Problems in Abstract Algebra

Transcription

1 Problems in Abstract Algebra Omid Hatami [Version 0.3, 25 November 2008]

2 2 Introduction The heart of Mathematics is its problems. Paul Halmos The purpose of this book is to present a collection of interesting and challenging problems in Algebra. The book is available at http : //omidhatami.googlepages.com This is a primary version of the book. I would greatly like to hear about interesting problems in Abstract Algebra. I also would appreciate hearing about any errors in the book, even minor ones. You can send all comments to the author at omidhatami@gmail.com.

3 Contents 1 Group Theory Problems First Section Second Section Third Section Fourth Section Extra Problems Ring Theory Problems 17 3

4 4 CONTENTS

5 Chapter 1 Group Theory Problems 1.1 First Section 1. Let (G, ) be a group, and a 1, a 2,..., a n G. Prove that: (a 1 a 2... a n ) 1 = a 1 n... a For each a, b Z, we define a b = a + b ab. Prove that (Z,, 0) is a monoid. 3. Prove that R\{ 1} is a group under multiplication. 4. Let M be a monoid. Prove that a M has an inverse, if and only if there is a b M such that aba = a and ab 2 a = e. 5. Prove that each group of size 5 is abelian. 6. (G,.) is a semigroup such that: G has 1 r which is an element such that for each a G, a.1 r = a. Each a G has a right inverse.(a.b = 1 r ) 7. Suppose (G, ) is a group. For each a G, let L a : G G be L a (x) = a x. Prove that L a is one to one. 8. Prove that the equation x 3 = e has odd solutions in group (G,., e). 9. Suppose a, b are two elements of group G, which don t commute. Prove that elements of subset {1, a, b, ab, ba} of G are all distinct. Conclude that order of each nonabelian group is at least Prove that in group (G,., e) number of elements that a 2 e is even. Conclude that in each group of even order, there exists a e, such that a 2 = e. 11. A, B are subgroups of G, such that A + B > G. Prove that AB = G. 12. Prove that a finite monoid M is a group the set I = {x M x 2 = x} has only one element. 5

6 6 CHAPTER 1. GROUP THEORY PROBLEMS 13. Let G be a group and x, y G, such that xy 2 = y 3 x, and yx 2 = x 3 y. Prove that x = y = e. 14. Prove that the equation x 2 ax = a 1 has a solution in G, if and only if there is y G, such that y 3 = x. 15. (a) G is a group and for each a, b G, a 2 b 2 = (ab) 2. Prove that G is abelian. (b) If for each a G, a 2 = e, prove that G is abelian. 16. (G,., e) is a group and there exists n N, such that for each i {n, n + 1, n + 2}, a i b i = (ab) i. Prove that G is abelian. 17. G is a finite semigroup such that for each x, y, z, if xy = yz, then x = z. Prove that G is abelian. 18. G is a finite semigroup such that for each x e, c 2 e. We know that for each a, b G, (ab) 2 = (ba) 2. Prove that G is abelian. 19. G is a finite semigroup such that for each for each x G, there exists a unique y, such that xyx = x. Prove that G is a group. 20. A semigroup S is called a regular semigroup if for each y S, there is a a S, such that yay = y. Let S be a semigroup with at least 3 elements, and x S is an element such that S\{x} is a group. Prove that S is regular, if and only if x 2 = x.

7 1.2. SECOND SECTION Second Section 21. Find all subgroups of Z G is an abelian group. Prove that H = {a G o(a) < } is a subgroup of G. 23. Prove that group G is not union of two of its proper subgroups. Is the statement true, when two is replaced by three? 24. Let G be a group and H be a subset of G. Prove that H < G, if and only if HH = H. 25. Let G be a group that does not have any nonobvious subgroups. Prove that G is a cyclic group of order p, which p is a prime number. 26. Prove that a group G has exactly 3 subgroups if and only if G = p 2, for a prime p. 27. G is a group, and H is a subgroup of G. Prove that xhx 1 = {xhx 1 h H} is a subgroup of G. 28. Suppose that G is a group of order n. Prove that G is cyclic, if and only if for each divisor d of n, G has exactly one subgroup of order d. 29. Suppose G = x be a cyclic group. Prove that G = x m, if and only if gcd(m, o(x)) = Let G be a group, and for each a, b G, we know that a 3 b 3 = (ab) 3, and a 5 b 5 = (ab) 5. Prove that G is abelian. 31. G is a group, and X is a subgroup of G, such that X 1 X. Prove that if for k > 2, X k X, then X G 1 < G. 32. Let G be a finite group, and A is subgroup of G such that AxA is constant for each x. Prove that for each g G : gag 1 = A. 33. G is a finite group abelian group, such that for each a e, a 2 e. Evaluate a 1 a 2... a n which G = {a 1, a 2,..., a n }. 34. Prove Wilson s Theorem. If p is a prime number: (p 1)! 1 (mod p). 35. Let p be a prime number, and let a 1, a 2,..., a p 1 be a permutation of {1, 2,..., p 1}. Prove that there exists i j such that ia i ja j (mod p). 36. m, n are two coprime numbers. a is an element of G, such that a n = 1. Prove that there exists b such that b n = a. 37. Suppose that S is a proper subgroup of G. Prove that G\S = G.

