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

21. Let \(\mathbb{Z}\) be the ring of integers under the usual addition and multiplication. Then every nontrivial ring homomorphism \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) is

(a.) Both injective and surjective
(b.) Injective but not surjective
(c.) Surjective but not injective
(d.) Neither injective nor surjective

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

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

23. Let \(\mathrm{R}\) be a ring. If \(\mathrm{R}[\mathrm{x}]\) is a principal ideal domain, then \(\mathrm{R}\) is necessarily a

(a.) Unique Factorization Domain
(b.) Principal Ideal Domain
(c.) Euclidean Domain
(d.) Field

24. Let \(R\) be the ring of polynomials over \(\mathbb{Z}_{2}\) and let \(I\) be the ideal of \(R\) generated by the polynomial \(x^{3}+x+1\). Then the number of elements in the quotient ring \(\frac{R}{I}\) is

(a.) 2
(b.) 4
(c.) 8
(d.) 16

25. Let \(m\) and \(n\) be coprime natural numbers. Then the kernel of the ring homomorphism \(\phi: \mathbb{Z} \rightarrow \mathbb{Z}_{m} \times \mathbb{Z}_{n}\), defined by \(\phi(x)=(\bar{x}, \bar{x})\) is

(a.) \(m \mathbb{Z}\)
(b.) \(m n \mathbb{Z}\)
(c.) \(n \mathbb{Z}\)
(d.) \(\mathbb{Z}\)

26. The ring \(\mathbb{Z}[x]\) is a

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

27. Let \(R\) be a Principal Ideal Domain and \(a b\) any two non-unit elements of \(R\). Then the ideal generated by \(a\) and \(b\) is also generated by

(a.) \(a+b\)
(b.) \(a b\)
(c.) \(\operatorname{gcd}(a, b)\)
(d.) \(\operatorname{lcm}(a, b)\)

28. Which of the following rings is a PID?

(a.) \(\mathbb{Q}[X, Y] /\langle X\rangle\)
(b.) \(\mathbb{Z} \oplus \mathbb{Z}\)
(c.) \(\mathbb{Z}[X]\)
(d.) \(M_{2}(\mathbb{Z})\), the ring of \(2 \times 2\) matrices with entries in \(\mathbb{Z}\)

29. Let \(f(x) \in \mathbb{Z}_{5}[x]\) be a polynomial such that \(\frac{\mathbb{Z}_{5}[x]}{\langle f(x)\rangle}\) is a field, where \(\langle f(x)\rangle\) denotes the ideal generated by \(f(x)\). Then one of the choices for \(f(x)\) is

(a.) \(x+1\)
(b.) \(x^{2}+3\)
(c.) \(x^{2}+1\)
(d.) \(x^{3}+3\)

30. Consider \(\mathbb{Z}_{5}\) and \(\mathbb{Z}_{20}\) as rings modulo 5 and 20, respectively. Then the number of homomorphisms \(\phi: \mathbb{Z}_{5} \rightarrow \mathbb{Z}_{20}\) is

(a.) 1
(b.) 2
(c.) 4
(d.) 5