwise addition and multiplication. Then * 24. Consider the polynomial f(x) = x^4 - x^3 + 14x^2 + 5x +16. Also for a prime number p, let F_p denote the field with p elements. Which of the following are always true? * 25. Which of the following is/are true: * 26. Let R be a commutative ring (with unity). Let I and J be ideal in R. Pick out the true statements: * 27. Let p and q be two distinct primes. Pick the correct statements from the following * 28. Let w = cos^(2π/10) + i sin^(2π/10) . Let K = Q(w^2) and let L = Q(w). Then * 29. Let R be a ring. If R[x] is a principal ideal domain, then R is necessarily a * 30. What is the degree of the following numbers over Q? * a - √2 + √3 * b - √2 - √3 * 31. Which of the following rings is a PID? * 32. Let f(x) ∈ Z_5[x] be a polynomial such that Z_5[x]/(f(x)) is a field, where (f(x)) denotes the ideal generated by f(x). Then one of the choices for f(x) is * 33. Consider Z_5 and Z_20 as ring modulo 5 and 20, respectively. Then the number of homomorphism φ: Z_5 → Z_20 is * 34. The polynomial ring Z[x] is * 35. The polynomial f(x) = x^2 + 5 is * 36. The number of non-trivial ring homomorphisms from Z_12 to Z_28 is * 37. Let I denote the ideal generated by x^2 + x^3 + x^2 + x + 1 in Z_2 [x] and F = Z_3[x] /I. Then * 38. The polynomial x^3 - 7x^2 + 15x - 9 is * 39. The polynomial f(X) = X^2 +aX + 1 in Z_3[X] is * 40. Let'f(x) = x^3 + 2x^2 +1 and g(x) = 2x^2 + x + 2. Then over Z_3, * 41. Let F_p denote the field Z_p^2, where p is a prime. Let F[x] be the associated polynomial ring. Which of the following quotient rings are fields? * 42. Let R be a commutative ring and R[x] be the polynomial ring in one variable over R. * 43. Let (p(x)) denote the ideal generated by the polynomial p(x) in Q[x]. If f(x) = x^3 + x^2 + x + 1 and g(x) = x^2 - x^2 + x - 1, then * 44. Determine which of the following polynomials are irreducible over the indicated rings. * a . x^3 - 3x^2 + 2x^3 - 5x + 8 over R. * b . x^3 + 2x^2 + x + 1 over Q. * c . x^3 + 3x^2 - 6x + 3 over Z. * d . x^2 + x^2+1 over Z/2Z. * 45. Which of the following polynomials are irreducible in the ring Z[x] of polynomials in one variable with integer coefficients - * a - x^2-5 * b - 1 + (x + 1) = (x + 1)^2 + (x+1)+(x+1)+ * c - 1 + x + x^2 + x^3 + x^4. * d - 1+x+x^2+x^3. * 46. Let F and F' be two finite fields of order q and q' respectively. Then: * 47. Which of the polynomials are irreducible over the given rings? * a - X^5+3X^4 + 9X+15 over Q, field of rationals * b - X^3 +2X^2 +X+1 over Z/7Z, the ring of integers modulo 7 * c - X^3 + X^2 + X + 1 over Z, the ring of integers * d - X^4 + X^3 + X^2 + X +1 over Z, the ring of integers. * 48. Consider the ring R = Z[√5]={a+b√ - 5: a, b ∈ Z} and the element a= 3 + √-5 of R. Then * 49. Let f(x) = x^3 + x^2 + x + 1 and g(x) = x^2 +1. Then in Q[x] * 50. Let R[x] be the polynomial ring over R in one variable. Let I⊆R[x] be an ideal. Then * 51. Let f(x) = x^4 + 3x^3 - 9x^2 + 7x + 27 and let p be a prime. Let f_p(x) denote the corresponding polynomial with coefficients in Z/p_Z. Then * 52. Which of the following is an irreducible factor of x^12-1 over Q? * 53. Let R be a Euclidean domain such that R is not a field. Then the polynomial ring R[x] is always * 54. Which of the following quotient rings are fields? * a . F_3[X]/(x^2 + x + 1)), where F_3 is the finite field with 3 elements. * b . Z[x]/((x-3)) * c . Z[x]/((x^2 + X + 1)) * d . F_2[X]/((x^2 + x + 1)) where F_2 is the finite field with 2 elements. * 55. Let A denote the quotient ring Z[X]/(x^3+cx+1) . Then * 56. Let c∈Z_3 be such that Z_3[X]/(x^3+cx+1) is a field. Then c is equal to * 57. The number of ring homeomorphisms form Z_2×Z_2 to Z_4 is equal to * 58. Pick out the true statements: * 59. Pick out the true statement(s): * 60. Let C(R) denote the ring of all continuous real-valued functions on R with the operations of pointwise addition and pontwise multiplication. Which of the following form an ideal in this ring? * 61. The number of roots of the equation x^6 + x^5 + x^4 + x^3 +x^2 + x + 1 = 0 in Z_7 is * 62. Which one of the following statements is correct where R[x] denotes the polynomial ring in the one variable x over a ring R: * 63. Let (R, +) be an abelian group. If multiplication (-) is defined on R by setting a.b = 0 for all a, b, ∈ R, then which one of the following statements is correct? * 64. Consider the following assertions * i. The characteristic of the ring (Z, +) is zero. * ii. For every composite number, n, Z_n, the ring of residue classes modulo n, is a field. * iii. Z_5, the ring of residue classes modulo 5, is an integral domain. * iv. The ring of all complex numbers is a field. Which of the above assertions are correct? ## Assignment Sheet 4 * 1. If p is prime, and Z_p^4 denote the ring of integers modulo p^4, then the number of