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 challengingproblems 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 inter-esting problems in Abstract Algebra. I also would appreciate hearing about anyerrors in the book, even minor ones. You can send all comments to the authorat [email protected].

http://omidhatami.googlepages.com

mailto:[email protected]

Contents

1 Group Theory Problems 51.1 First Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Second Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Third Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Fourth Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 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 a1, a2, . . . , an ∈ G. Prove that:

(a1 ∗ a2 ∗ . . . an)−1 = a−1n ∗ . . . a−11

2. For each a, b ∈ Z, we define a ? b = a + b − ab. Prove that (Z, ?, 0) is amonoid.

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 thereis a b ∈M such that aba = a and ab2a = 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. Provethat elements of subset {1, a, b, ab, ba} of G are all distinct. Conclude thatorder 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 thata2 = 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} hasonly one element.

5

6 CHAPTER 1. GROUP THEORY PROBLEMS

13. Let G be a group and x, y ∈ G, such that xy2 = y3x, and yx2 = x3y.Prove that x = y = e.

14. Prove that the equation x2ax = a−1 has a solution in G, if and only ifthere is y ∈ G, such that y3 = x.

15. (a) G is a group and for each a, b ∈ G, a2b2 = (ab)2. Prove that G isabelian.

(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}, aibi = (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 thatfor 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 aunique 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 aa ∈ 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 isregular, if and only if x2 = x.

1.2. SECOND SECTION 7

1.2 Second Section

21. Find all subgroups of Z6.

22. G is an abelian group. Prove that H = {a ∈ G|o(a) < ∞} is a subgroupof G.

23. Prove that group G is not union of two of its proper subgroups. Is thestatement 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 onlyif HH = H.

25. Let G be a group that does not have any nonobvious subgroups. Provethat 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, fora 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 onlyif 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 = 〈xm〉, if and only ifgcd(m, o(x)) = 1.

30. Let G be a group, and for each a, b ∈ G, we know that a3b3 = (ab)3, anda5b5 = (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 thatif for k > 2, Xk ⊂ X, then X |G|−1 < G.

32. Let G be a finite group, and A is subgroup of G such that |AxA| is constantfor 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

a1a2 . . . 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 〈G\S〉 = G.


38. Prove that union of two subgroups of G is a subgroup of G, if and only ifone 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. Provethat o(ab) = o(a)o(b).

40. Suppose that G is a simple nonabelian group. Prove that if f is an auto-morphism 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 eachx ∈ K, y ∈ H, xy = yx.

42. G is a group of odd order and x is multiplication of all elements in anarbitrary order. Prove that x ∈ G′.

43. Prove that an infinite group is cyclic, if and only if it is isomorphic to allof its nonobvious subgroups.

44. Let G be a group. We know that the function f : G −→ G, f(x) = x3 isa monomorphism. Prove that G is abelian.

45. We call a normal subgroup N of G a maximal normal subgroup if theredoes 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 issimple.

46. G,H are cyclic groups. Prove that G×H is a cyclic group, if and only ifgcd(|G|, |H|) = 1.

47. {Gi|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 offinite index, such that gcd(|N |, [G : H]) = 1. Prove that N ⊂ H.

49. M,N are normal subgroups of G. Prove that GM∩N is isomorphic to a

subgroup of GM ×

GN .

50. A,B are subgroups of G, such that gcd([G : A], [G : B]) = 1. Prove thatG = AB.

51. H is a proper subgroup of G. Prove that:

G 6=⋃x∈G

xHx−1

52. G is a finite group, and f : G −→ G is an automorphism of G such thatat for at least 3

4 of elements of G such as x, f(x) = x−1. Prove thatf(x) = x−1, and G is abelian.

53. Let G be a group of order 2n. Suppose that if half of elements of G areof order 2, the remaining elements form a group of order n, like H. Provethat n is odd, and H is abelian.

54. Let G be a group that has a subgroup of order m, and also has a subgroupof 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 manysubgroups of form xHx−1.


56. Consider the group (R,+) and it subgroup Z. Prove that RZ is a group

ismomorphic to complex numbers with norm 1 with the multiplicationoperation.

