Practice Questions for CSIR NET Ring Theory : UFD, PID and ED VII

61. Consider \(S=C[x^5]\), complex polynomials in \(x^5\), as a subset of \(T=C[x]\), the ring of all complex polynomials. Then

(a.) \(S\) is neither an ideal nor a subring of \(T\)
(b.) \(S\) is an ideal, but not a subring of \(T\)
(c.) \(S\) is a subring but not an ideal of \(T\)
(d.) \(S\) is both a subring and an ideal of \(T\)

62. Which of the following statements are true?

(a.) There exists a finite field in which the additive group is not cyclic.
(b.) If \(F\) is a finite field, there exists a polynomial \(p\) over \(F\) such that \(p(x) \neq 0\) for all \(x \in F\), where 0 denotes the zero in \(F\).
(c.) Every finite field is isomorphic to a subfield of the field of complex numbers.
(d.) None of these

63. Let \(p, q\) be distinct primes. Then

(a.) \(\mathbb{Z} /_{p^2 q \mathbb{Z}}\) has exactly 3 distinct ideals
(b.) \(\mathbb{Z} /_{p^2 q \mathbb{Z}}\) has exactly 3 distinct maximal ideals
(c.) \(\mathbb{Z} /_{p^2 q \mathbb{Z}}\) has exactly 2 distinct maximal ideals
(d.) None of these

64. Let \(a, b, c, d\) be real numbers with \(a

(a.) \(S\) is NOT an ideal of \(C[a, b]\)
(b.) \(S\) is an ideal of \(C[a, b]\) but NOT a prime ideal of \(C[a, b]\)
(c.) \(S\) is a prime ideal of \(C[a, b]\) but NOT a maximum ideal of \(C[a, b]\)
(d.) \(S\) is a maximum ideal of \(C[a, b]\)

65. The possible values for the degree of an irreducible polynomial in \(\mathbb{R}[x]\).

(a.) 2
(b.) 3
(c.) 4
(d.) 5

66. Consider \(\mathbb{Z}[x]\), the set of all polynomials with integer coefficients and \(\mathbb{Q}[\sqrt{2}]\), the set of all real numbers of the form \(a+b \sqrt{2}\) with \(a, b\) rational numbers. Which of the following is correct about \(\mathbb{Z}[x]\) and \(\mathbb{Q}[\sqrt{2}]\) ?

(a.) Both are rings, but only one has unity
(b.) Both are commutative rings, but only one is an integral domain
(c.) Both are integral domains, but only one is a field
(d.) Both are fields

67. Pick out the rings which are integral domains:

(a.) \(\mathbb{R}[x]\), the ring of all polynomials in one variable with real coefficients
(b.) \(C^1[0,1]\), the ring of continuously differentiable real-valued function on the interval \([0,1]\) (with respect to point-wise addition and point-wise multiplication)
(c.) \(M_n(\mathbb{R})\), the ring of all \(n \times n\) matrices with real entries.
(d.) None of these

68. Pick out the correct statements from the following list:

(a.) A homomorphic image of a UFD (unique factorization domain) is again a (UFD).
(b.) The element \(2 \in \mathbb{Z}[\sqrt{-5}]\) is irreducible in \(\mathbb{Z}[\sqrt{-5}]\).
(c.) Units of the ring \(\mathbb{Z}[\sqrt{-5}]\) are the units of \(\mathbf{Z}\).
(d.) The element 2 is a prime element in \(\mathbf{Z}[\sqrt{-5}]\).

69. Suppose that \(R\) is a unique factorization domain and that \(a, b \in R\) are distinct irreducible elements. Which of the following statements is TRUE?

(a.) The ideal \(\langle 1+a\rangle\) is a prime ideal
(b.) The ideal \(\langle a+b\rangle\) is a prime ideal
(c.) The ideal \((1+a b\rangle\) is a prime ideal
(d.) The ideal \(\langle a\rangle\) is not necessarily a maximal ideal

70. Which of the following is a field?

(a.) \(\frac{\mathrm{C}[x]}{\left(x^2+2\right\rangle}\)
(b.) \(\frac{\mathrm{Z}[x]}{\left\langle x^2+2\right\rangle}\)
(c.) \(\frac{\mathbb{Q}[x]}{\left(x^2-2\right\rangle}\)
(d.) \(\frac{\mathrm{R}[x]}{\left\langle x^2-2\right\rangle}\)

71. If \(\mathbb{Z}[i]\) is the ring of Gaussian integers, the quotient \(\mathbb{Z}[i] /\langle(3-i)\rangle\) is isomorphic to

(a.) \(\mathbb{Z}\)
(b.) \(\frac{\mathbb{Z}}{3 \mathbb{Z}}\)
(c.) \(\frac{\mathbb{Z}}{4 \mathbb{Z}}\)
(d.) \(\frac{\mathbb{Z}}{10 \mathbb{Z}}\)

72. Let \(\mathbb{Z}[i]\) denote the ring of Gaussian integers. For which of the following values of \(n\) is the quotient ring \(\mathbb{Z}[i] / n \mathbb{Z}[i]\) an integer domain?

(a.) 2
(b.) 13
(c.) 19
(d.) 7

73. Which of the following statements are true?

(a.) \(\mathbb{Z}[x]\) is a principal ideal domain.
(b.) \(\mathbb{Z}[x] /\langle y+1\rangle\) is a unique factorization domain.
(c.) If \(\mathrm{R}\) is a principal ideal domain and \(p\) is a non-zero prime ideal, then \(\mathrm{R} / p\) has finitely many prime ideals.
(d.) If \(\mathrm{R}\) is a principal ideal domain, then any subring of \(\mathrm{R}\) containing 1 is again a principal ideal domain.