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

21. Suppose \(a\) and \(b\) are elements in \(R\), a commutative ring with unity. Then the equation \(a x=b\)

(a.) always has exactly one solution
(b.) has a solution only if \(a\) is a unit
(c.) has more than one solution only if \(b=0\)
(d.) may have more than one solution

22. Pick out the true statements:

(a.) The set \(\left\{\begin{bmatrix}a & a \\ a & a\end{bmatrix} \mid a \in \mathbb{R}, a \neq 0\right\}\) is a group with respect to matrix multiplication.
(b.) The set \(\left\{\begin{bmatrix}a & b \\ c & d\end{bmatrix} \mid a, b, c, d \in \mathbb{R}\right\}\) is a commutative ring with identity with respect to matrix addition and matrix multiplication.
(c.) The set \(\left\{\begin{bmatrix}a & b \\ -b & a\end{bmatrix} \mid a, b \in \mathbb{R}\right\}\) is a field with respect to matrix addition and matrix multiplication.
(d.) None of these

23. Let \(R\) be a (commutative) ring (with unity). Let \(I\) and \(J\) be ideal in \(R\). Pick out the true statements:

(a.) \(I \cup J\) is an ideal in \(R\)
(b.) \(I \cap J\) in an ideal in \(R\)
(c.) \(I+J=\{x+y: x \in I, y \in J\}\) is an ideal in \(R\).
(d.) None of these

24. Pick out the units in \(\mathbb{Z}[\sqrt{3}]\)

(a.) \(-7+4 \sqrt{3}\)
(b.) \(5+3 \sqrt{3}\)
(c.) \(2-\sqrt{3}\)
(d.) \(-3-2 \sqrt{3}\)

25. Pick out the integral domains from the following list of rings:

(a.) \(\{a+b \sqrt{5} \mid a, b \in \mathbb{Q}\}\)
(b.) The ring of continuous functions from \([0,1]\) into \(\mathbb{R}\)
(c.) The ring of complex analytic functions on the disc \(\{z \in \mathbb{C}|| z \mid<1\}\)
(d.) The polynomial ring \(\mathbb{Z}[x]\).

26. Let \(P\) be a prime ideal in a commutative ring \(R\) and let \(S=R \setminus P\), i.e., the complement of \(P\) in \(R\). Pick out the true statements:

(a.) \(S^{-}\) is closed under addition
(b.) \(S\) is closed under multiplication
(c.) \(S\) is closed under addition and multiplication
(d.) None of these

27. The number of elements of a principal ideal domain can be

(a.) 15
(b.) 25
(c.) 35
(d.) 36

28. Let \(m\) be an odd integer \(>6\). Then the multiplicative inverse of 2 in the ring \(\left(\mathbb{Z}_{m}, +_{m}, \cdot_{m}\right)\) (where \(+_{m}\) and \(\cdot_{m}\) denote the addition and multiplication modulo \(m\) respectively).

(a.) does not exist.
(b.) is \(\frac{m-1}{2}\).
(c.) is \(\frac{m+1}{2}\).
(d.) is \(m-2\).

29. Let \(F\) be a field of 8 elements and \(F=\left\{x \in F \mid x^{7}=1\right.\) and \(x^{k} \neq 1\) for all natural numbers \(k<7\}\). Then the number of elements in \(A\) is

(a.) 1
(b.) 2
(c.) 3
(d.) 6

30. Which of the following cannot be the cardinality of a field?

(a.) 6
(b.) 4
(c.) 27
(d.) 7