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

1. Consider the following statements:

1. Every PID is ED
2. The group of units in the ring \(\frac{\mathbb{Z}}{37 \mathbb{Z}}\) is cyclic
3. There is a field with \(6^{5}\) elements.

(a.) (1) is true but (2) and (3) are false.
(b.) (2) is true but (1) and (3) are false.
(c.) (3) is true but (1) and (2) are false.
(d.) All these statements are false.

2. Let \(R\) be the ring \(\mathbb{Z}[x] / \langle\left(x^{2}+x+1\right)\left(x^{3}+x+1\right)\rangle\). What is the cardinality of the ring \(R\) ?

(a.) 27
(b.) 32
(c.) 64
(d.) Infinite

3. Pick out the cases where the given ideal is a maximal ideal.

(a.) The ideal \(15 \mathbb{Z}\) in \(\mathbb{Z}\).
(b.) The ideal \(I=\{f: f(0)=0\}\) in the ring \(C[0,1]\) of all continuous real valued functions on the interval \([0,1]\).
(c.) The ideal generated by \(x^{3}+x+1\) in the ring of polynomials \(\mathbb{F}_{3}[x]\), where \(\mathbb{F}_{3}\) is a field of three elements.
(d.) None of these

4. Consider the polynomial ring \(\mathbb{Q}[x]\). The ideal of \(\mathbb{Q}[x]\) generated by \(x^{2}-3\) is

(a.) Maximal but not prime
(b.) Prime but not maximal
(c.) Both maximal and prime
(d.) Neither maximal nor prime

5. The number of subfields of a field of cardinality \(2^{100}\) is

(a.) 2
(b.) 4
(c.) 9
(d.) 100

6. \(\mathbb{Z}_{2}[x] /\left\langle x^{3}+x^{2}+1\right\rangle\) is

(a.) a field having 8 elements
(b.) a field having 9 elements
(c.) an infinite field
(d.) NOT a field

7. Let \(\mathbb{R}[X]\) be the ring of real polynomials in the variable \(X\). The number of ideals in the quotient ring \(\mathbb{R} X] /\left\langle\left(X^{2}-3 X+2\right)\right\rangle\) is

(a.) 2
(b.) 3
(c.) 4
(d.) 6

8. Let \(R=\mathbb{Q}[x]\). Let \(I\) be the principal ideal \(\left\langle x^{2}+1\right\rangle\) and \(J\) be the principal ideal \(\left\langle x^{2}\right\rangle\). Then

(a.) \(\frac{R}{I}\) is a field and \(\frac{R}{J}\) is a field
(b.) \(\frac{R}{I}\) is an integral domain and \(\frac{R}{J}\) is a field
(c.) \(\frac{R}{I}\) is a field and \(\frac{R}{J}\) is a PID
(d.) \(\frac{R}{I}\) is a field and \(\frac{R}{J}\) is not an integral domain.

9. Let \(\mathbb{Z}_{n}\) be the ring of integers modulo \(n\), where \(n\) is an integer \(\geq 2\). Then Choose the correct in the following:

(a.) \(\mathbb{Z}_{n}\) is a field then \(n\) is prime
(b.) If \(\mathbb{Z}_{n}\) is an integral domain then \(n\) is prime
(c.) If there is an injective ring homomorphism of \(\mathbb{Z}_{5}\) to \(\mathbb{Z}_{n}\) then \(n\) is a multiple of 5
(d.) None of these

10. Let \(F\) be a finite field. If \(f: F \rightarrow F\), given by \(f(x)=x^{3}\) is a ring homomorphism, then

(a.) \(F=\mathbb{Z} / 3 \mathbb{Z}\)
(b.) \(F=\mathbb{Z} / 2 \mathbb{Z}\) or characteristic of \(F=3\)
(c.) \(F=\mathbb{Z} / 2 \mathbb{Z}\) or \(\mathbb{Z} / 3 \mathbb{Z}\)
(d.) Characteristic \(F\) is 3