8 8 CHAPTER 1. GROUP THEORY PROBLEMS 38. Prove that union of two subgroups of G is a subgroup of G, if and only if one of these subgroups is subset of the other subgroup. 39. G is an abelian group and a, b G, such that gcd(o(a), o(b)) = 1. Prove that o(ab) = o(a)o(b). 40. Suppose that G is a simple nonabelian group. Prove that if f is an automorphism of G such that x.f(x) = f(x).x for every x G, then f = 1.

9 1.3. THIRD SECTION Third Section 41. H, K are normal subgroups of G, and H K = {1}. Prove that for each x K, y H, xy = yx. 42. G is a group of odd order and x is multiplication of all elements in an arbitrary order. Prove that x G. 43. Prove that an infinite group is cyclic, if and only if it is isomorphic to all of its nonobvious subgroups. 44. Let G be a group. We know that the function f : G G, f(x) = x 3 is a monomorphism. Prove that G is abelian. 45. We call a normal subgroup N of G a maximal normal subgroup if there does not exist a nonobvious a normal subgroup K, such that N K G. Prove that N is a maximal normal subgroup of G, if and only if G N is simple. 46. G, H are cyclic groups. Prove that G H is a cyclic group, if and only if gcd( G, H ) = {G i i I} is a family of groups. Prove that order of each element of i I G i is finite. 48. N is a normal subgroup of G of finite order, and H is a subgroup of G of finite index, such that gcd( N, [G : H]) = 1. Prove that N H. G 49. M, N are normal subgroups of G. Prove that M N subgroup of G M G N. is isomorphic to a 50. A, B are subgroups of G, such that gcd([g : A], [G : B]) = 1. Prove that G = AB. 51. H is a proper subgroup of G. Prove that: G xhx 1 x G 52. G is a finite group, and f : G G is an automorphism of G such that at for at least 3 4 of elements of G such as x, f(x) = x 1. Prove that f(x) = x 1, and G is abelian. 53. Let G be a group of order 2n. Suppose that if half of elements of G are of order 2, the remaining elements form a group of order n, like H. Prove that n is odd, and H is abelian. 54. Let G be a group that has a subgroup of order m, and also has a subgroup of order n. Prove that G has a subgroup of order lcm(m, n). 55. H is a subgroup of G with finite index. Prove that G has finitely many subgroups of form xhx 1.

10 10 CHAPTER 1. GROUP THEORY PROBLEMS 56. Consider the group (R, +) and it subgroup Z. Prove that R Z is a group ismomorphic to complex numbers with norm 1 with the multiplication operation. 57. G is a finite group with n elements. K is a subset of G with more than n 2 elements. Prove that for every g G, we can find h, k K such that g = h.k. 58. Let p > 3 be a prime number, and: Prove that p 2 a p 1 = a b 59. Let G be a finitely generated group. Prove that for each n, G has finitely many groups of index n. 60. Let G be a finitely generated group, and H be a subgroup of G of finite index. Prove that H is finitely generated. 61. Let m and n be coprime. Assume that G is a group such that m-powers and n-powers commute. Then G is abelian. 62. H is a subgroup of index r of G. Prove that there exists z 1, z 2,..., z r G such that: r r z i H = Hz i = G i=1 63. G is a group of order 2k, in which k is an odd number. Prove that G has subgroup of index Prove that there does not exist any group satisfying the following conditions: (a) G is simple and finite. i=1 (b) G has at least two maximal subgroups. (c) For each two maximal subgroups such as G 1, G 2, G 1 G 2 = {e}.

