Rings - Euclidean rings, polynomial rings, unique factorization domains, quotient rings, ideals, maximal ideals - One Line Questions

1. In the ring Z, the ideal generated by 6 and 9 is: — (3)
2. Which of the following is an example of a maximal ideal in Z? — (p) where p is prime
3. 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? — (x^2 + 1)
4. What is the degree of the zero polynomial? — Undefined or -∞
5. Which of the following is an irreducible element in the ring of integers Z? — 7
6. Which of the following is a unit in the ring of integers Z? — -1
7. Let R be an integral domain. If R is a PID, then R is also: — A UFD
8. In a ring R, what is an ideal I? — A subset of R that is an additive subgroup and absorbs multiplication from R.
9. In a UFD, an element is irreducible if and only if it is: — Prime.
10. In an integral domain, what is the relationship between irreducible and prime elements? — All prime elements are irreducible, but not vice-versa.
11. Consider the polynomial ring R[x]. If R is a UFD, then R[x] is: — A UFD, but not necessarily a PID.
12. What is a principal ideal in a ring R? — An ideal generated by a single element.
13. What is a prime ideal in a commutative ring R with unity? — An ideal I such that if ab ∈ I, then a ∈ I or b ∈ I.
14. What is a maximal ideal in a commutative ring R with unity? — An ideal I such that if J is an ideal with I ⊆ J ⊆ R, then J = I or J = R.
15. If R is a commutative ring with unity and I is a maximal ideal, then R/I is: — A field
16. If R is a commutative ring with unity and I is a prime ideal, then R/I is: — An integral domain
17. What is a Unique Factorization Domain (UFD)? — An integral domain where every non-zero, non-unit element can be uniquely factored into irreducible elements, up to order and units.
18. What is the relationship between maximal ideals and prime ideals in a commutative ring with unity? — Every maximal ideal is prime.
19. What is the relationship between PIDs and UFDs? — Every PID is a UFD.
20. Which of the following statements about ideals is FALSE? — In a commutative ring with unity, every prime ideal is maximal.
21. A UFD is an integral domain where every non-zero, non-unit element can be factored into: — Irreducible elements uniquely, up to order and units.
22. What is the defining property of a Euclidean domain? — It possesses a Euclidean function (or norm) that allows for a division algorithm.
23. Which property guarantees that an integral domain is a UFD? — Every non-zero non-unit element has a unique factorization into irreducibles.
24. In a principal ideal domain (PID), every prime ideal is also: — Maximal
25. Let I be an ideal in a commutative ring R with unity. R/I is an integral domain if and only if I is: — Prime
26. Consider the ring Z. The ideal (0) is: — Prime but not maximal.
27. The quotient ring Z_n is a field if and only if: — n is prime.
28. Consider the ring Z. Is the ideal (2) a maximal ideal? —
29. Is the ring of polynomials F[x] over a field F a UFD? — Yes, it is a UFD.
30. Consider the ring Z[i] (Gaussian integers). Is it a Euclidean domain? — Yes, with the norm function N(a+bi) = a^2 + b^2.
31. A principal ideal domain (PID) is an integral domain where every ideal is: — Principal
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: — S
33. Which condition is necessary for a ring R to be a Euclidean domain? — R must be commutative and have a unity element.
34. What is the ideal generated by the polynomial x in the ring R[x]? — The set of polynomials with zero constant term.
35. For a commutative ring R with unity, the ideal I is maximal if and only if: — R/I is a field.
36. For a commutative ring R with unity, the ideal I is prime if and only if: — R/I is an integral domain.
37. In the context of Euclidean rings, the 'division algorithm' refers to: — 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).
38. The ring of polynomials F[x] over a field F is a Euclidean domain with the Euclidean function being: — The degree of the polynomial.
39. Which of the following is NOT necessarily a Euclidean domain? — The ring of Eisenstein integers, Z[ω]
40. Which of the following is a UFD? — The ring of integers Z
41. What is the quotient ring R/I formed by an ideal I in a ring R? — The set of cosets of I in R, with appropriate ring operations.
42. Which property is shared by all Euclidean domains? — They are principal ideal domains.
43. If R is a commutative ring with unity, and I is an ideal, when is R/I a field? — When I is a maximal ideal.
44. Consider the ring Z_4. The ideal I = (2) = {0, 2}. Is I a maximal ideal in Z_4? — Yes, because Z_4/(2) is isomorphic to Z_2, which is a field.
45. Let R = Z_6. The ideal I = (3) = {0, 3}. Is I a maximal ideal in Z_6? — Yes, because Z_6/(3) is isomorphic to Z_2, which is a field.
46. In the ring Z[x], is the ideal (x) a maximal ideal? — No, because Z[x]/(x) is isomorphic to Z, which is not a field.
47. Let R = Z and I = 4Z. The quotient ring Z/4Z is isomorphic to: — Z_4
48. Which of the following is a UFD but not a PID? — Z[x]
49. Which of the following is NOT a polynomial ring? — Z_n
50. In a Euclidean domain, every non-zero non-unit element is: — Irreducible
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