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
1.1 First Section . . . . .
1.2 Second Section . . . .
1.3 Third Section . . . . .
1.4 Fourth Section . . . .
1.5 Extra Problems . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2 Ring Theory Problems
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
5
7
9
11
14
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:
−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.
Miklós 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.
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 = 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 .
Miklós 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.
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 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 Möbius 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 Möbius 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.
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 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.
Miklós 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] 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