11 1.4. FOURTH SECTION Fourth Section 65. Let G be a group and H be a subgroup of G. Prove that if G = Ha 1 Ha 2... Ha n. Prove that: 66. Prove that Aut(Q) = Q. G = a 1 1 H a 1 2 H... a 1 n H 67. Let G = (Z n, +). Prove that Aut(G) = GL n (Z). 68. G 1, G 2 are simple groups. Find all normal subgroups of G 1 G Let G be a group. Prove that Aut(G) is abelian, if and only if G is cyclic. 70. a is the only element of G which is of order n. Prove that a Z(G). 71. G has exactly one subgroup of index n. Prove that the subgroup of order n is normal. 72. Prove that if every cyclic subgroup T of G, is a normal subgroup, then for every subgroup of G, is a normal subgroup. 73. A, B are two subgroups of G, and [G : A] is finite. Prove that: [A : A B] [G : B] and equality occurs, if and only if G = AB. 74. Let G be a group. We know that G = k i=1 H i, which H i G, and H i H j = {e}. Prove that G is abelian. 75. S is a nonempty subset of G, and G = n. For each k, let S k be: Prove that S n G. k { s i s i S} i=1 76. H, K are subgroups of G. For each a, b G, prove that Ha Kb = or Ha Kb = (H K)c for some c G. 77. Let S = n=1s n, which S n is n-th symmetric group. Prove that only nonobvious subgroup of S is A = n=1a n. 78. Prove that there does not exist a finite nonobvious group such that each of G except the unit, commutes with exactly half of elements of G. 79. Prove that for groups G 1, G 2,..., G n : Z(G 1 ) Z(G 2 ) Z(G n ) = Z(G 1 G 2 G n ). 80. Prove that ( ) and ( ) are conjugate in S 5, but they are not conjugate in A 5.

12 12 CHAPTER 1. GROUP THEORY PROBLEMS 81. G is an infinite simple group. Prove that: (a) Each x e has infinitely many conjugates. (b) Each H {e} has infinitely many conjugates. 82. G is a group of order pq, which p < q, p, q are prime numbers and p q 1. Prove that G is abelian. 83. Let N be a normal subgroup of a finite p-group, G. Prove that N Z(G) = {e}. 84. Let H be a normal subgroup of G, and H G = {e}. Prove that H Z(G). 85. G is a nonabelian group of order p 3, which p is a prime number. Prove that Z(G) = G. 86. G is a finite nonabelian p-group. Prove that Aut(G) is divisible by p Prove that the number of elements of S n with no fixed point is equal to: ( 1 n! 2! 1 3! ) ( 1)n n! 88. Let X = {1, 2,... }, and A be the sungroup of S X generated by 3-cycles. Prove that A is an infinte, simple group. 89. Let {N i i I} be a family of normal subgroups G, and N = i I N i. Prove that G/N is isomorphic to a subgroup of i I G/N i. Prove that if [G : N i ] <, for each i, all elements of G/N are of finite order. Conclude that if G is a group that each element of G has finitely many conjugates, [G : Z(G)] <. 90. G is an arbitray finite nonabelian group, and P (G) is the probabilty that two arbitray elements of G commute. Prove that P (G) 5 8 American Mathematical Monthly, Nov. 1973, pp G has two maximal subgroups H, K. Prove that if H, K are abelian, and Z(G) = {e}, H K = {e}. IMS G is a finite group, and p is a prime number. Let a, b be two elements of order p, such that b a. Prove that G has at least p 2 1 elements of order p. IMS G is a group, such that each of its subgroups are in a proper subgroup of finite index. Prove that G is cyclic. 94. G is a nonobvious group such that for each two subgroups H, K of G, H K or K H. Prove that G is abelian p-group, for a prime p.

