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 linear 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 [email protected].

Contents 1 Group Theory Problems 1.1 First Section . . . . . 1.2 Second Section . . . . 1.3 Third Section . . . . . 1.4 Fourth Section . . . . 1.5 Extra Problems . . . .

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2 Ring Theory Problems

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5 5 7 9 11 14 17

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4

CONTENTS

Chapter 1

Group Theory Problems 1.1

First Section

1. Let (G, ∗) be a group, and a1 , a2 , . . . , an ∈ G. Prove that: −1 (a1 ∗ a2 ∗ . . . an )−1 = a−1 n ∗ . . . a1

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 ab2 a = e. 5. Prove that each group of size 5 is abelian. 6. (G, .) is a semigroup such that: • G has 1r which is an element such that for each a ∈ G, a.1r = a. • Each a ∈ G has a right inverse.(a.b = 1r ) 7. Suppose (G, ∗) is a group. For each a ∈ G, let La : G −→ G be La (x) = a ∗ x. Prove that La is one to one. 8. Prove that the equation x3 = 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 a2 6= e is even. Conclude that in each group of even order, there exists a = 6 e, such that a2 = 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 |x2 = 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 yx2 = x3 y. Prove that x = y = e. 14. Prove that the equation x2 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, a2 b2 = (ab)2 . Prove that G is abelian. (b) If for each a ∈ G, a2 = 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}, ai bi = (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 6= e, c2 6= 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 x2 = x.

1.2. SECOND SECTION

1.2

7

Second Section

21. Find all subgroups of Z6 . 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| = p2 , 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 = hxi be a cyclic group. Prove that G = hxm i, if and only if gcd(m, o(x)) = 1. 30. Let G be a group, and for each a, b ∈ G, we know that a3 b3 = (ab)3 , and a5 b5 = (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 6= e, a2 6= e. Evaluate a1 a2 . . . an which G = {a1 , a2 , . . . , an }. 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 a1 , a2 , . . . , ap−1 be a permutation of {1, 2, . . . , p−1}. Prove that there exists i 6= j such that iai ≡ jaj (mod p). 36. m, n are two coprime numbers. a is an element of G, such that an = 1. Prove that there exists b such that bn = a. 37. Suppose that S is a proper subgroup of G. Prove that hG\Si = 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

1.3

9

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 ∈ G0 . 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) = x3 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. G Prove that N is a maximal normal subgroup of G, if and only if 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 Q i |i ∈ I} is a family of groups. Prove that order of each element of i∈I Gi 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. 49. M, N are normal subgroups of G. Prove that G G ×N . subgroup of M

G M ∩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 6= 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 34 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 RZ 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: 1+

1 a 1 1 + + ··· + = 2 3 p−1 b

Prove that p2 |a. 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 z1 , z2 , . . . , zr ∈ G such that: r r [ [ zi H = Hzi = G i=1

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. (b) G has at least two maximal subgroups. (c) For each two maximal subgroups such as G1 , G2 , G1 ∩ G2 = {e}.

1.4. FOURTH SECTION

1.4

11

Fourth Section

65. Let G be a group and H be a subgroup of G. Prove that if G = Ha1 ∪ Ha2 ∪ . . . Han . Prove that: −1 −1 G = a−1 1 H ∪ a2 H ∪ . . . an H

66. Prove that Aut(Q) = Q∗ . 67. Let G = (Zn , +). Prove that Aut(G) ∼ = GLn (Z). 68. G1 , G2 are simple groups. Find all normal subgroups of G1 × G2 . 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 = ∪ki=1 Hi , which Hi E G, and Hi ∩ Hj = {e}. Prove that G is abelian. 75. S is a nonempty subset of G, and |G| = n. For each k, let S k be: {

k Y

si |si ∈ S}

i=1

Prove that S n E G. 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=1 Sn , which Sn is n-th symmetric group. Prove that only nonobvious subgroup of S is A = ∪∞ n=1 An . 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 G1 , G2 , . . . , Gn : Z(G1 ) × Z(G2 ) × · · · × Z(Gn ) ∼ = Z(G1 × G2 × · · · × Gn ). 80. Prove that (1 2 3 4 5) and (1 2 3 5 4) are conjugate in S5 , but they are not conjugate in A5 .