57. G is a finite group with n elements. K is a subset of G with more thann2 elements. Prove that for every g ∈ G, we can find h, k ∈ K such thatg = h.k.

58. Let p > 3 be a prime number, and:

1 +1

2+

1

3+ · · ·+ 1

p− 1=a

b

Prove that p2|a.

59. Let G be a finitely generated group. Prove that for each n, G has finitelymany groups of index n.

60. Let G be a finitely generated group, and H be a subgroup of G of finiteindex. Prove that H is finitely generated.

61. Let m and n be coprime. Assume that G is a group such that m-powersand n-powers commute. Then G is abelian.

62. H is a subgroup of index r of G. Prove that there exists z1, z2, . . . , zr ∈ Gsuch that:

r⋃i=1

ziH =

r⋃i=1

Hzi = G

63. G is a group of order 2k, in which k is an odd number. Prove that G hassubgroup of index 2.

64. Prove that there does not exist any group satisfying the following condi-tions:

(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 11

1.4 Fourth Section

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

G = a−11 H ∪ a−12 H ∪ . . . a−1n 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 ordern is normal.

72. Prove that if every cyclic subgroup T of G, is a normal subgroup, then forevery 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=1Hi, which Hi E G, andHi ∩Hj = {e}. Prove that G is abelian.

75. S is a nonempty subset of G, and |G| = n. For each k, let Sk be:

{k∏

i=1

si|si ∈ S}

Prove that Sn EG.

76. H,K are subgroups of G. For each a, b ∈ G, prove that Ha ∩Kb = ∅ orHa ∩Kb = (H ∩K)c for some c ∈ G.

77. Let S = ∪∞n=1Sn, which Sn is n-th symmetric group. Prove that onlynonobvious subgroup of S is A = ∪∞n=1An.

78. Prove that there does not exist a finite nonobvious group such that eachof 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 arenot conjugate in A5.


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 ∩ G′ = {e}. Prove that H ⊂Z(G).

85. G is a nonabelian group of order p3, which p is a prime number. Provethat Z(G) = G′.

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:

n!

(1

2!− 1

3!+ · · ·+ (−1)n

1

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 G, and N = ∩i∈INi.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. Concludethat 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 thattwo 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, andZ(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 oforder p, such that b 6∈ 〈a〉. Prove that G has at least p2 − 1 elements oforder p.

IMS 2001

93. G is a group, such that each of its subgroups are in a proper subgroup offinite 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 naturalnumber.) Prove that there exists a finite abelian group with exactly nsubgroups 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 = G′, 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 nonabeliangroup 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−1Hx = {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\Hand each y ∈ G, there is a u ∈ H that y−1xy = u−1xu. Prove that HEG,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) ∼= Zp × Zp.

IMS 1992

102. G is a finite p-group. Prove that G 6= G′.

IMS 1989


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.

Miklos Schweitzer Competition

104. Determine all finite groups G that have an automorphism f such thatH 6⊆ f(H) for all proper subgroups H of G.


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 canbe written as a product of m (not necessarily distinct) elements if K.

• G is equal to its own commutator subgroup.


106. Let n = pk (p a prime number, k ≥ 1), and let G be a transitive subgroupof the symmetric group Sn. Prove that the order of normalizer of G in Sn

is at most |G|k+1.


107. Let G,H be two countable abelian groups. Prove that if for each naturaln, pnG = pn+1G, H is a homomorphic image of G.


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 aproper 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 abeliangroup 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 or1x ∈ A. Prove that H = {0}.

1.5. EXTRA PROBLEMS 15

115. Let n be an even number greater than 2. Prove that if the symmetricgroup Sn contains an element of order m, then GLn−2(Z) contains anelement of order m.

116. Prove that ∀n ∈ N, group(QZ ,+

)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 ⊂ Nand G

N is cyclic and [N : M ] = 2. Prove that GM is abelian.

119. Let G be a finite abelian group, and H is a subgroup of G. Prove that Ghas a subgroup isomorphic to G

H .

120. Let G be a group, and let H be a maximal subgroup of G. Prove that ifH 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 offinite 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 aFibonacci number.

