Practice Questions for CSIR NET Ring Theory : Ring and Ideals - V

41. The number of non-zero elements in the field \(\mathbb{Z}_{p}\), where \(p\) is an odd prime number, which are squares, i.e., of the form \(m^{2}, m \in \mathbb{Z}_{p}, m \neq 0\) ____.

(a.) \((p-1)/2\)
(b.) \(p-1\)
(c.) \((p+1)/2\)
(d.) \(p\)

42. Let \(\mathrm{M}\) denote the set of all \(2 \times 2\) matrices over the reals. Addition and multiplication on \(\mathrm{M}\) are as follows:

\(A=\left(a_{i j}\right)\) and \(B=\left(b_{i j}\right)\), then \(A+B=\left(c_{i j}\right)\), where \(c_{i j}=a_{i j}+b_{i j}\), and \(A \cdot B=\left\langle d_{i j}\right)\), where \(d_{i j}=a_{i j} b_{i j}\)

Then which one of the following is valid for \((\mathrm{M},+, \cdot)\) ?

(a.) \(\mathrm{M}\) is a field
(b.) \(\mathrm{M}\) is an integral domain which is not a field
(c.) \(\mathrm{M}\) is a commutative ring which is not an integral domain
(d.) \(\mathrm{M}\) is a non-commutative ring

43. Let \((R,+)\) be an abelian group. If multiplication \((\cdot)\) is defined on \(\mathbf{R}\) by setting \(a \cdot b=0\) for all \(a, b, \in R\), then which one of the following statements is correct?

(a.) \((R,+, \cdot)\) is not a ring.
(b.) \((R,+, \cdot)\) is a ring, but not commutative.
(c.) \((R,+, \cdot)\) is a commutative ring, but has no unity.
(d.) \((R,+, \cdot)\) is not a field.

44. Consider the following assertions

i. The characteristic of the ring \((\mathbb{Z},+, \cdot)\) is zero.

ii. For every composite number, \(\mathrm{n}, \mathbb{Z}_{n}\), the ring of residue classes modulo \(\mathrm{n}\), is a field.

iii. \(\mathbb{Z}_{5}\), the ring of residue classes modulo 5, is an integral domain.

iv. The ring of all complex numbers is a field.

Which of the above assertions are correct?

(a.) i, iii and iv
(b.) i, ii and iii
(c.) i, ii and iv
(d.) i, iii and iv

45. Let \(\mathrm{F}\) be a finite field with \(\mathrm{n}\) elements. What is the possible value of \(n\) ?

(a.) 1
(b.) 36
(c.) 37
(d.) 125

46. If \(\mathrm{R}\) is a finite integral domain with \(\mathrm{n}\) element, then what is the number of invertible elements under multiplication in \(\mathrm{R}\) ?

(a.) 1
(b.) \(n\)
(c.) \(\mathrm{n}-1\)
(d.) \([\mathrm{n} / 2]\) where \([\,]\) is the bracket function

47. Consider the ring \(\mathbb{Z}_{n}=\{\overline{0}, \overline{1}, \overline{2}, \ldots ., \bar{n}\}\) of congruent modulo \(n\) classes. Under addition and multiplication modulo \(\mathrm{n}\), consider the following statements is/are are correct:

(a.) In \(\mathbb{Z}_{6}, \overline{4}\) divides \(\overline{2}\)
(b.) In \(\mathbb{Z}_{8}, \overline{3}\) divides \(\overline{7}\)
(c.) In \(\mathbb{Z}_{15}, \overline{9}\) divides \(\overline{12}\)
(d.) None of these

48. The zero divisors in \(\mathbb{Z}_{8}\) are

(a.) \(\overline{3}, \overline{4}, \overline{5}\)
(b.) \(\overline{1}, \overline{2}, \overline{4}\)
(c.) \(\overline{2}, \overline{4}, \overline{7}\)
(d.) \(\overline{2}, \overline{4}, \overline{6}\)

49. The characteristic of the ring \(\mathbb{Z}_{4} \times \mathbb{Z}_{6}\) is

(a.) 0
(b.) 6
(c.) 12
(d.) 24

50. Let \(\mathrm{F}\) be a field containing 11 elements. Which one of the following is correct?

(a.) If \(\alpha\) is a non-zero element of \(F\), then \(5 \alpha \neq\) 0
(b.) \(\alpha^5=1\) for every non-zero \(\alpha \in \mathrm{F}\) where 1 is the multiplicative identity of \(\mathrm{F}\)
(c.) \(\alpha^{10}=1\) for all \(\alpha \in F\)
(d.) Let \(\alpha\) be a non-zero element of \(F\). It is possible to find a proper subset \(S\) of \(F\) such that \(\alpha \in S\) and \(\beta s \in S\) for any \(\beta \in F\), \(s \in \mathrm{S}\)