12

CHAPTER 1. GROUP THEORY PROBLEMS

81. G is an infinite simple group. Prove that: (a) Each x 6= e has infinitely many conjugates. (b) Each H 6= {e} has infinitely many conjugates. 82. G is a group of order pq, which p < q, p, q are prime numbers and p 6 |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 ∩ G0 = {e}. Prove that H ⊂ Z(G). 85. G is a nonabelian group of order p3 , which p is a prime number. Prove that Z(G) = G0 . 86. G is a finite nonabelian p-group. Prove that |Aut(G)| is divisible by p2 . 87. Prove that the number of elements of Sn with no fixed point is equal to:   1 1 n 1 n! − + · · · + (−1) 2! 3! n! 88. Let X = {1, 2, . . . }, and A be the sungroup of SX generated by 3-cycles. Prove that A is an infinte, simple group. 89. Let {Ni |i ∈ I} be a family of normal subgroups Q G, and N = ∩i∈I Ni . Prove that G/N is isomorphic to a subgroup of i∈I G/Ni . Prove that if [G : Ni ] < ∞, 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) ≤ 85 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 6∈ hai. Prove that G has at least p2 − 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. IMS 2007 96. Let K be a subgroup of group G. • Prove that

NG (K) CG (K)

is isomorphic to a subgroup of Aut(K).

• Prove that if K is abelian, and K E G = G0 , then K ≤ Z(G). 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. IMS 1999 99. Let G be a finite group, and H be a subgroup of G, such that: ∀x(x 6∈ H =⇒ H ∩ x−1 Hx = {eG }) Prove that |H| and [G : H] are coprime. 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 E G, G is abelian. and H 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) ∼ = Zp × Zp . IMS 1992 102. G is a finite p-group. Prove that G 6= G0 . IMS 1989

14

CHAPTER 1. GROUP THEORY PROBLEMS

1.5

Extra Problems

103. Let G be a transitive subgroup of symmetric group S25 different from S25 and A25 . Prove that order of G is not divisible by 23. Mikl´ os Schweitzer Competition 104. Determine all finite groups G that have an automorphism f such that H 6⊆ f (H) for all proper subgroups H of G. Mikl´ os 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. Mikl´ os Schweitzer Competition 106. Let n = pk (p a prime number, k ≥ 1), and let G be a transitive subgroup of the symmetric group Sn . Prove that the order of normalizer of G in Sn is at most |G|k+1 . Mikl´ os Schweitzer Competition 107. Let G, H be two countable abelian groups. Prove that if for each natural n, pn G = pn+1 G, H is a homomorphic image of G. Mikl´ os 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 Sa+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 Rn , such that for each bounded sunset B ⊂ Rn , |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 1 x ∈ A. Prove that H = {0}.

1.5. EXTRA PROBLEMS

15

115. Let n be an even number greater than 2. Prove that if the symmetric group Sn contains an element of order m, then GLn−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 An has a subgroup of order n. 118. Let G be a group and M, N be normal subgroups of G such that M ⊂ N G G is cyclic and [N : M ] = 2. Prove that M is abelian. and N 119. Let G be a finite abelian group, and H is a subgroup of G. Prove that G G has a subgroup isomorphic to 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 a2 b2 = c2 . Prove that ab and c are in the same coset of commutator group(G0 ). 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. Berkeley P5-Spring 1988 127. Prove that only subgroup of index 2 of Sn is An . 128. Prove that if (n, ϕ(n)) = 1, each group of order n is abelian. 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 SX does not have proper subgroup of finite index. 133. Let G be a group of order pm n, such that m < 2p. Prove that G has a normal subgroup of order pm or pm−1 . 134. Let p be a prime number and H is a subgroup of Sp , and contains a transposition and a p-cycle. Prove that H = Sp . n

135. Prove that the largest abelian subgroup of Sn contains at most 3 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 6= 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 7→ xn is an isomorphism. Prove that for all a ∈ G, an−1 ∈ Z(G). Hungary-Israel Binational 1993

Chapter 2

Ring Theory Problems 1. Prove that all of continuous functions on R, such that Z |f (x)| < ∞ R

form a ring. 2. Prove that the only subring of Z is Z. 3. An element a of ring R is called idempotent, if and only if a2 = 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 I1 ⊂ I2 ⊂ . . . there exists k ∈ N, such that Ik = Ik+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) = an xn + an−1 xn−1 + · · · + ax + a0 ∈ U (R[x]) , if and only if a0 ∈ U (R) and ai ’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 ∈ Mn (R). Prove that: U (Mn (R)) = {A ∈ Mn (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 M¨obius function, on natural numbers. µ(1) = 1, and for non-freesquare numbers n, we have µ(n) = 0. Also if n = p1 p2 . . . ps , in which p1 , . . . , ps are different primes, µ(n) = (−1)s . Prove that µ(n) is multiplicative, i.e. if (n1 , n2 ) = 1, µ(n1 n2 ) = µ(n1 )µ(n2 ). Also prove that  X 1 if n = 1 µ(d) = 0 if n = 0 d|n

14. Prove the M¨ obius inversion formula. If f (n) is a function and defined on natural numbers, and X g(n) = f (n) d|n

Prove that f (n) =

X n g(d) µ d d|n

15. Prove that if ϕ(n) is the Euler function: ϕ(n) =

X 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: 1 X n d N (n, q) = µ q n d d|n

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. Cartan-Brauer-Hua 18. Prove that Z

h

√ 1+ −19 2

i

is not Euclidean.