13 1.4. FOURTH SECTION Let G be a group with exactly n subgroups of index 2.(n is a natural number.) Prove that there exists a finite abelian group with exactly n subgroups of order Let K be a subgroup of group G. Prove that N G(K) C G (K) is isomorphic to a subgroup of Aut(K). Prove that if K is abelian, and K G = G, then K Z(G). IMS 2007 IMS Let G be a finite group of order n. Prove that if [G : Z(G)] = 4, then 8 n. For each 8 n find a group satisfying the condition [G : Z(G)] = 4. IMS G is a nonabelian group. Prove that Inn(G) can not be a nonabelian group of order Let G be a finite group, and H be a subgroup of G, such that: x(x H = H x 1 Hx = {e G }) Prove that H and [G : H] are coprime. IMS 1999 IMS Let G be a group and H be a subgroup of G such that for each x G\H and each y G, there is a u H that y 1 xy = u 1 xu. Prove that H G, and G H is abelian. IMS G is an abelian group and A, B are two different abelian subgroups of G, such that [G : A] = [G : B] = p, and p is the smallest integer dividing G. Prove that Inn(G) = Z p Z p G is a finite p-group. Prove that G G. IMS 1992 IMS 1989

14 14 CHAPTER 1. GROUP THEORY PROBLEMS 1.5 Extra Problems 103. Let G be a transitive subgroup of symmetric group S 25 different from S 25 and A 25. Prove that order of G is not divisible by 23. Miklós Schweitzer Competition 104. Determine all finite groups G that have an automorphism f such that H f(h) for all proper subgroups H of G. Miklós Schweitzer Competition 105. Let G be a finite group, and K a conjugacy class of G that generates G. Prove that the following two statements are equivalent: There exists a positive integer m such that every element of G can be written as a product of m (not necessarily distinct) elements if K. G is equal to its own commutator subgroup. Miklós Schweitzer Competition 106. Let n = p k (p a prime number, k 1), and let G be a transitive subgroup of the symmetric group S n. Prove that the order of normalizer of G in S n is at most G k+1. Miklós Schweitzer Competition 107. Let G, H be two countable abelian groups. Prove that if for each natural n, p n G = p n+1 G, H is a homomorphic image of G. Miklós Schweitzer Competition 108. Let G be a finite group, and p be the smallest prime number that divides G. Prove that if A < G is a group of order p, A < Z(G) Let a, b > 1 be two integers. Prove that S a+b has a subgroup of order ab Let G be an infinite group such that index of each of its subgroups is finite. Prove that G is cyclic Let H be a subgroup of group G, and [G : H] = 4. Prove that G has a proper subgroup K that [G : K] < Let A be a subgroup of R n, such that for each bounded sunset B R n, A B <. Prove that there exists m n, such that A is an abelian group generated by m elements Prove that each group of order 144 is not simple Let H be an additive subgroup of Q such that for each x Q, x A or A. Prove that H = {0}. 1 x

15 1.5. EXTRA PROBLEMS Let n be an even number greater than 2. Prove that if the symmetric group S n contains an element of order m, then GL n 2 (Z) contains an element of order m Prove that n N, group ( Q Z, +) has exactly one subgroup of order n Find all n such that A n has a subgroup of order n Let G be a group and M, N be normal subgroups of G such that M N and G N is cyclic and [N : M] = 2. Prove that G M is abelian Let G be a finite abelian group, and H is a subgroup of G. Prove that G has a subgroup isomorphic to G H Let G be a group, and let H be a maximal subgroup of G. Prove that if H is abelian G (3) = e Let f : G G be a homomorphism. Prove that: f(g) 2 G f(f(g)) 122. Prove that a simple group G does not have a proper, simple subgroup of finite index Let G be a finite group, and for each a, b G\{e}, there exists f Aut(G) such that f(a) = b. Prove that G is abelian Prove that there is no nonabelian finite simple group whose order is a Fibonacci number Let a, b, c be elements of odd order in group G, and a 2 b 2 = c 2. Prove that ab and c are in the same coset of commutator group(g ) Let n be an odd number, and G be a group of order 2n. H is a subgroup of G of order n such that for each x G\H, xhx 1 = h 1. Prove that H is abelian, and each element of G\H is of order Prove that only subgroup of index 2 of S n is A n Prove that if (n, ϕ(n)) = 1, each group of order n is abelian. Berkeley P5-Spring Prove that each uncountable abelian group has a proper subgroup of the same cardinal. David Hammer 130. Let G be a group, and H is a subgroup and H be a subgroup of index 2. Prove that there is a permutation group isomorphic with G, such that its alternating subgroup is isomorphic to H We say that the permutation satisfies the condition T, if and only if it is abelian, and for each i, j {1, 2,..., n} there is a permutation σ such that σ(i) = j. Prove that if n is free-square, then each group satisfying condition T is abelian.

