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

41. Let \(R\) be a Euclidean domain such that \(R\) is not a field. Then the polynomial ring \(R[X]\) is always

(a.) a Euclidean domain.
(b.) a principal ideal domain, but not a Euclidean domain.
(c.) a unique factorization domain, but not a principal ideal domain.
(d.) not a unique factorization domain.

42. Which of the following quotient rings are fields?

(a.) \(\mathbf{F}_{3}[X] /\left\langle\left(X^{2}+X+1\right)\right\rangle\), where \(\mathbf{F}_{3}\) is the finite field with 3 elements.
(b.) \(\mathbb{Z}[X] /\langle(X-3)\rangle\)
(c.) \(\mathbb{q} X] /\left\langle\left(X^{2}+X+1\right)\right\rangle\)
(d.) \(\mathbf{F}_{2}[X] /\left(\left(X^{2}+X+1\right)\right\rangle\), where \(\mathbf{F}_{2}\) is the finite field with 2 elements.

43. Let \(A\) denote the quotient ring \(\frac{\mathbb{Q}[x]}{\left\langle x^{3}\right\rangle}\) Then

(a.) There are exactly three distinct proper ideals in \(A\).
(b.) There is only one prime ideal in \(A\).
(c.) \(A\) is an integral domain.
(d.) Let \(f, g\) be in \(\mathbb{Q}[X]\) such that \(\bar{f} \cdot \bar{g}=0\) in \(A\). Here \(\bar{f}\) and \(\bar{g}\) denote the image of \(f\) and \(g\) respectively in \(A\). Then \(f(0) \cdot g(0)=0\).

44. 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 \(M=\{f \in R \mid f(1 / 2)=0\}\). Then \(M\) is a maximal ideal.
(d.) None of these

45. 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 with 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

46. Let \(C(\mathbb{R})\) denote the ring of all continuous real-valued functions on \(\mathbb{R}\), with the operations of pointwise addition and pointwise multiplication. Which of the following form an ideal in this ring?

(a.) The set of all \(C^{\infty}\) functions with compact support.
(b.) The set of all continuous functions with compact support.
(c.) The set of all continuous functions that vanish at infinity, i.e., functions \(f\) such that \(\lim _{|x| \rightarrow \infty} f(x)=0\).
(d.) None of these

47. Let \(\mathrm{R}\) be a finite non-zero commutative ring with unity. Then which of the following statements are necessarily true?

(a.) Any non-zero element of \(\mathrm{R}\) is either a unit or a zero divisor.
(b.) There may exist a non-zero element of \(\mathrm{R}\) which is neither a unit nor a zero divisor.
(c.) Every prime ideal of \(\mathrm{R}\) is maximal.
(d.) If \(\mathrm{R}\) has no zero divisors then the order of any additive subgroup of \(\mathrm{R}\) is a prime power.

48. Which one of the following statements is correct where \(\mathrm{R}[\mathrm{x}]\) denotes the polynomial ring in the one variable \(x\) over a ring \(\mathrm{R}\) :

(a.) If \(\mathrm{R}\) is a field then \(\mathrm{R}[x]\) is a field
(b.) If \(\mathrm{R}\) is an integral domain, then \(\mathrm{R}[x]\) is a field
(c.) If \(\mathrm{R}\) is a field then \(\mathrm{R}[x]\) is an integral domain
(d.) Every integral domain is a field.

49. Let \(C([0,1])\) be the ring of all real-valued continuous functions on \([0,1]\). Which of the following statements are true?

(a.) \(C([0,1])\) is an integral domain.
(b.) The set of all functions vanishing at 0 is a maximal ideal.
(c.) The set of all functions vanishing at both 0 and 1 is a prime ideal.
(d.) If \(f \in C([0,1])\) is such that \((f(x))^n=0\) for all \(x \in[0,1]\) for some \(n>1\), then \(f(x)=0\) for all \(x \in[0,1]\).

50. Let \(R\) be the ring obtained by taking the quotient of \((\mathbb{Z} / 6 \mathbb{Z})[X]\) by the principal ideal \(I=\langle 2 X+4\rangle\). Then

(a.) \(R\) has infinitely many elements.
(b.) \(R\) is a field.
(c.) 5 is a unit in \(R\).
(d.) 4 is a unit in \(R\).