Problems in Abstract Algebra

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

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.

Contents 1 Group Theory Problems 5 1.1 First Section............................. 5 1.2 Second Section............................ 7 1.3 Third Section............................. 9 1.4 Fourth Section............................ 11 1.5 Extra Problems............................ 14 2 Ring Theory Problems 17 3

4 CONTENTS

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 1 1 2. 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 6. 10. 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 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.

1.2. SECOND SECTION 7 1.2 Second Section 21. Find all subgroups of Z 6. 22. 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)) = 1. 30. 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 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.

1.3. THIRD SECTION 9 1.3 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 ) = 1. 47. {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 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. 1 + 1 2 + 1 3 + + 1 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 2. 64. 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}.

1.4. FOURTH SECTION 11 1.4 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 2. 69. 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 (1 2 3 4 5) and (1 2 3 5 4) are conjugate in S 5, but they are not conjugate in A 5.

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 2. 87. Prove that the number of elements of S n with no fixed point is equal to: ( 1 n! 2! 1 3! + + 1 ) ( 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. 1031-1034 91. G has two maximal subgroups H, K. Prove that if H, K are abelian, and Z(G) = {e}, H K = {e}. IMS 2002 92. 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 2001 93. 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.

1.4. FOURTH SECTION 13 95. 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 2. 96. 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 2005 97. 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 2001 98. G is a nonabelian group. Prove that Inn(G) can not be a nonabelian group of order 8. 99. 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 1993 100. 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 2003 101. 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. 102. G is a finite p-group. Prove that G G. IMS 1992 IMS 1989

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). 109. Let a, b > 1 be two integers. Prove that S a+b has a subgroup of order ab. 110. Let G be an infinite group such that index of each of its subgroups is finite. Prove that G is cyclic. 111. Let H be a subgroup of group G, and [G : H] = 4. Prove that G has a proper subgroup K that [G : K] < 4. 112. 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. 113. Prove that each group of order 144 is not simple. 114. Let H be an additive subgroup of Q such that for each x Q, x A or A. Prove that H = {0}. 1 x

1.5. EXTRA PROBLEMS 15 115. 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. 116. Prove that n N, group ( Q Z, +) has exactly one subgroup of order n. 117. Find all n such that A n has a subgroup of order n. 118. 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. 119. Let G be a finite abelian group, and H is a subgroup of G. Prove that G has a subgroup isomorphic to G H. 120. Let G be a group, and let H be a maximal subgroup of G. Prove that if H is abelian G (3) = e. 121. 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. 123. 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. 124. Prove that there is no nonabelian finite simple group whose order is a Fibonacci number. 125. 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 ). 126. 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 2. 127. Prove that only subgroup of index 2 of S n is A n. 128. Prove that if (n, ϕ(n)) = 1, each group of order n is abelian. Berkeley P5-Spring 1988 129. 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. 131. 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 CHAPTER 1. GROUP THEORY PROBLEMS 132. X is an infinite set. Prove that S X does not have proper subgroup of finite index. 133. 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 1. 134. 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. 135. Prove that the largest abelian subgroup of S n contains at most 3 n 3 elements. 136. 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. 137. 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

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 2. 4. 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 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 1 + + a x + a 0 U(R[x]), if and only if a 0 U(R) and a i s are nilpotent for i > 0. 11. 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 [ ] 1+ 19 2 is not Euclidean. Cartan-Brauer-Hua 19. Prove that the polynomial det(a) 1 k[x 11, x 12,..., x nn ] is irreducible.

19 20. 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 2007 25. 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 2006 26. 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 2003 27. 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 2002 28. 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 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 2004 34. 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 2. 36. 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 1998 37. 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 1996 39. 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 40. 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 1999 41. 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 3 42. 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 2003 47. 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 2000 48. 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 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 1. 61. 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 71. 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 1994 73. 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 1 + 1 and 1 + 1 + 1 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 1 76. Find all n 1 such that there exists (A, +,.) such that for each x A\{0}, x 2n +1 = 1 Romanian National Mathematics Olympiad 2007 77. 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 = 0 79. 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 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.

Bibliography [1] Jacobson N. Basic Algebra I, W. H. Freeman and Company 1974 [2] Sahai V., Bist V., Algebra, Alpha Science International Ltd. 2003 [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 1962-1991, Springer-Verlag 1996 [7] AoPS& Mathlinks The largest online problem solving community 25