16 16 CHAPTER 1. GROUP THEORY PROBLEMS 132. X is an infinite set. Prove that S X does not have proper subgroup of finite index Let G be a group of order p m n, such that m < 2p. Prove that G has a normal subgroup of order p m or p m Let p be a prime number and H is a subgroup of S p, and contains a transposition and a p-cycle. Prove that H = S p Prove that the largest abelian subgroup of S n contains at most 3 n 3 elements We call an element x of finite group G, a good element, if and only if, there are two elements u, v e, such that uv = vu = x. Prove that if x is not a good element, x has order 2, and G = 2(2k 1) for some k N Let n 1 and x x n is an isomorphism. Prove that for all a G, a n 1 Z(G). Hungary-Israel Binational 1993

17 Chapter 2 Ring Theory Problems 1. Prove that all of continuous functions on R, such that f(x) < form a ring. 2. Prove that the only subring of Z is Z. R 3. An element a of ring R is called idempotent, if and only if a 2 = a: (a) Let R be a ring with 1, and a be an idempotent element. Prove that 1 a is also idempotent. (b) Prove that if R is an integral domain, the only idempotent elements of R are 0, 1. (c) Let R be ring and each of its elements are idempotent. Prove that R is commutative with characteristic Give an example of ideal such that is not a subring and give an example of a subring that is not an ideal. 5. Prove that the following statements are equivalent: (a) Each ideal of ring R is finitely generated. (b) For every sequence of ideals I 1 I 2... there exists k N, such that I k = I k+1 =... A ring R with the previous conditions is called a Noetherian ring. 6. Let A be a Noetherian ring. Prove that A[x] is a Noetherian ring. 7. Let R be a commutative ring, and u, v are two nilpotent elements. Prove that u + v is also nilpotent. 8. Let R be a ring. Prove that if a has more than one right inverses, then it has infinitely many right inverses. 9. R is a ring with 1. Prove that if R does not contain any nilpotent elements, then all of its idempotent elements are in center of R. 17

18 18 CHAPTER 2. RING THEORY PROBLEMS 10. Let R be a ring with 1. Prove that if p(x) = a n x n + a n 1 x n a x + a 0 U(R[x]), if and only if a 0 U(R) and a i s are nilpotent for i > Let R be a commutative ring with 1. We see that we can det(a) is welldefined for each A M n (R). Prove that: U(M n (R)) = {A M n (R) det(a) U(R)} 12. Let R be a ring with 1. Prove that if 1 ab is invertible, 1 ba is also invertible. 13. We µ(n) be the Möbius function, on natural numbers. µ(1) = 1, and for non-freesquare numbers n, we have µ(n) = 0. Also if n = p 1 p 2... p s, in which p 1,..., p s are different primes, µ(n) = ( 1) s. Prove that µ(n) is multiplicative, i.e. if (n 1, n 2 ) = 1, µ(n 1 n 2 ) = µ(n 1 )µ(n 2 ). Also prove that { 1 if n = 1 µ(d) = 0 if n = 0 d n 14. Prove the Möbius inversion formula. If f(n) is a function and defined on natural numbers, and g(n) = f(n) d n Prove that f(n) = d n ( n ) µ g(d) d 15. Prove that if ϕ(n) is the Euler function: ϕ(n) = ( n ) µ d d n 16. F be a finite field with q elements. Prove that if N(n, q) is the number of irreducible polynomials of degree n: N(n, q) = 1 n d n ( n ) µ q d d 17. Let D be division ring, and C is its center. S is a sub-division ring of D such that is invariant under each of the mappings x dxd 1, which d is a non-zero element of D. Prove that S = D or S C. 18. Prove that Z [ ] is not Euclidean. Cartan-Brauer-Hua 19. Prove that the polynomial det(a) 1 k[x 11, x 12,..., x nn ] is irreducible.

