book of problems

7/24/2019 Book of Problems

1/25

Problems in Abstract Algebra

Omid Hatami

[Version 0.3, 25 November 2008]


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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 inter-

esting 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

[email protected].
http://omidhatami.googlepages.com/http://omidhatami.googlepages.com/http://omidhatami.googlepages.com/mailto:[email protected]:[email protected]://omidhatami.googlepages.com/


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Contents

1 Group Theory Problems 5

1.1 First Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Second Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Third Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Fourth Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Extra Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Ring Theory Problems 17

3


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4 CONTENTS


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Chapter 1

Group Theory Problems

1.1 First Section

1. Let (G, ) be a group, and a1, a2, . . . , anG. Prove that:

(a1 a2 . . . an)1 =a1n . . . a11

2. For eacha, b Z, we define a b = a+b ab. Prove that (Z, , 0) is amonoid.

3. Prove thatR\{1} is a group under multiplication.4. LetMbe a monoid. Prove thata

Mhas an inverse, if and only if there

is a bM such that aba = a and ab2a= e.5. Prove that each group of size 5 is abelian.

6. (G, .) is a semigroup such that:

Ghas 1r which is an element such that for each aG, a.1r =a. Each aG has a right inverse.(a.b= 1r)

7. Suppose (G, ) is a group. For eachaG, let La :GG be La(x) =a x. Prove thatLa is one to one.

8. Prove that the equationx3 =e has odd solutions in group (G,.,e).

9. Supposea, b are two elements of group G, which dont commute. Provethat elements of subset{1,a,b,ab,ba}ofG 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 = e is even.Conclude that in each group of even order, there exists a= e, such thata2 =e.

11. A, B are subgroups ofG, such that|A| + |B|>|G|. Prove that AB = G.12. Prove that a finite monoidM is a group the set I={xM|x2 =x} has

only one element.

5


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6 CHAPTER 1. GROUP THEORY PROBLEMS

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

14. Prove that the equation x2ax = a1 has a solution in G, if and only ifthere is yG, such thaty3 =x.

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

(b) If for eachaG, a2 =e, prove thatG is abelian.16. (G,.,e) is a group and there exists nN, 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, ifxy =yz, then x= z .

Prove thatGis abelian.

18. G is a finite semigroup such that for each x= e, c2 = e. We know thatfor each a, bG, (ab)2 = (ba)2. Prove thatG is abelian.

19. G is a finite semigroup such that for each for each x G, there exists auniquey , such that xy x= x. Prove that G is a group.

20. A semigroupS is called a regular semigroup if for each yS, there is aaS, such thaty ay= y. LetSbe 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 ifx2 =x.


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1.2. SECOND SECTION 7

1.2 Second Section

21. Find all subgroups ofZ6.

22. G is an abelian group. Prove thatH={aG|o(a)2, Xk X, then X|G|1 < G.

32. LetGbe a finite group, andAis subgroup ofGsuch that |AxA| is constantfor each x. Prove that for each gG : gAg1 =A.

33. G is a finite group abelian group, such that for each a= e, a2 = e.Evaluate

a1a2 . . . an

which G =

{a1, a2, . . . , an

}.

34. Prove Wilsons Theorem. Ifp is a prime number:

(p 1)! 1 (mod p).

35. Let p be a prime number, and let a1, a2, . . . , ap1 be a permutation of{1, 2, . . . , p1}. Prove that there existsi=j such thatiaij aj (mod p).

36. m, n are two coprime numbers. a is an element ofG, such that an = 1.Prove that there exists b such thatbn =a.

37. Suppose thatSis a proper subgroup ofG. Prove thatG\S= G.


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38. Prove that union of two subgroups ofG is a subgroup ofG, if and only if

one of these subgroups is subset of the other subgroup.39. G is an abelian group and a, bG, such that gcd(o(a), o(b)) = 1. Prove

thato(ab) =o(a)o(b).

40. Suppose thatG is a simple nonabelian group. Prove that iff is an auto-morphism ofG such that x.f(x) =f(x).x for every xG, then f= 1.


