Rings - Euclidean rings, polynomial rings, unique factorization domains, quotient rings, ideals, maximal ideals - Question Bank

1. Which of the following statements about ideals is FALSE?
A) In a PID, every prime ideal is maximal.
B) In a UFD, every irreducible element is prime.
C) In a commutative ring with unity, every maximal ideal is prime.
D) In a commutative ring with unity, every prime ideal is maximal.
2. Consider the ring Z_4. The ideal I = (2) = {0, 2}. Is I a maximal ideal in Z_4?
A) Yes, because Z_4/(2) is isomorphic to Z_2, which is a field.
B) No, because Z_4/(2) is isomorphic to Z_2, which is not a field.
C) Yes, because Z_4/(2) is isomorphic to Z, which is a field.
D) No, because Z_4/(2) is isomorphic to Z, which is not a field.
3. Let R be an integral domain. If R is a PID, then R is also:
A) A Euclidean Domain
B) A UFD
C) A Field
D) An Artinian Ring
4. In the context of Euclidean rings, the 'division algorithm' refers to:
A) The ability to divide any element by any non-zero element.
B) For any a, b in R with b != 0, there exist q, r such that a = bq + r with r = 0 or φ(r) < φ(b).
C) The existence of a unique quotient and remainder.
D) The property that every non-zero element is a unit.
5. Which of the following is a UFD but not a PID?
A) Z
B) F[x]
C) Z[x]
D) Q
6. Let R = Z_6. The ideal I = (3) = {0, 3}. Is I a maximal ideal in Z_6?
A) Yes, because Z_6/(3) is isomorphic to Z_3, which is a field.
B) No, because Z_6/(3) is isomorphic to Z_3, which is not a field.
C) Yes, because Z_6/(3) is isomorphic to Z_2, which is a field.
D) No, because Z_6/(3) is isomorphic to Z_2, which is not a field.
7. What is the degree of the zero polynomial?
A) 0
B) 1
C) Undefined or -∞
D) Depends on the ring
8. Which of the following is NOT a polynomial ring?
A) Z[x]
B) Q[x]
C) R[x, y]
D) Z_n
9. If R is a commutative ring with unity, and I is an ideal, when is R/I a field?
A) When I is a prime ideal.
B) When I is a maximal ideal.
C) When I is a principal ideal.
D) When I is the zero ideal.
10. Which property guarantees that an integral domain is a UFD?
A) It is a field.
B) It satisfies the ascending chain condition on ideals (ACC).
C) Every non-zero non-unit element has a unique factorization into irreducibles.
D) It is a principal ideal domain.
11. In a UFD, an element is irreducible if and only if it is:
A) A unit.
B) Zero.
C) Prime.
D) A zero divisor.
12. Consider the ring Z. The ideal (0) is:
A) Maximal but not prime.
B) Prime but not maximal.
C) Both maximal and prime.
D) Neither maximal nor prime.
13. What is the relationship between maximal ideals and prime ideals in a commutative ring with unity?
A) Every maximal ideal is prime.
B) Every prime ideal is maximal.
C) Maximal and prime ideals are the same.
D) They are completely unrelated.
14. In the ring Z[x], is the ideal (x) a maximal ideal?
A) Yes, because Z[x]/(x) is isomorphic to Z, which is an integral domain.
B) No, because Z[x]/(x) is isomorphic to Z, which is not a field.
C) Yes, because Z[x]/(x) is isomorphic to Z, which is a field.
D) No, because Z[x]/(x) is isomorphic to Z, which is not an integral domain.
15. If R is a commutative ring with unity and I is a prime ideal, then R/I is:
A) An integral domain
B) A field
C) The zero ring
D) Isomorphic to R
16. If R is a commutative ring with unity and I is a maximal ideal, then R/I is:
A) An integral domain
B) A field
C) The zero ring
D) Isomorphic to R
17. Which of the following is an example of a maximal ideal in Z?
A) (4)
B) (6)
C) (p) where p is prime
D) (0)
18. The quotient ring Z_n is a field if and only if:
A) n is composite.
B) n is prime.
C) n is even.
D) n is odd.
19. Consider the ring Z. Is the ideal (2) a maximal ideal?
A) No, because Z/(2) is not a field.
B) Yes, because Z/(2) is isomorphic to Z_2, which is a field.
C) No, because (2) is not a prime ideal.
D) Yes, because Z/(2) is isomorphic to Z.
20. Which condition is necessary for a ring R to be a Euclidean domain?
A) R must be finite.
B) R must be commutative and have a unity element.
C) R must be a field.
D) R must have exactly two ideals.
21. What is the ideal generated by the polynomial x in the ring R[x]?
A) R[x]
B) The set of constant polynomials.
C) The set of polynomials with zero constant term.
D) The zero ideal.
22. In the ring Z, the ideal generated by 6 and 9 is:
A) (3)
B) (15)
C) (54)
D) (9)
23. Let I be an ideal in a commutative ring R with unity. R/I is an integral domain if and only if I is:
A) Maximal
B) Principal
C) Prime
D) Zero
24. The ring of polynomials F[x] over a field F is a Euclidean domain with the Euclidean function being:
A) The absolute value of the polynomial.
B) The degree of the polynomial.
C) The number of terms in the polynomial.
D) The leading coefficient of the polynomial.