19 Prove that in the ring R, the number of units is larger or equal than the number of nilpotents. 21. Let R be an Artinian ring with 1. Prove that each idempotent element of R commutes with every element such that its square is equal to zero. Suppose that we can write R as sum of two ideals A and B. Prove that AB = BA. Miklós Schweitzer Competition 22. Let R be an infinite ring such that each of its subrings except {0} has finite index (index of a subring is the index of its additive group). Prove that the additive group of R is cyclic. Miklós Schweitzer Competition 23. Let R be a finite ring. Prove that R contains 1, if and only if the only annihilator of R is 0. Miklós Schweitzer Competition 24. Let R be a commutative ring with 1. Prove that R[x] contains infinitely many maximal ideals. IMS Let R be a commutative ring with 1, containing an element such as a, such that a 3 a 1 = 0. Prove that if J is an ideal of R such that R/J contains at most 4 elements. Prove that J = R. IMS Let R, R be two rings such that all of their elements are nilpotent. Let f : R R be a bijective function such that for each x, y R, f(xy) = f(x)f(y). Prove that R R. IMS Let R be a commutative ring with 1, such that each of its ideals is principal. Prove that if R has a unique maximal ideal, then for each x, y R, we have Rx Ry or Ry Rx. IMS Prove that intersection of all of left maximal ideals of a ring is a two-sided ideal. 29. Let I be an ideal of Z[x] such that: (a) gcd of coefficients of each element of I is 1. (b) For each R Z, I contains an element with constant coefficient equal to R.

20 20 CHAPTER 2. RING THEORY PROBLEMS Prove that I contains an element of form 1+x+ +x r 1 for some r N. Miklós Schweitzer Competition 30. Let R be a finite ring and for each a, b R, there is an element c R such that a 2 + b 2 = c 2. Prove that for each a, b, c R, there is a d R such that 2abc = d 2. Vojtec Jarnick Competition 31. Ring R has at least one divisor of zero, and the number of its zero divisors is finite. Prove that R is finite. Vojtec Jarnick Competition 32. Let n be an odd number. Prove that for each ideal of ring I 2 = I. 33. Let A be ring with 2 n + 1 elements. Let M := {k N x k = x, x A} Z 2 [x] (x n 1), Prove that A is a field, if and only if M is not empty, and the least element of M is equal to 2 n + 1. Romanian District Olympiad Let I be an irreducible ideal of commutative ring R containing 1. For each r R, we define (I : r) = {x R rx I}. Let r R be an element such that (I : r) I. Also suppose that {(I : r i )} i=1 is a finite set. Prove that there is a n N, such that (I : r n ) = R. 35. Let (A, +, ) be a finite ring in which 0 1. If a, b A are such that ab = 0, then a = 0 or b {ka k Z}. Prove that there is a prime p such that A = p Let R be a ring, and for each x R, x 2 = 0. Prove that x = 0. Suppose that M = {a A a 2 = a}. Prove that if a, b M, a + b 2ab M. Romanian Olympiad Prove that in each boolean ring, every finitely generated ideal is principal. 38. Let R be a ring in which 0 1. R contains 2 n 1 invertible elements, and at least half of its elements are invertible. Prove that R is a field. Romanian Olympiad Let (A, +, ) be a ring with characteristic 2. For each x A, there is a k such that x 2k +1 = x. Prove that for each x A, x 2 = x.

21 Let (A, +, ) be a ring in which 1 0. The mapping f : A A, f(x) = x 10 is group homomorphism of (A, +). Prove that A contains 2 or 4 elements. Romanian Olympiad Let A be a ring and x 2 = 1 or x 2 = x for each x A. Prove that if A contains at least two invertible elements, A = Z Let R be a ring, and x n = x for each x R. Prove that for each x, y, xy n 1 = y n 1 x. 43. Let A be a finite ring in which 0 1. Prove that A is not a field if and only if for each n, x n + y n = z n has a solution. 44. Let A be a finite commutative ring with at least 2 elements and n is a natural number. Prove that there exists p A[x], such that p does not have any roots in A. 45. Let n be an integer, and ζ = e 2πi n. Prove that: n ζ k2 = n k=1 Romanian District Olympiad 46. Let R be a ring, in which a 2 = 0 for each a A. Prove that for each a, b, c R, abc + abc = 0. IMC Let R be a ring of characteristic zero, and e, f, g are three idempotent elements, such that e + f + g = 0. Prove that e = f = g = 0. IMC Let R be a Noetherian ring, and f : A A is surjective. Prove that f is injective. 49. Let A be a ring such that ab = 1 implies ba = 1. Prove that we have the same property for R[x]. 50. Prove that in each Noetherian ring, there are only finitely many minimal ideals. 51. Let R be an Euclidean ring, with a unique Euclidean division. Prove that this ring is isomorphic to a ring of form K[x] which K is a field. 52. Let K be a field, and A is a ring containing K, which is finite dimensional as a K-vector space. Prove that A is Artinian and Noetherian ring. 53. Let R be a commutative ring with 1, and P 1, P 2,..., P n are prime ideals of R. If I P 1 P 2 P n, then i, I P i.