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1.3. THIRD SECTION 9

1.3 Third Section

41. H, Kare normal subgroups ofG, and H K={1}. Prove that for eachxK, yH, xy = yx.

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

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 functionf :GG, f(x) =x3 isa monomorphism. Prove that G is abelian.

45. We call a normal subgroupN ofG a maximal normal subgroup if theredoes not exist a nonobvious a normal subgroup K, such thatN K G.Prove that N is a maximal normal subgroup of G, if and only if G

N is

simple.

46. G, Hare cyclic groups. Prove that G His 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 ofiIGi is finite.

48. Nis a normal subgroup ofG of finite order, and His a subgroup ofG offinite index, such thatg cd(|N|, [G: H]) = 1. Prove that NH.

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

subgroup of GM

GN

.

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

51. His a proper subgroup ofG. Prove that:

G=xG

xHx1

52. G is a finite group, and f : G G is an automorphism ofG such thatat for at least

34 of elements of G such as x, f(x) = x

1

. Prove thatf(x) =x1, and G is abelian.

53. Let G be a group of order 2n. Suppose that if half of elements ofG areof order 2, the remaining elements form a group of order n, likeH. Provethatn is odd, and His abelian.

54. LetG be a group that has a subgroup of orderm, and also has a subgroupof ordern. Prove that G has a subgroup of order lcm(m, n).

55. H is a subgroup ofG with finite index. Prove thatG has finitely manysubgroups of form xH x1.


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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 ofG with more thann2

elements. Prove that for everyg G, we can find h, k Ksuch thatg= h.k.

58. Letp >3 be a prime number, and:

1 +1

2+

1

3+ + 1

p 1 =a

b

Prove thatp2|a.

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

60. Let G be a finitely generated group, and H be a subgroup ofG of finiteindex. Prove thatH 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. His a subgroup of index r ofG. Prove that there existsz1, z2, . . . , zrGsuch that:

ri=1

ziH=

ri=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) Gis simple and finite.

(b) Ghas at least two maximal subgroups.

(c) For each two maximal subgroups such asG1, G2, G1 G2={e}.


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1.4. FOURTH SECTION 11

1.4 Fourth Section

65. Let G be a group and Hbe a subgroup ofG. Prove that ifG = Ha1Ha2 . . . H an. Prove that:

G= a11 H a12 H . . . a1n H

66. Prove thatAut(Q) = Q.

67. LetG = (Zn, +). Prove that Aut(G)=GLn(Z).68. G1, G2 are simple groups. Find all normal subgroups ofG1 G2.69. LetG be a group. Prove that Aut(G) is abelian, if and only ifG is cyclic.

70. a is the only element ofG which is of order n. Prove that aZ(G).71. Ghas exactly one subgroup of index n. Prove that the subgroup of order

n is normal.

72. Prove that if every cyclic subgroupT ofG, is a normal subgroup, then forevery subgroup ofG, is a normal subgroup.

73. A, B are two subgroups ofG, and [G: A] is finite. Prove that:

[A: A B][G: B ]

and equality occurs, if and only ifG = AB .

74. Let G be a group. We know that G =ki=1Hi, which Hi G, andHi Hj ={e}. Prove that G is abelian.

75. Sis a nonempty subset ofG, and|G|= n. For each k, letSk be:

{ki=1

si|siS}

Prove thatSn G.

76. H, Kare subgroups ofG. For each a, bG, prove that Ha Kb= orHa Kb = (H K)c for somecG.

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

78. Prove that there does not exist a finite nonobvious group such that eachofG except the unit, commutes with exactly half of elements ofG.

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.


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81. Gis an infinite simple group. Prove that:

(a) Each x=e has infinitely many conjugates.(b) Each H={e} has infinitely many conjugates.

