Are $\mathbb C[x , y]$ and $ \frac {\mathbb R[x , y]}{ \left<x^2+1 , y\right>}$ PIDs?

Question

Are $\mathbb C[x , y]$ and $ \frac {\mathbb R[x , y]}{ \left<x^2+1 , y\right>}$ PIDs?

Can anyone please educate me how to handle two variable stuff?

I know how to check this kind of stuff in one variable.

Answer 1

Hints : if $R$ is an integral domain, can $(X,Y)$ be principal in $R[X,Y]$ ?

And $\mathbb{R}[X,Y]/(X^2+1) \simeq \mathbb{C}[Y]$

Answer 2

Hint:

1. $\mathbb C[x , y] \cong \mathbb C[x][y]$. Now use that $D[y]$ is a PID iff $D$ is a field.

2. $\mathbb{R}[x,y]/(x^2+1,y) \cong \mathbb{R}[x]/(x^2+1)$