Prove that $R = \Bbb C \left [X,Y,Z \right ]/ \left \langle 2X^2Y - 3XY^3 + Z \right \rangle$ is a unique factorization domain.

Question

Let $R = \Bbb C \left [X,Y,Z \right ]/ \left \langle 2X^2Y - 3XY^3 + Z \right \rangle.$ Show that $R$ is a unique factorization domain.

How do I show that? Can anybody please help me proving that?

Thanks in advance.

EDIT $:$ Let $S = \Bbb C \left [X,Y \right ].$ Then $\alpha : = 2X^2Y - 3XY^3 \in S.$Therefore $$R = S \left [Z \right ] / \left \langle Z + \alpha \right \rangle \cong S$$ since $\alpha \in S.$ Hence $R$ is isomorphic to $\Bbb C \left [X,Y \right ]$ as rings and since $\Bbb C \left [X,Y \right ]$ is a unique factorization domain (by Gauss's theorem) it follows that $R$ is also a unique factorization domain, as required.

Can anybody please check my reasoning above? Thanks in advance.

Answer 1

Your reasoning is entirely correct.