82. Gis a group of orderpq, whichp < q,p, qare prime numbers andp|q1.Prove thatGis abelian.

83. LetNbe a normal subgroup of a finitep-group,G. Prove thatNZ(G) ={e}.

84. Let H be a normal subgroup ofG, and H G ={e}. Prove thatHZ(G).

85. G is a nonabelian group of order p3, which p is a prime number. Prove

that Z(G) =G.86. Gis a finite nonabelian p-group. Prove that|Aut(G)| is divisible by p2.87. Prove that the number of elements ofSn 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 ofSX generated by 3-cycles.Prove thatAis an infinte, simple group.

89. Let{Ni|i I} be a family of normal subgroups G, and N =iINi.Prove thatG/Nis isomorphic to a subgroup ofiIG/Ni. Prove that if[G: Ni]


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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 nsubgroups of order 2.

IMS 2007

96. LetKbe a subgroup of group G.

Prove that NG(K)CG(K)

is isomorphic to a subgroup ofAut(K).

Prove that ifKis abelian, and K G= G, then KZ(G).

IMS 2005

97. LetG be a finite group of ordern. Prove that if [G: Z(G)] = 4, then 8|n.For each 8|nfind 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. LetG be a finite group, and Hbe a subgroup ofG, such that:

x(x

H=

H

x1Hx =

{eG

})

Prove that|H| and [G: H] are coprime.

IMS 1993

100. LetG be a group and Hbe a subgroup ofG such that for each xG\Hand eachyG, there is a uHthaty1xy= u1xu. Prove thatHG,and G

H is abelian.

IMS 2003

101. Gis an abelian group and A, B are two different abelian subgroups ofG,such that [G: A] = [G: B ] =p, andp is the smallest integer dividing

|G

|.

Prove thatInn(G)= ZpZp.

IMS 1992

102. G is a finite p-group. Prove thatG=G.

IMS 1989


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1.5 Extra Problems

103. LetG be a transitive subgroup of symmetric group S25 different fromS25and A25. Prove that order ofG is not divisible by 23.

Miklos Schweitzer Competition

104. Determine all finite groups G that have an automorphism f such thatHf(H) for all proper subgroups H ofG.


105. LetG be a finite group, andK a conjugacy class ofG that generates G.Prove that the following two statements are equivalent:

There exists a positive integer m such that every element ofG canbe written as a product ofm (not necessarily distinct) elements ifK.

Gis equal to its own commutator subgroup.Miklos Schweitzer Competition

106. Letn = pk (pa prime number, k1), and let G be a transitive subgroupof the symmetric groupSn. Prove that the order of normalizer ofG in Snis at most|G|k+1.


107. LetG, Hbe two countable abelian groups. Prove that if for each naturaln, pnG= pn+1G, His a homomorphic image ofG.


108. LetG be a finite group, and p be the smallest prime number that divides|G|. Prove that ifA < Gis a group of order p, A < Z(G).

109. Leta,b >1 be two integers. Prove that Sa+b has a subgroup of orderab.

110. LetGbe an infinite group such that index of each of its subgroups is finite.Prove thatG is cyclic.

111. LetHbe a subgroup of group G, and [G : H] = 4. Prove that G has aproper subgroup K that [G: K]< 4.

112. LetA be a subgroup ofRn, such that for each bounded sunset B Rn,|A B|


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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 GLn2(Z) contains anelement of orderm.

116. Prove thatnN, group QZ

, +

has exactly one subgroup of order n.

117. Find alln such that An has a subgroup of order n.

118. LetG be a group andM , Nbe normal subgroups ofG such that MNand G

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

M is abelian.

119. LetG be a finite abelian group, and H is a subgroup ofG. Prove thatGhas a subgroup isomorphic to G

H.

120. LetG be a group, and let Hbe a maximal subgroup ofG. Prove that ifHis abelian G(3) =e.

121. Letf :GG be a homomorphism. Prove that:|f(G)|2 |G| |f(f(G))|

122. Prove that a simple groupG does not have a proper, simple subgroup offinite index.

123. LetGbe a finite group, and for each a, bG\{e}, there existsfAut(G)such that f(a) =b. Prove that Gis abelian.

124. Prove that there is no nonabelian finite simple group whose order is aFibonacci number.

125. Leta, b, cbe elements of odd order in groupG, anda2

b2

=c2

. Prove thataband c are in the same coset of commutator group(G).

