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

1. We denote the characteristic of \(R\) by char \((R)\). In the following, let \(R\) and \(S\) be nonzero commutative rings with unity. Then

(a.) Char \((R)\) is always a prime number.
(b.) if \(S\) is a quotient ring of \(R\), then either char \((S)\) divides char \((R)\), or char \((S)=0\) .
(c.) if \(S\) is a subring of \(R\) containing \(1_{R}\) then \(\operatorname{char}(S)=\operatorname{char}(R)\).
(d.) if char \((R)\) is a prime number, then \(R\) is a field.

2. For which of the following values of \(n\), does the finite field \(\mathbb{F}_{5^{n}}\) with \(5^{n}\) elements contain a non-trivial \(93^{\text {rd }}\) root of unity?

(a.) 92
(b.) 30
(c.) 15
(d.) 6

3. Let \(\mathrm{R}\) be a commutative ring. Let \(I\) and \(J\) be ideals of \(R\).

Let \(I-J=\{x-y \mid x \in I, y \in J\}\) and

\(IJ=\{xy \mid x \in I, y \in J\}\) . Then

(a.) \(I-J\) is an ideal and \(IJ\) is an ideal in \(R\)
(b.) \(I-J\) is an ideal and IJ need not be an ideal in \(R\)
(c.) \(I-J\) need not be an ideal but \(IJ\) is an ideal in \(R\)
(d.) Neither \(I-J\) nor \(I \cdot J\) need to be an ideal in \(R\)

4. In the ring of Gaussian integers \(\mathbb{Z}[i]\),

(a.) 5 and 7 are irreducible
(b.) 5 is irreducible but 7 is reducible
(c.) 5 is reducible but 7 is irreducible
(d.) Neither 5 nor 6 is irreducible

5. Let \(I_{1}, I_{2}\) be two ideals of a commutative ring \(R\) with identity. Which one of the following is true?

(a.) \(I_{1}+I_{2}\) and \(I_{1} \cap I_{2}\) are ideals of \(R\)
(b.) \(I_{1}+I_{2}\) is an ideal of \(R\), but \(I_{1} \cap I_{2}\) is not an ideal of \(R\)
(c.) \(I_{1}+I_{2}\) is not an ideal of \(R\), but \(I_{1} \cap I_{2}\) is an ideal of \(R\)
(d.) Neither \(I_{1}+I_{2}\) nor \(I_{1} \cap I_{2}\) is an ideal of \(R\)

6. Let \(R\) be the ring of all \(2 \times 2\) matrices

\(E\) the ring of all even integers.

\(T\) the ring of integers \((\bmod 10)\) and

\(S\) the ring of all multiples of 6 .

Then

(a.) Only \(E\) has no zero divisors
(b.) Only \(R\) and \(S\) have no zero divisors
(c.) Only \(E\) and \(S\) have no zero divisors
(d.) Only \(E\) and \(T\) have no zero divisors

7. Let \(I\) be any ideal in the ring \(\mathbb{Z}\) of integers. Then

(a.) \(I\) can be generated by one element
(b.) \(I=\langle 0\rangle\) or \(I=\mathbb{Z}\)
(c.) There is an ideal \(I^{\prime}\) such that \(I \oplus I^{\prime}=\mathbb{Z}\)
(d.) \(I\) is generated by a prime number

8. Let \(A\) be a finite integral domain with unity \(l \in A\). Then the order of 1 in the additive group \((A,+)\) is

(a.) A prime
(b.) Zero
(c.) An arbitrary integer
(d.) A power of a prime number

9. A non-zero element ' \(a\) ' of a ring is said to be nilpotent if \(a^{n}=0\) for some \(n\in \mathbb{N}\) $\cdots$. In the ring of integers \((\bmod 8)\) all nilpotent elements are

(a.) 2,4 and 6
(b.) 2 and 4
(c.) 4
(d.) 0,2,4 and 6

10. Which one of the following ideals of the ring \(\mathbb{Z}[i]\) of Gaussian integers is NOT maximal?

(a.) \(\langle 1+i\rangle\)
(b.) \(\langle 1-i\rangle\)
(c.) \(\langle 2+i\rangle\)
(d.) \(\langle 3+i\rangle\)