Which of the following are principal ideal domains?
(A) $\mathbb{Z}[X]/\langle X^2+1 \rangle$
(B) $\mathbb{Z}[X]$
(C) $\mathbb{C}[X,Y]$
(D) $\mathbb{R}[X,Y]/\langle X^2+1,Y \rangle$
My answer: (In brief)
$\mathbb{Z}[x]/\langle x^2+1 \rangle$ is the ring $\mathbb{Z}[i]$ which we know is a Euclidean domain so clearly a PID.
$\mathbb{Z}[X]$ is not a PID
$\mathbb{C}[X,Y]$ is PID (for every field $F$, $F[X,Y]$ is a PID).
$\mathbb{R}[X,Y]/\langle X^2+1,Y \rangle$ is the field $\mathbb{C}$ which is a PID.
Can anyone please critique my answers. Thanks.
You are correct in each, for valid reasons, except for part $C)$, as mentioned in the comments.
If $A$ is a field, then $A[x]$ is a PID, and the converse holds. However, when $B$ is only a PID (and not a field), it is not the case that $B[x]$ will be a PID.
Here, I mean to say that $B=A[y]$ being a PID doesn't make $B[x]\cong A[x,y]$ a PID. If it did, then due to our converse $A[y]$ would be a field.