25. A UFD is an integral domain where every non-zero, non-unit element can be factored into:
A) Irreducible elements uniquely, up to order and units.
B) Prime elements uniquely, up to order and units.
C) Units uniquely, up to order.
D) Zero elements uniquely.
26. In an integral domain, what is the relationship between irreducible and prime elements?
A) All irreducible elements are prime, and all prime elements are irreducible.
B) All irreducible elements are prime, but not vice-versa.
C) All prime elements are irreducible, but not vice-versa.
D) Irreducible and prime elements are unrelated concepts.
27. In a Euclidean domain, every non-zero non-unit element is:
A) Zero
B) A unit
C) Irreducible
D) Prime
28. Which of the following is a unit in the ring of integers Z?
A) 2
B) -1
C) 0
D) 3
29. Which of the following is an irreducible element in the ring of integers Z?
A) 1
B) 4
C) 7
D) 0
30. Consider the ring Z[i] (Gaussian integers). Is it a Euclidean domain?
A) No, it is not an integral domain.
B) Yes, with the norm function N(a+bi) = a^2 + b^2.
C) Yes, but the norm function is more complex.
D) No, it is a UFD but not a Euclidean domain.
31. In the ring of polynomials F[x], what is the ideal generated by the polynomial x^2 + 1 over the field of real numbers R?
A) (x)
B) (x^2)
C) (x^2 + 1)
D) (x - 1)
32. The First Isomorphism Theorem for Rings states that if φ: R → S is a surjective ring homomorphism, then R/ker(φ) is isomorphic to:
A) R
B) S
C) ker(φ)
D) Im(φ)
33. Let R = Z and I = 4Z. The quotient ring Z/4Z is isomorphic to:
A) Z
B) Z_2
C) Z_4
D) Z_5
34. What is the quotient ring R/I formed by an ideal I in a ring R?
A) The set of all ideals of R.
B) The set of cosets of I in R, with appropriate ring operations.
C) The set of all units in R.
D) The set of all zero divisors in R.
35. Consider the polynomial ring R[x]. If R is a UFD, then R[x] is:
A) Always a PID.
B) A UFD, but not necessarily a PID.
C) Never a UFD.
D) A field.
36. What is the relationship between PIDs and UFDs?
A) Every UFD is a PID.
B) Every PID is a UFD.
C) PIDs and UFDs are unrelated concepts.
D) A ring is a PID if and only if it is a UFD.
37. Is the ring of polynomials F[x] over a field F a UFD?
A) No, it is not an integral domain.
B) Yes, it is a UFD.
C) Only if F is a finite field.
D) Only if F is the field of real numbers.
38. Which of the following is a UFD?
A) The ring of polynomials in two variables over a field, F[x, y]
B) The ring of integers Z
C) The ring of polynomials Z[x]
D) The ring of polynomials Z[x, y]
39. What is a Unique Factorization Domain (UFD)?
A) An integral domain where every non-zero, non-unit element can be uniquely factored into irreducible elements, up to order and units.
B) A ring where every ideal is principal.
C) A ring where every element is either a unit or nilpotent.
D) A field.
40. In a principal ideal domain (PID), every prime ideal is also:
A) Maximal
B) Principal
C) A field
D) An idempotent ideal
41. For a commutative ring R with unity, the ideal I is prime if and only if:
A) R/I is a field.
B) R/I is an integral domain.
C) R/I is isomorphic to R.
D) R/I is the zero ring.
42. What is a prime ideal in a commutative ring R with unity?
A) An ideal I such that if ab ∈ I, then a ∈ I or b ∈ I.
B) An ideal I such that if ab ∈ I, then a ∈ I and b ∈ I.
C) An ideal I such that R/I is a field.
D) An ideal I such that I = R.
43. For a commutative ring R with unity, the ideal I is maximal if and only if:
A) R/I is a field.
B) R/I is an integral domain.
C) R/I is isomorphic to R.
D) R/I is the zero ring.
44. What is a maximal ideal in a commutative ring R with unity?
A) An ideal I such that if J is an ideal with I ⊆ J ⊆ R, then J = I or J = R.
B) An ideal I such that R/I is an integral domain.
C) An ideal I such that I is a prime ideal.
D) An ideal I such that I = (0).
45. Which property is shared by all Euclidean domains?
A) They are fields.
B) They are principal ideal domains.
C) They are commutative rings with unity.
D) They are finite rings.
46. A principal ideal domain (PID) is an integral domain where every ideal is:
A) Prime
B) Maximal
C) Principal
D) Idempotent
47. What is a principal ideal in a ring R?
A) An ideal generated by a single element.
B) An ideal that is also a subring.
C) An ideal that contains all zero divisors.
D) An ideal that is maximal.
48. In a ring R, what is an ideal I?
A) A subring of R.
B) A subset of R that is an additive subgroup and absorbs multiplication from R.
C) A multiplicative subset of R.
D) A ring homomorphism from R to itself.
49. Which of the following is NOT necessarily a Euclidean domain?
A) The ring of integers, Z
B) The ring of Gaussian integers, Z[i]
C) The ring of polynomials F[x] over a field F
D) The ring of Eisenstein integers, Z[ω]
50. What is the defining property of a Euclidean domain?
A) It is a field.
B) It possesses a Euclidean function (or norm) that allows for a division algorithm.
C) It is a principal ideal domain.
D) It is a unique factorization domain.
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