19. Prove that the polynomial det(A) − 1 ∈ k[x11 , x12 , . . . , xnn ] 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. Mikl´ os 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. Mikl´ os 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. Mikl´ os 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 a3 − 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, R0 be two rings such that all of their elements are nilpotent. Let f : R0 → R be a bijective function such that for each x, y ∈ R0 , f (xy) = f (x)f (y). Prove that R ' R0 . 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 + · · · + xr−1 for some r ∈ N. Mikl´ os Schweitzer Competition

30. Let R be a finite ring and for each a, b ∈ R, there is an element c ∈ R such that a2 + b2 = c2 . Prove that for each a, b, c ∈ R, there is a d ∈ R such that 2abc = d2 . 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

Z2 [x] , (xn − 1)

I 2 = I. 33. Let A be ring with 2n + 1 elements. Let M := {k ∈ N|xk = x, ∀x ∈ A} Prove that A is a field, if and only if M is not empty, and the least element of M is equal to 2n + 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) 6= I. Also suppose that {(I : ri )}∞ i=1 is a finite set. Prove that there is a n ∈ N, such that (I : rn ) = R. 35. Let (A, +, ∗) be a finite ring in which 0 6= 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| = p2 . 36. Let R be a ring, and for each x ∈ R, x2 = 0. Prove that x = 0. Suppose that M = {a ∈ A|a2 = 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 6= 1. R contains 2n − 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 k such that x2 +1 = x. Prove that for each x ∈ A, x2 = x.

21 40. Let (A, +, ∗) be a ring in which 1 6= 0. The mapping f : A −→ A, f (x) = x10 is group homomorphism of (A, +). Prove that A contains 2 or 4 elements. Romanian Olympiad 1999 41. Let A be a ring and x2 = 1 or x2 = x for each x ∈ A. Prove that if A contains at least two invertible elements, A ∼ = Z3 42. Let R be a ring, and xn = 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 6= 1. Prove that A is not a field if and only if for each n, xn + 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. Romanian District Olympiad 2πi

45. Let n be an integer, and ζ = e n . Prove that: n X 2 √ k ζ = n k=1

46. Let R be a ring, in which a2 = 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 P1 , P2 , . . . , Pn are prime ideals of R. If I ⊂ P1 ∪ P2 ∪ · · · ∪ Pn , then ∃i, I ⊂ Pi .

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 am = bm and an = bn 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 Mn (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, x3 = 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 Mn (R) ∼ = S? = Mn (S). Does it imply R ∼ 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 6= n, such that for each x ∈ R, xm = xn . 64. Let R be a commutative ring. For each ideal I we define: √ I = {x ∈ R|∃n, xn ∈ I} Prove that

√

\

I=

J

J is prime,I⊂J 65. Prove that if F is a field, then F [x] is not a field. 66. Let I1 , I2 , . . . , In be ideals of commutative ring R, such that for each j 6= k, Ij + Ik = R. Prove that I1 ∩ I2 ∩ · · · ∩ In = I1 I2 . . . In . 67. Let R be a commutative ring with identity element. Prove that hxi 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 |m, n 6= 0 ∈ Z, p1 , p1 , . . . , pk - n} n

such that pi are prime numbers. Prove R has exactly k maximal ideals.

23 71. Let R be a ring. Prove that: p(x) = an xn + an−1 xn−1 + d · · · + a1 x + a0 is nilpotent if and only if ai is nilpotent for each i. 72. Let A be a ring, such that: (a) x + x = 0 for each x ∈ A. k

(b) For each x ∈ A, there is a k ≥ 1 such that x2

+1

= x.

Prove that x2 = 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 x3 = y 3 then x = y. Prove that if for a, b, c ∈ A a2 + b2 + c2 = ab + bc + ac then a = b = c. 75. Let (A, +, .) be a commutative ring with n ≥ 6 elements, which is a not field: (a) Prove that u : A −→ A  u(x) =

1, 1,

x 6= 0 x=0

is not a polynomial function. (b) Let P be the number of polynomial functions f : A −→ A of degree n. Prove that: n2 ≤ P ≤ nn−1 76. Find all n ≥ 1 such that there exists (A, +, .) such that for each x ∈ A\{0}, n x2 +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: x3 + ax2 + bx = 0 Prove that A is a commutative ring. 79. Let A be a commutative ring with 2n + 1 elements such that n > 4. Prove that for every non-invertible element such as, a2 ∈ {−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] Sz´ekely G.J., Contests in Higher Mathematics: Mikl´ os Schweitzer Competitions 1962-1991, Springer-Verlag 1996 [7] AoPS& Mathlinks The largest online problem solving community

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