125. Let a, b, c be elements of odd order in group G, and a2b2 = c2. Prove thatab 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 subgroupof G of order n such that for each x ∈ G\H, xhx−1 = h−1. Prove that His 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 thesame 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 itsalternating subgroup is isomorphic to H.

131. We say that the permutation satisfies the condition T , if and only if itis abelian, and for each i, j ∈ {1, 2, . . . , n} there is a permutation σ suchthat σ(i) = j. Prove that if n is free-square, then each group satisfyingcondition T is abelian.


132. X is an infinite set. Prove that SX does not have proper subgroup of finiteindex.

133. Let G be a group of order pmn, such that m < 2p. Prove that G has anormal subgroup of order pm or pm−1.

134. Let p be a prime number and H is a subgroup of Sp, and contains atransposition and a p-cycle. Prove that H = Sp.

135. Prove that the largest abelian subgroup of Sn contains at most 3n3 ele-

ments.

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 isnot 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∫R|f(x)| <∞

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 that1− a is also idempotent.

(b) Prove that if R is an integral domain, the only idempotent elementsof R are 0, 1.

(c) Let R be ring and each of its elements are idempotent. Prove thatR is commutative with characteristic 2.

4. Give an example of ideal such that is not a subring and give an exampleof 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, suchthat 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. Provethat u+ v is also nilpotent.

8. Let R be a ring. Prove that if a has more than one right inverses, then ithas 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) = anxn + an−1x

n−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 well-defined 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 alsoinvertible.

13. We µ(n) be the Mobius function, on natural numbers. µ(1) = 1, and fornon-freesquare numbers n, we have µ(n) = 0. Also if n = p1p2 . . . ps, inwhich p1, . . . , ps are different primes, µ(n) = (−1)s. Prove that µ(n) ismultiplicative, i.e. if (n1, n2) = 1, µ(n1n2) = µ(n1)µ(n2). Also prove that∑

d|n

µ(d) =

{1 if n = 10 if n = 0

14. Prove the Mobius inversion formula. If f(n) is a function and defined onnatural numbers, and

g(n) =∑d|n

f(n)

Prove thatf(n) =

∑d|n

µ(nd

)g(d)

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

ϕ(n) =∑d|n

µ(nd

)

16. F be a finite field with q elements. Prove that if N(n, q) is the number ofirreducible polynomials of degree n:

N(n, q) =1

n

∑d|n

µ(nd

)qd

17. Let D be division ring, and C is its center. S is a sub-division ring of Dsuch that is invariant under each of the mappings x→ dxd−1, which d isa non-zero element of D. Prove that S = D or S ⊂ C.

Cartan-Brauer-Hua

18. Prove that Z[1+√−192

]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 thenumber of nilpotents.

21. Let R be an Artinian ring with 1. Prove that each idempotent elementof 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 thatAB = BA.


22. Let R be an infinite ring such that each of its subrings except {0} hasfinite index (index of a subring is the index of its additive group). Provethat the additive group of R is cyclic.


23. Let R be a finite ring. Prove that R contains 1, if and only if the onlyannihilator of R is 0.


24. Let R be a commutative ring with 1. Prove that R[x] contains infinitelymany 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/Jcontains 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. Letf : 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. LetR 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, wehave Rx ⊂ Ry or Ry ⊂ Rx.

IMS 2002

28. Prove that intersection of all of left maximal ideals of a ring is a two-sidedideal.

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 equalto R.


Prove that I contains an element of form 1+x+ · · ·+xr−1 for some r ∈ N.


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

Vojtec Jarnick Competition

31. Ring R has at least one divisor of zero, and the number of its zero divisorsis finite. Prove that R is finite.

Vojtec Jarnick Competition

32. Let n be an odd number. Prove that for each ideal of ringZ2[x]

(xn − 1),

I2 = 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 elementof M is equal to 2n + 1.

