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

51. Let \(\mathbf{F}\) be a finite field such that for every \(a \in \mathbf{F}\) the equation \(x^2=a\) has a solution in \(\mathbf{F}\). Then

(a.) The characteristic of \(\mathbf{F}\) must be 2
(b.) \(\mathbf{F}\) must have a square number of elements
(c.) The order of \(\mathbf{F}\) is a power of 3
(d.) \(\mathbf{F}\) must be a field with a prime number of elements

52. \(R\) be a commutative ring with unity and \(I\) is a prime ideal which is true.

(a.) \(I\) must be a maximal ideal of \(R\).
(b.) If \(\frac{R}{I}\) is a finite ring then \(I\) is a maximal ideal.
(c.) \(I\) is never a maximal ideal.
(d.) If \(I\) is a maximal ideal then \(\frac{R}{I}\) must be a finite ring.

53. Let \(R=\mathbf{Z} \times \mathbf{Z} \times \mathbf{Z}\) and \(I=\mathbf{Z} \times \mathbf{Z} \times\{0\}\). Then which of the following statement is correct?

(a.) \(I\) is a maximal ideal but not a prime ideal of \(R\).
(b.) \(I\) is a prime ideal but not a maximal ideal of \(R\).
(c.) \(I\) is both a maximal ideal and a prime ideal of \(R\).
(d.) \(I\) is neither a maximal ideal nor a prime ideal of \(R\).

54. Let \(R\) be the ring of all real valued continuous functions on \([0,1]\). \(I=\{f \in R: f(0)=0\}\). Then

(a.) \(I\) is not an ideal of \(R\)
(b.) \(I\) is an ideal, but not a prime ideal of \(R\)
(c.) \(I\) is a prime ideal, but not a maximal ideal of \(R\)
(d.) \(I\) is a maximal ideal of \(R\)

55. Let \(S=\left\{\left(\begin{array}{ll}a & b \\ 0 & c\end{array}\right): a, b, c, \in R\right\}\) be the ring under matrix addition and multiplication. Then the subset \(\left\{\left(\begin{array}{ll}0 & p \\ 0 & 0\end{array}\right): p \in R\right\}\) is

(a.) Not an ideal of \(\mathrm{S}\)
(b.) An ideal but not a prime ideal of \(\mathrm{S}\)
(c.) A prime ideal but not a maximal ideal of \(\mathrm{S}\)
(d.) A maximal ideal of \(\mathrm{S}\)

56. Let \(C[0,1]\) be the ring of continuous real-valued functions on \([0,1]\), with addition and multiplication defined pointwise. For any subset \(S\) of \([0,1]\) let \(Z(S)=\{f \in C[0,1] \mid f(x)=0\) for all \(x \in S\}\). Then which of the following statements are true?

(a.) If \(Z(S)\) is an ideal in \(C[0,1]\) then \(S\) is closed in \([0,1]\).
(b.) If \(Z(S)\) is a maximal ideal then \(S\) has only one point.
(c.) If \(S\) has only one point then \(Z(S)\) is a maximal ideal.
(d.) None of these

57. Let \(\mathbb{C}[0,1]\) denote the ring of all continuous real-valued functions on \([0,1]\) with respect to pointwise addition and pointwise multiplication. Pick out the true statements:

(a.) \(\mathbf{C}[0,1]\) is an integral domain.
(b.) Let \(a \in[0,1]\). Set \(I=\{f \in \mathbb{C}[0,1] \mid f(a)=0\}\). Then \(I\) is an ideal in \(\mathbb{C}[0,1]\).
(c.) If \(I\) is any proper ideal in \(C[0,1]\), then there exists at least one point \(a \in[0,1]\) such that \(f(a)=0\) for all \(f \in I\).
(d.) None of these

58. Pick out the rings which are integral domains:

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

59. Pick out the true statement(s):

(a.) The set of all \(2 \times 2\) matrices with rational entries (with the usual operations of matrix addition and matrix multiplication) is a ring which has no non-trivial ideals.
(b.) Let \(R=C[0,1]\) be considered as a ring with the usual operations of \(I=\{f:[0,1] \rightarrow \mathbb{R} \mid f(1 / 2)=0\}\). Then \(I\) is a maximal ideal.
(c.) Let \(R\) be a commutative ring and let \(P\) be a prime ideal of \(R\). Then \(R / P\) is an integral domain.
(d.) None of these

60. Pick out the true statements:

(a.) Let \(R\) be a commutative ring with identity. Let \(M \) be an ideal such that every element of \(R\) not in \(M\) is a unit. Then \(R / M\) is a field.
(b.) Let \(R\) be as above and let \(M\) be an ideal such that \(R / M\) is an integral domain. Then \(M\) is a prime ideal.
(c.) Let \(R=C[0,1]\) be the ring of real-valued continuous functions on \([0,1]\) with respect to pointwise addition and pointwise multiplication. Let \(M=\{f \in R \mid f(0)=f(1)=0\}\). Then \(M\) is a maximal ideal.
(d.) None of these