List of top Mathematics Questions on Ring Theory asked in CPET

Consider the following statements:
(I) The kernel of the ring homomorphism \(f:\mathbb{Z}[x]\to\mathbb{Z}\) given by \(f(p(x))=p(1)\) is \(\{(x-1)q(x): q(x)\in\mathbb{Z}[x]\}\).
(II) The ring \(\mathbb{Z}_4\) is isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\).
Choose the correct answer:

Which one of the following options is incorrect?

Which of the statements given below is/are true?
(I) \(\mathbb{Z}[i]\) is a Principal ideal domain and Euclidean domain.
(II) \(\mathbb{Z}[\sqrt{-5}]\) is neither a Principal ideal domain nor a Unique factorization domain.

On \(R=\left\{\begin{pmatrix}a&b\\0&c\end{pmatrix}: a,b,c\in\mathbb{R}\right\}\) with the usual addition and multiplication of matrices, which of the following statements is true?

Which of the following statements is/are false?
(I) \(I=\left\{\begin{pmatrix}a&0\\c&d\end{pmatrix}: a,c,d\in\mathbb{R}\right\}\) is both a subring and an ideal of \(M_2(\mathbb{R})\).
(II) \(\mathbb{Z}[\sqrt2]=\{a+b\sqrt2 : a,b\in\mathbb{Z}\}\) is an integral domain, but not a field.

Which one of the following options is correct for the ideal \(I=\langle x^2+5\rangle\) in the ring \(\mathbb{Q}[x]\)?