Review Problems for Exam 2 - BYU Mathdoud/Math371/Exam2Review.pdf · Darrin Doud Brigham Young University Exam 2 Review. Written Answers 21. De ne the terms in boldface by completing

Review Problems for Exam 2

Darrin Doud

March 25, 2019

Darrin Doud Brigham Young University Exam 2 Review

True/False

1. Let R be a ring. Every prime ideal of R is maximal.

False


True/False

1. Let R be a ring. Every prime ideal of R is maximal.

False


True/False

2. Let R be a ring with unity and let I be an ideal of R. If Icontains a unit, then I = R.

True


True/False

2. Let R be a ring with unity and let I be an ideal of R. If Icontains a unit, then I = R.

True


True/False

3. Let G = Z3 × Z9 be an additive group, and let H be thesubgroup of G generated by (2, 6). The two cosets H + (2, 4)and H + (4, 1) are equal.

True


True/False

3. Let G = Z3 × Z9 be an additive group, and let H be thesubgroup of G generated by (2, 6). The two cosets H + (2, 4)and H + (4, 1) are equal.

True


True/False

4. Let G be an abelian group, and define f : G→ G byf(g) = g3. Then f is a group homomorphism.

True


True/False

4. Let G be an abelian group, and define f : G→ G byf(g) = g3. Then f is a group homomorphism.

True


True/False

5. Let R be a commutative ring with unity, and letr ∈ R− {0}. If I is an ideal of R and r ∈ I, then I = R.

False


True/False

5. Let R be a commutative ring with unity, and letr ∈ R− {0}. If I is an ideal of R and r ∈ I, then I = R.

False


True/False

6. Every ideal of Z is an ideal of Q.

False


True/False

6. Every ideal of Z is an ideal of Q.

False


True/False

7. If G is a cyclic group, and H is a subgroup, then H CGand G/H is cyclic.

True


True/False

7. If G is a cyclic group, and H is a subgroup, then H CGand G/H is cyclic.

True


True/False

8. If R is an integral domain, then R contains no idempotentelements.

False


True/False

8. If R is an integral domain, then R contains no idempotentelements.

False


True/False

9. Let R = Z, and let I = 15Z and J = 12Z. Then the idealI + J = 27Z.

False


True/False

9. Let R = Z, and let I = 15Z and J = 12Z. Then the idealI + J = 27Z.

False


True/False

10. Let G and H be groups, and let φ : G→ H be ahomomorphism. If H is cyclic, then G is cyclic.

False


True/False

10. Let G and H be groups, and let φ : G→ H be ahomomorphism. If H is cyclic, then G is cyclic.

False


Multiple Choice

11. Consider the following theorem.Theorem: Let G be abelian, and let D ⊆ G×G be the setD = {(a, a) ∈ G×G : a ∈ G}. Then D C (G×G) and(G×G)/D ∼= G.Which of the following would be most useful in proving thistheorem?a) Cayley’s Theoremb) The First Isomorphism Theoremc) The Second Isomorphism Theoremd) The Third Isomorphism Theorem

b) The First Isomorphism Theorem


Multiple Choice

11. Consider the following theorem.Theorem: Let G be abelian, and let D ⊆ G×G be the setD = {(a, a) ∈ G×G : a ∈ G}. Then D C (G×G) and(G×G)/D ∼= G.Which of the following would be most useful in proving thistheorem?a) Cayley’s Theoremb) The First Isomorphism Theoremc) The Second Isomorphism Theoremd) The Third Isomorphism Theorem

b) The First Isomorphism TheoremDarrin Doud Brigham Young University Exam 2 Review

Multiple Choice

12. Which of the following rings is not an integral domain?a) {a+ bi : a, b ∈ Z} b) Q c) Z/(2Z)d) M2(R) (two-by-two matrices with real entries)

d) M2(R)


Multiple Choice

12. Which of the following rings is not an integral domain?a) {a+ bi : a, b ∈ Z} b) Q c) Z/(2Z)d) M2(R) (two-by-two matrices with real entries)

d) M2(R)


Multiple Choice

13. Which of the following is an example of a noncommutativedivision ring?a) M2(R)b) The invertible three-by-three matrices with entries in Qc) The quaternionsd) C (the complex numbers)e) None of the above.

c) The quaternions


Multiple Choice

13. Which of the following is an example of a noncommutativedivision ring?a) M2(R)b) The invertible three-by-three matrices with entries in Qc) The quaternionsd) C (the complex numbers)e) None of the above.

c) The quaternions


Multiple Choice

14. Let G = GL(2,R) be the group of 2× 2 matrices withnonzero determinant under matrix multiplication, and letH = {A ∈ G : det(A) > 0}. Which of the followingstatements is true?a) H is not a subgroup of G.b) H is a subgroup of G, but is not a normal subgroup of G.c) G/H is isomorphic to R− {0} under multiplication.d) G/H is isomorphic to the positive real numbers undermultiplication.e) G/H is isomorphic to {1,−1} under multiplication.

e) G/H is isomorphic to {1,−1} under multiplication.


Multiple Choice

14. Let G = GL(2,R) be the group of 2× 2 matrices withnonzero determinant under matrix multiplication, and letH = {A ∈ G : det(A) > 0}. Which of the followingstatements is true?a) H is not a subgroup of G.b) H is a subgroup of G, but is not a normal subgroup of G.c) G/H is isomorphic to R− {0} under multiplication.d) G/H is isomorphic to the positive real numbers undermultiplication.e) G/H is isomorphic to {1,−1} under multiplication.

e) G/H is isomorphic to {1,−1} under multiplication.