Romanian District Olympiad 2004

34. Let I be an irreducible ideal of commutative ring R containing 1. For eachr ∈ R, we define (I : r) = {x ∈ R|rx ∈ I}. Let r ∈ R be an element suchthat (I : r) 6= I. Also suppose that {(I : ri)}∞i=1 is a finite set. Prove thatthere 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 thatab = 0, then a = 0 or b ∈ {ka|k ∈ Z}. Prove that there is a prime p suchthat |A| = p2.

36. Let R be a ring, and for each x ∈ R, x2 = 0. Prove that x = 0. Supposethat 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.


39. Let (A,+, ∗) be a ring with characteristic 2. For each x ∈ A, there is a k

such that x2k+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 or4 elements.


41. Let A be a ring and x2 = 1 or x2 = x for each x ∈ A. Prove that if Acontains 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,xyn−1 = yn−1x.

43. Let A be a finite ring in which 0 6= 1. Prove that A is not a field if andonly if for each n, xn + yn = zn has a solution.

44. Let A be a finite commutative ring with at least 2 elements and n is anatural number. Prove that there exists p ∈ A[x], such that p does nothave any roots in A.

Romanian District Olympiad

45. Let n be an integer, and ζ = e2πin . Prove that:∣∣∣∣∣n∑

k=1

ζk2

∣∣∣∣∣ =√n

46. Let R be a ring, in which a2 = 0 for each a ∈ A. Prove that for eacha, b, c ∈ R, abc+ abc = 0.

IMC 2003

47. Let R be a ring of characteristic zero, and e, f, g are three idempotentelements, 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 fis injective.

49. Let A be a ring such that ab = 1 implies ba = 1. Prove that we have thesame property for R[x].

50. Prove that in each Noetherian ring, there are only finitely many minimalideals.

51. Let R be an Euclidean ring, with a unique Euclidean division. Prove thatthis 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 dimensionalas 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 idealsof R. If I ⊂ P1 ∪ P2 ∪ · · · ∪ Pn, then ∃i, I ⊂ Pi.


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 suchthat 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 thatfor each a ∈ R there is a unique element b ∈ R such that aba = a. Provethat R is a division ring.

57. Let F be a field and n > 1. Let R be the ring of all upper-triangularmatrices 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 iscommutative.

59. Let R be a commutative and contains only one prime ideal. Prove thateach element of R is nilpotent or unit.

60. Prove tha each boolean ring without 1, can be embedded into a booleanring with 1.

61. Let R,S be two rings such that Mn(R) ∼= Mn(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 6= n, suchthat 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 is prime,I⊂J

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 = I1I2 . . . In.

67. Let R be a commutative ring with identity element. Prove that 〈x〉 is aprime 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 = {mn|m,n 6= 0 ∈ Z, p1, p1, . . . , pk - 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) = anxn + an−1x

n−1 + d · · ·+ a1x+ 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.

(b) For each x ∈ A, there is a k ≥ 1 such that x2k+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 gen-erated. Prove that R is Noetherian.

74. (A,+, .) is a commutative ring in which 1 + 1 and 1 + 1 + 1 are invertible,and if x3 = y3 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 notfield:

(a) Prove that u : A −→ A

u(x) =

{1, x 6= 01, x = 0

is not a polynomial function.

(b) Let P be the number of polynomial functions f : A −→ A of degreen. Prove that:

n2 ≤ P ≤ nn−1

76. Find all n ≥ 1 such that there exists (A,+, .) such that for each x ∈ A\{0},x2

n+1 = 1

Romanian National Mathematics Olympiad 2007

77. Let D be division ring, and a ∈ D. Prove that if a has finitely manyconjugates, 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. Provethat for every non-invertible element such as, a2 ∈ {−a, a}. Prove that Ais a ring.


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, Sec-ond Edition, 1990

[4] Bhattacharya P.B., Jain S.K., Nagpaul S.R., Basic abstract algebra, SecondEdition, 1994

[5] Rotman J.J. An Introduction to The Theory of Groups, Fourth Edition,Springer-Verlag 1995

[6] Szekely G.J., Contests in Higher Mathematics: Miklos Schweitzer Competi-tions 1962-1991, Springer-Verlag 1996

[7] AoPS& Mathlinks The largest online problem solving community

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http://mathlinks.ro

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