TITLE: Is $\mathbb Z[i]/n\mathbb Z[i]$ an integral domain?
QUESTION [6 upvotes]: Let $ \mathbb{Z}[i]$ denote the ring of the Gaussian intergers. For which of the following value of n is the quotient ring $ \mathbb{Z}[i]/n\mathbb{Z}[i]$ an integral domain?
$ a. 2$
$ b. 13$
$ c. 19$
$ d. 7$
I'm doubtful with the following attempt I made.

I think all 4 options are correct: It suffices to show $n\mathbb Z[i]$ is a prime ideal of $\mathbb Z[i]$ if $n$ is prime. Now $(n)=n\mathbb Z[i].$ So $n$ is prime element of $n\mathbb Z[i]\implies(n)$ is a prime ideal of $\mathbb Z[i].$
Let $n$ be a prime integer. Of course then $n$ is non zero and non unit. Let $n|(a+ib)(c+id).$ That's $n|(ac-bd)+i(ad+bc)\\\implies\dfrac{ac-bd}{n},\dfrac{ad+bc}{n}\in\mathbb Z\\\implies n|ac,bd,ad,bc\\\implies n|\{a~or~c\}~and~\{b~or~d\}~and~\{a~or~d\}~and~\{b~or~c\}\\\implies n\text{ divides at least $3$ of }a,b,c,d.$
WLG let $n|a,b\implies n|a+ib.$

Is my attempt correct?

REPLY [4 votes]: The following theorem is well known (see a book on algebraic number theory)
Theorem: Let $p$ be a rational prime (that is, a prime in $\mathbb Z$). Then, $p$ is a prime in the Gaussian integers $\mathbb Z[i]$ if and only if $p\equiv 3 \pmod 4$.
Since $\mathbb Z[i]/n \mathbb Z[i]$ is an integral domain if and only if $n$ is prime in $\mathbb Z[i]$,  it follows that $(3)$ and $(4)$ are correct.