126. Letn be an odd number, and G be a group of order 2n. His a subgroupofG of ordern such that for each xG\H, xhx1 =h1. Prove thatHis abelian, and each element ofG\H is of order 2.

Berkeley P5-Spring 1988

127. Prove that only subgroup of index 2 ofSn isAn.

128. Prove that if (n, (n)) = 1, each group of order nis abelian.

129. Prove that each uncountable abelian group has a proper subgroup of the

same cardinal.

David Hammer

130. LetGbe a group, and H is a subgroup and Hbe a subgroup of index 2.Prove that there is a permutation group isomorphic withG, 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 suchthat (i) = j. Prove that ifn is free-square, then each group satisfyingconditionT is abelian.


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132. Xis an infinite set. Prove that SX does not have proper subgroup of finite

index.133. LetG be a group of order pmn, such that m < 2p. Prove that G has a

normal subgroup of order pm orpm1.

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 3n

3 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=e, such thatuv = vu = x. Prove that ifx isnot a good element, x has order 2, and|G|= 2(2k 1) for some k N.

137. Let n 1 and x xn is an isomorphism. Prove that for all a G,an1 Z(G).

Hungary-Israel Binational 1993


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Chapter 2

Ring Theory Problems

1. Prove that all of continuous functions on R, such thatR

|f(x)|


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18 CHAPTER 2. RING THEORY PROBLEMS

10. LetR be a ring with 1. Prove that if

p(x) =anxn + an1xn1 + + ax+ a0U(R[x])

, if and only ifa0U(R) and ais 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 AMn(R). Prove that:

U(Mn(R)) ={AMn(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 (n) be the Mobius function, on natural numbers. (1) = 1, and for

non-freesquare numbers n, we have (n) = 0. Also ifn = p1p2 . . . ps, inwhich p1, . . . , ps are different primes, (n) = (1)s. Prove that (n) ismultiplicative, i.e. if (n1, n2) = 1,(n1n2) =(n1)(n2). Also prove that

d|n

(d) =

1 if n= 10 if n= 0

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

g(n) =d|n

f(n)

Prove that f(n) =d|n

nd

g(d)

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

(n) =d|n

n

d

16. Fbe a finite field with q elements. Prove that ifN(n, q) is the number ofirreducible polynomials of degreen:

N(n, q) = 1

nd|n

n

d qd

17. Let D be division ring, and Cis its center. S is a sub-division ring ofDsuch that is invariant under each of the mappings xdxd1, which d isa non-zero element ofD . Prove that S= D orSC.

Cartan-Brauer-Hua

18. Prove thatZ1+

192

is not Euclidean.

19. Prove that the polynomial det(A) 1k[x11, x12, . . . , xnn] is irreducible.


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19

20. Prove that in the ringR, 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

ofR 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 ofR is cyclic.


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


24. Let R be a commutative ring with 1. Prove thatR[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 ifJis an ideal ofR 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 RR.

IMS 2003

27. Let Rbe 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 RxRy orRyRx.

IMS 2002

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

29. LetIbe an ideal ofZ[x] such that:

(a) gcd of coefficients of each element ofI is 1.

(b) For eachR Z,Icontains an element with constant coefficient equalto R.


20/25


Prove thatIcontains an element of form 1 +x + +xr1 for somer 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. RingR 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 ring Z2[x]

(xn 1) ,I2 =I.

33. LetA be ring with 2n + 1 elements. Let

M :={k N|xk =x, xA}

Prove thatAis a field, if and only ifMis not empty, and the least elementofM is equal to 2n + 1.

Romanian District Olympiad 2004

34. LetIbe an irreducible ideal of commutative ringRcontaining 1. For eachrR, we define (I :r) ={xR|rxI}. LetrR be an element suchthat (I :r)=I. Also suppose that{(I :ri)}i=1 is a finite set. Prove thatthere is a nN, such that (I :rn) =R.

