49. Let R and S be rings. a. Show that the mapping from R O S onto R given by (a, b) -> a is a ring homomorphism. b. Show that the mapping from R to R O S given by a -> (a, 0) is a one-to-one ring homomorphism. c. Show that R O S is ring-isomorphic to SR. 50. Show that if m and n are distinct positive integers, then mZ is not ring-isomorphic to nZ. 51. Prove or disprove that the field of real numbers is ring-isomorphic to the field of complex numbers. 52. Show that the only ring automorphism of the real numbers is the identity mapping. 53. Determine all ring homomorphisms from R to R. 54. Suppose that n divides m and that a is an idempotent of Z, (that is, a2 = a). Show that the mapping x - ax is a ring homomorphism from Z to Z . Show that the same correspondence need not yield a ring homomorphism if n does not divide m. 55. Show that the operation of multiplication defined in the proof of Theorem 15.6 is well-defined. 56. Let Q[V2] = (a + bV2 1 a, b E Q} and ([V5] = (a + by5 1 a, b E Q). Show that these two rings are not ring-isomorphic. 57. Let Z[i] = (a + bila, b E Z). Show that the field of quotients of Z[i] is ring-isomorphic to Q[i] = {r + silr, s E Q). (This exercise is referred to in Chapter 18.) 58. Let F be a field. Show that the field of quotients of F is ring- isomorphic to F. 59. Let D be an integral domain and let F be the field of quotients of D. Show that if E is any field that contains D, then E contains a subfield that is ring-isomorphic to F. (Thus, the field of quotients of an integral domain D is the smallest field containing D.) 60. Explain why a commutative ring with unity that is not an integral do- main cannot be contained in a field. (Compare with Theorem 15.6.) 61. Show that the relation = defined in the proof of Theorem 15.6 is an equivalence relation. 62. Give an example of a ring without unity that is contained in a field. 63. Prove that the set T in the proof of Corollary 3 to Theorem 15.5 is49. Let R and S be rings. a. Show that the mapping from R O S onto R given by (a, b) -> a is a ring homomorphism. b. Show that the mapping from R to R O S given by a -> (a, 0) is a one-to-one ring homomorphism. c. Show that R O S is ring-isomorphic to SR. 50. Show that if m and n are distinct positive integers, then mZ is not ring-isomorphic to nZ. 51. Prove or disprove that the field of real numbers is ring-isomorphic to the field of complex numbers. 52. Show that the only ring automorphism of the real numbers is the identity mapping. 53. Determine all ring homomorphisms from R to R. 54. Suppose that n divides m and that a is an idempotent of Z, (that is, a2 = a). Show that the mapping x - ax is a ring homomorphism from Z to Z . Show that the same correspondence need not yield a ring homomorphism if n does not divide m. 55. Show that the operation of multiplication defined in the proof of Theorem 15.6 is well-defined. 56. Let Q[V2] = (a + bV2 1 a, b E Q} and ([V5] = (a + by5 1 a, b E Q). Show that these two rings are not ring-isomorphic. 57. Let Z[i] = (a + bila, b E Z). Show that the field of quotients of Z[i] is ring-isomorphic to Q[i] = {r + silr, s E Q). (This exercise is referred to in Chapter 18.) 58. Let F be a field. Show that the field of quotients of F is ring- isomorphic to F. 59. Let D be an integral domain and let F be the field of quotients of D. Show that if E is any field that contains D, then E contains a subfield that is ring-isomorphic to F. (Thus, the field of quotients of an integral domain D is the smallest field containing D.) 60. Explain why a commutative ring with unity that is not an integral do- main cannot be contained in a field. (Compare with Theorem 15.6.) 61. Show that the relation = defined in the proof of Theorem 15.6 is an equivalence relation. 62. Give an example of a ring without unity that is contained in a field. 63. Prove that the set T in the proof of Corollary 3 to Theorem 15.5 is