Problem related to the polynomial ring of several variables.

Question

I have to show that :

1. $\mathbb Z[x,y]/<y+1>$ is an unique factorisation domain.
2. $\mathbb C[x,y]/<x^2+1,y>$ is neither a prime nor a maximal ideal.

I hardly understand the polynomial rings of several variables and that's why I asked this question , but unfortunately I didn't get any response from anyone. That's why I am posting my specific problems about the polynomial rings.

Any helpful response will be appreciated. Thank you.

Answer 1

Hints:

1. $\mathbf Z[X,Y]/(Y+1)\simeq\mathbf (Z[X])[Y]/(Y+1)\simeq \mathbf Z[X]$.
2. $\begin{aligned}[t]\mathbf C[X,Y]/(X^2+1, Y)&\simeq\bigl(\mathbf C[X]/(X^2+1)\bigr)[Y]/(Y)\simeq\mathbf C[X]/(X^2+1)\\&\simeq\mathbf C[X]/(X+i)(X-i)\simeq\mathbf C\times\mathbf C.\end{aligned}$