maximal ideals in Z_p^4 is * 2. For n ≥ 1, let (Z/nZ)* be the group of units of (Z/nZ). Which of the following groups are cyclic? * 3. Let a,b,c,d be real numbers with a<c<d<b. Consider the ring C[a,b] with pointwise addition and multiplication. If S = {f ∈ C[a,b]: f(x) = 0 for all x ∈ [c,d]}, then * 4. Let F_125 be the field of 125 elements. The number of non-zero elements a∈F_125 such that a^5 = a is * 5. The possible valued for the degree of an irreducible polynomial in R[x]. * 6. The number of non-zero elements in the field Z_p^m, where p is an odd prime number, which are squares, i.e., of the form m^2, m∈Z_p, m≠0. * 7. Let M denote the set of all 2 × 2 matrices over the reals. Addition and multiplication on M are as follows: A=(a_ij) and B = (b_ij), then A + B = (c_ij), where c_ij = a_ij + b_ij, and AB = (d_ij), where d_ij = a_ijb_ij. Then which one of the following is valid for (M, +, ·)? * 8. Consider Z[x], the set of all polynomials with integer coefficients and Q[√2], the set of all real numbers of the form a + b√2 with a, b rational numbers. Which of the following is correct about Z[x] and Q[√2]? * 9. Let (R, +) be an abelian group. If multiplication (-) is defined on R by setting a.b = 0 for all a, b, ∈ R, then which one of the following statements is correct? * 10. Consider the following assertions * i. The characteristic of the ring (Z, +) is zero. * ii. For every composite number, n, Z_n, the ring of residue classes modulo n, is a field. * iii. Z_5, the ring of residue classes modulo 5, is an integral domain. * iv. The ring of all complex numbers is a field. Which of the above assertions are correct? * 11. Let F be a finite field with n elements. What is the possible value of n? * 12. If R is a finite integral domain with n element, then what is the number of invertible elements under multiplication in R? * 13. Consider the ring Z_n = {0,1,2,...., n} of congruent modulo n classes. Under addition and multiplication modulo n, consider the following statements is/are are correct: * 14. Consider the algebraic extension E = Q(√2, √3, √5) of the field Q of rational numbers. Then [E:Q] the degree of E over Q, is * 15. Let w be a complex number such that w^3 = 1 and w≠1. Suppose L is the field Q(√2, w) generated by √2 and w over the field of rational numbers. Then the number of subfields K of L such that K⊆L is * 16. Find the degree of the field extension Q(√2, √2, √2) over Q * 17. For a positive integer n let f_n(x) = x^n - 1 + x^(n - 2) + … + x + 1. Then * 18. Which of the following is true * 19. For a positive integer m, let a_m denote the number of distinct prime ideals of the ring Q[x]/(x^m-1) . Then * 20. Let Z[i] denote the ring of Gaussian integers. For which of the following values of n is the quotient ring Z[i] / nZ[i] as integer domain? * 21. Let F = F_3[X] / ((x^2 + 2x - 1)), where F_3 is the field with 3 elements. Which of the following statements are true? * 22. Let G be the Galois group of the splitting field of x^5-2 over Q. Then, which of the following statements are true? * 23. Let R be the ring of all entire functions, i.e., R is the ring of functions f: C → C that are analytic at every point of C , with respect to pointwise addition and multiplication. Then * 24. Consider the polynomial f(x) = x^4 - x^3 + 14x^2 + 5x +16. Also for a prime number p, let F_p denote the field with p elements. Which of the following are always true? * 25. Which of the following is/are true: * 26. Let R be a commutative ring (with unity). Let I and J be ideal in R. Pick out the true statements: * 27. Let p and q be two distinct primes. Pick the correct statements from the following * 28. Let w = cos^(2π/10) + i sin^(2π/10) . Let K = Q(w^2) and let L = Q(w). Then * 29. Let R be a ring. If R[x] is a principal ideal domain, then R is necessarily a * 30. What is the degree of the following numbers over Q? * a - √2 + √3 * b - √2 - √3 * 31. Which of the following rings is a PID? * 32. Let f(x) ∈ Z_5[x] be a polynomial such that Z_5[x]/(f(x)) is a field, where (f(x)) denotes the ideal generated by f(x). Then one of the choices for f(x) is * 33. Consider Z_5 and Z_20 as ring modulo 5 and 20, respectively. Then the number of homomorphism φ: Z_5 → Z_20 is * 34. The polynomial ring Z[x] is * 35. The polynomial f(x) = x^2 + 5 is * 36. The number of non-trivial ring homomorphisms from Z_12 to Z_28 is * 37. Let I denote the ideal generated by x^2 + x^3 + x^2 + x + 1 in Z_2 [x] and F = Z_3[x] /I. Then * 38. The polynomial x^3 - 7x^2 + 15x - 9 is * 39. The polynomial f(X) = X^2 +aX + 1 in Z_3[X] is * 40. Let'f(x) = x^3 + 2x^2 +1 and g(x) = 2x^2 + x + 2. Then over Z_3, * 41. Let F_p denote the field Z_p^2, where p is a prime. Let F[

Dips Modern Algebra Ring Theory Notes PDF

Document Details

Tags

Related

Summary

Full Transcript

Upgrade to continue