22 22 CHAPTER 2. RING THEORY PROBLEMS 54. K is an infinite field. Find all of the automorphisms of K. 55. Let R be a ring with no nilpotent non-zero element. Let a, b R such that a m = b m and a n = b n for some coprime m, n. Prove that a = b. 56. Let R be a ring with 1, and containing at least two elements, such that for each a R there is a unique element b R such that aba = a. Prove that R is a division ring. 57. Let F be a field and n > 1. Let R be the ring of all upper-triangular matrices in M n (F ), such that all of the elements on its diagonal are equal. Prove that R is a local ring. 58. Let R be a ring such that for each x R, x 3 = x. Prove that R is commutative. 59. Let R be a commutative and contains only one prime ideal. Prove that each element of R is nilpotent or unit. 60. Prove tha each boolean ring without 1, can be embedded into a boolean ring with Let R, S be two rings such that M n (R) = M n (S). Does it imply R = S? 62. Let K be a field. Can K[x] have finitely many irreducible polynomials? 63. Let R be a finite commutative ring. Prove that there are m n, such that for each x R, x m = x n. 64. Let R be a commutative ring. For each ideal I we define: I = {x R n, x n I} Prove that I = J is prime,i J 65. Prove that if F is a field, then F [x] is not a field. 66. Let I 1, I 2,..., I n be ideals of commutative ring R, such that for each j k, I j + I k = R. Prove that I 1 I 2 I n = I 1 I 2... I n. 67. Let R be a commutative ring with identity element. Prove that x is a prime ideal in R[x], if and only if R is an integral domain. 68. Prove that each finite ring without zero divisor is a field. 69. Prove that in every finite ring, each prime ideal is maximal. 70. Let m, n be coprime numbers. Let R = { m n m, n 0 Z, p 1, p 1,..., p k n} such that p i are prime numbers. Prove R has exactly k maximal ideals. J

23 Let R be a ring. Prove that: p(x) = a n x n + a n 1 x n 1 + d + a 1 x + a 0 is nilpotent if and only if a i is nilpotent for each i. 72. Let A be a ring, such that: (a) x + x = 0 for each x A. (b) For each x A, there is a k 1 such that x 2k +1 = x. Prove that x 2 = x for each x A. RMO Let R be a commutative ring that all of its prime ideals are finitely generated. Prove that R is Noetherian. 74. (A, +,.) is a commutative ring in which and are invertible, and if x 3 = y 3 then x = y. Prove that if for a, b, c A then a = b = c. a 2 + b 2 + c 2 = ab + bc + ac 75. Let (A, +,.) be a commutative ring with n 6 elements, which is a not field: (a) Prove that u : A A u(x) = is not a polynomial function. { 1, x 0 1, x = 0 (b) Let P be the number of polynomial functions f : A A of degree n. Prove that: n 2 P n n Find all n 1 such that there exists (A, +,.) such that for each x A\{0}, x 2n +1 = 1 Romanian National Mathematics Olympiad Let D be division ring, and a D. Prove that if a has finitely many conjugates, a Z(D). 78. Let (A, +,.) be a ring and a, b A such that for each x A: Prove that A is a commutative ring. x 3 + ax 2 + bx = Let A be a commutative ring with 2n + 1 elements such that n > 4. Prove that for every non-invertible element such as, a 2 { a, a}. Prove that A is a ring.

24 24 CHAPTER 2. RING THEORY PROBLEMS 80. (A, +,.) is a ring such that: (a) A contains the identity element, and Char(A) = p. (b) There is a subset B of A such that B = p, and for all x, y A, there is an element b A such that xy = byx. Prove that A is commutative.

25 Bibliography [1] Jacobson N. Basic Algebra I, W. H. Freeman and Company 1974 [2] Sahai V., Bist V., Algebra, Alpha Science International Ltd [3] Singh S., Zameerudding Q., Modern Algebra, Vikas Publishing House, Second Edition, 1990 [4] Bhattacharya P.B., Jain S.K., Nagpaul S.R., Basic abstract algebra, Second Edition, 1994 [5] Rotman J.J. An Introduction to The Theory of Groups, Fourth Edition, Springer-Verlag 1995 [6] Székely G.J., Contests in Higher Mathematics: Miklós Schweitzer Competitions , Springer-Verlag 1996 [7] AoPS& Mathlinks The largest online problem solving community 25