Part CCSIR NET December 2023gaussian-integers

Gaussian integers

Let $R = Z [X] / (X^{2} + 1)$ and $ψ : Z [X] \to R$ be the natural quotient map. Which of the following statements are true?

A.R is isomorphic to a subring of $C$ .
B.For any prime number $p \in Z$ , the ideal generated by $ψ (p)$ is a proper ideal of R.
C.R has infinitely many prime ideals.
D.The ideal generated by $ψ (X)$ is a prime ideal in R.

The answer and working come with the PYQ pack.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

More on this topic