Multiple Choice

15. Let G be a group, and let H be a subgroup of G. Fora, b ∈ G, which of the following is equivalent to saying

Ha = Hb?

a) a = b.b) ab−1 ∈ H.c) a−1b ∈ H.d) There is some g ∈ G such that a = gbg−1.e) None of the above.

b) ab−1 ∈ H.


Multiple Choice

15. Let G be a group, and let H be a subgroup of G. Fora, b ∈ G, which of the following is equivalent to saying

Ha = Hb?

a) a = b.b) ab−1 ∈ H.c) a−1b ∈ H.d) There is some g ∈ G such that a = gbg−1.e) None of the above.

b) ab−1 ∈ H.


Multiple Choice

16. Let G be the additive group Z× Z, and letφ : Z× Z→ Z5 be defined by φ(a, b) = b̄. Find the kernel ofφ.a) {(a, b) : a, b ∈ Z, b = 0}.b) {(a, 5b) : a, b ∈ Z}.c) {(5a, b) : a, b ∈ Z}.d) {(5a, 5b) : a, b ∈ Z}.

b) {(a, 5b) : a, b ∈ Z}.


Multiple Choice

16. Let G be the additive group Z× Z, and letφ : Z× Z→ Z5 be defined by φ(a, b) = b̄. Find the kernel ofφ.a) {(a, b) : a, b ∈ Z, b = 0}.b) {(a, 5b) : a, b ∈ Z}.c) {(5a, b) : a, b ∈ Z}.d) {(5a, 5b) : a, b ∈ Z}.

b) {(a, 5b) : a, b ∈ Z}.


Multiple Choice

17. Let G = S3 and let H = 〈(1, 2, 3)〉 be the subgroup of Ggenerated by the permutation (1, 2, 3). Which of the followingstatements is false?a) G/H is cyclic.b) G/H is abelian.c) |G/H| = 2d) H is not normal in G.

d) H is not normal in G.


Multiple Choice

17. Let G = S3 and let H = 〈(1, 2, 3)〉 be the subgroup of Ggenerated by the permutation (1, 2, 3). Which of the followingstatements is false?a) G/H is cyclic.b) G/H is abelian.c) |G/H| = 2d) H is not normal in G.

d) H is not normal in G.


Multiple Choice

18. How many idempotent elements are there in Z12?a) 1.b) 2.c) 3.d) 4.e) 6.f) 12.

d) 4.


Multiple Choice

18. How many idempotent elements are there in Z12?a) 1.b) 2.c) 3.d) 4.e) 6.f) 12.

d) 4.


Multiple Choice

19. Let R be the ring of real valued functions on the real line.Which of the following is a prime ideal of R?a) {f : f(x) = 0 for all x ∈ R}b) {f : f(1) = 0 and f(2) = 0}.c) {f : f(1) = 0}.d) {f : f(1) = 1}.

c) {f : f(1) = 0}.


Multiple Choice

19. Let R be the ring of real valued functions on the real line.Which of the following is a prime ideal of R?a) {f : f(x) = 0 for all x ∈ R}b) {f : f(1) = 0 and f(2) = 0}.c) {f : f(1) = 0}.d) {f : f(1) = 1}.

c) {f : f(1) = 0}.


Multiple Choice

20. Let R be a ring with unity and let S be a subring of Rcontaining 1R. which of the following is true.a) If r ∈ R is a unit of R, and r ∈ S, then r is a unit of S.b) If r ∈ R is a nilpotent element of R, and r ∈ S, then r is anilpotent element of S.c) If S is an integral domain, then R is an integral domain.

b) If r ∈ R is a nilpotent element of R, and r ∈ S, thenr is a nilpotent element of S.


Multiple Choice

20. Let R be a ring with unity and let S be a subring of Rcontaining 1R. which of the following is true.a) If r ∈ R is a unit of R, and r ∈ S, then r is a unit of S.b) If r ∈ R is a nilpotent element of R, and r ∈ S, then r is anilpotent element of S.c) If S is an integral domain, then R is an integral domain.

b) If r ∈ R is a nilpotent element of R, and r ∈ S, thenr is a nilpotent element of S.


Written Answers

21. Define the terms in boldface by completing the sentences:

An integral domain is

A maximal ideal of a ring R is an ideal I such that

An element r of a ring R is nilpotent if

The kernel of a ring homomorphism ϕ : R→ S is

Two groups G and H are isomorphic if


Written Answers

22. Let R be a commutative ring with unity and let I be anideal. Prove that R/I is a field if and only if I is a maximalideal.


Written Answers

23. Find the units, the zero divisors, the nilpotent elementsand the maximal ideals of the ring Z18.


Written Answers

24. Let G = Z8 × Z4 and let H = 〈(2, 1)〉 be a subgroup ofG. Determine the size of G/H, and whether G/H is cyclic.Justify your answer.


Written Answers

25. Let R and S be rings, and let ϕ : R→ S be a ringhomomorphism. Prove that ϕ is injective if and only ifkerϕ = {0}.


Written Answers

26. Let G and K be groups. Let ϕ : G→ K be a surjectivehomomorphism, and let J be a normal subgroup of K. Provethat there is a normal subgroup H of G such thatG/H ∼= K/J .


Written Answers

27. Let R be the ring of real valued functions on the realnumbers, with pointwise addition and multiplication. Give anexample of an ideal I of R that is not maximal, and an ideal Iof R that is maximal. In each case, prove that the ideal yougive has the desired property. (You may use any theoremsfrom the book that you wish in your proof.)


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Review Problems for Exam 2 - BYU Mathdoud/Math371/Exam2Review.pdf · Darrin Doud Brigham Young University Exam 2 Review. Written Answers 21. De ne the terms in boldface by completing