35. Let (A, +, ) be a finite ring in which 0= 1. If a, b A are such thatab= 0, then a = 0 orb {ka|kZ}. Prove that there is a primep suchthat|A|= p2.

36. Let R be a ring, and for each xR, x2 = 0. Prove thatx = 0. SupposethatM={aA|a2 =a}. Prove that ifa, bM, a + b 2abM.

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 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 xA, there is a ksuch that x2

k+1 =x. Prove that for each xA, x2 =x.


21/25

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 thatA 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,xyn1 =yn1x.

43. Let A be a finite ring in which 0= 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. Letn be an integer, and =e2i

n . Prove that:n

k=1

k2

=

n

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

IMC 2003

47. Let R be a ring of characteristic zero, and e,f,g are three idempotentelements, such thate + f+ g= 0. Prove that e = f=g = 0.

IMC 2000

48. Let R be a Noetherian ring, and f :AA is surjective. Prove thatfis injective.

49. Let A be a ring such thatab= 1 implies ba= 1. Prove that we have the

same property forR[x].50. Prove that in each Noetherian ring, there are only finitely many minimal

ideals.

51. LetR 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. LetKbe a field, and A is a ring containingK, which is finite dimensionalas aK-vector space. Prove that Ais Artinian and Noetherian ring.

53. Let R be a commutative ring with 1, and P1, P2, . . . , P n are prime idealsofR. IfIP1 P2 Pn, theni, IPi.


22/25


54. K is an infinite field. Find all of the automorphisms ofK.

55. Let R be a ring with no nilpotent non-zero element. Let a, b R suchthatam =bm and an =bn for some coprime m, n. Prove thata= b.

56. Let R be a ring with 1, and containing at least two elements, such thatfor each aR there is a unique elementbR such that aba = a. ProvethatRis a division ring.

57. Let F be a field and n > 1. Let R be the ring of all upper-triangularmatrices inMn(F), such that all of the elements on its diagonal are equal.Prove thatR 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 ofRis nilpotent or unit.

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

61. LetR, Sbe two rings such that Mn(R)=Mn(S). Does it imply R=S?62. LetKbe a field. Can K[x] have finitely many irreducible polynomials?

63. Let R be a finite commutative ring. Prove that there are m= n, suchthat for each xR, xm =xn.

64. LetR be a commutative ring. For each ideal Iwe define:

I={xR|n, xn I}

Prove that I=

J is prime,IJ

J

65. Prove that ifF is a field, then F[x] is not a field.

66. LetI1, I2, . . . , I nbe ideals of commutative ringR, such that for each j=k,Ij+ Ik =R. Prove thatI1 I2 In = I1I2 . . . I n.

67. Let R be a commutative ring with identity element. Prove thatx is aprime ideal in R[x], if and only ifR 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. Letm, nbe coprime numbers. Let

R={mn

|m, n= 0 Z, p1, p1, . . . , pk n}

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


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71. LetR be a ring. Prove that:

p(x) =anxn + an1xn1 + d + a1x + a0is nilpotent if and only ifai is nilpotent for each i.

72. LetA be a ring, such that:

(a) x + x= 0 for each xA.(b) For eachxA, there is a k1 such thatx2k+1 =x.

Prove thatx2 =x for each xA.

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 ifx3 =y3 then x = y. Prove that if for a, b, cA

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 : AA

u(x) =

1, x= 01, x= 0

is not a polynomial function.

(b) Let Pbe the number of polynomial functions f :AA of degreen. Prove that:

n2 P nn1

76. Find alln1 such that there exists (A, +, .) such that for each xA\{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,aZ(D).

78. Let (A, +, .) be a ring and a, bA such that for each xA:x3 + ax2 + bx= 0

Prove thatAis a commutative ring.

79. LetA be a commutative ring with 2n +1 elements such thatn >4. Provethat for every non-invertible element such as, a2 {a, a}. Prove thatAis a ring.


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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 bA such that xy = byx.

Prove thatAis commutative.


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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& MathlinksThe largest online problem solving community

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http://mathlinks.ro/http://mathlinks.ro/

book of problems

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