Show that the ring $\Bbb C[x,y,z]/\langle 2x^2y-5xy^3+z\rangle $ is a UFD

Question

I need to show that the ring $\Bbb C[x,y,z]/\langle 2x^2y-5xy^3+z\rangle $ is unique factorisation domain.

I think this ring is isomorphic to $\Bbb C[x,y]/\langle 2x^2y-5xy^3\rangle $ so it's enough to show the later ring is a UFD. I am stuck at this part.

Answer 1

Write $\Bbb C[x,y,z]=D[z]$, where $D=\Bbb C[x,y]$. Let $a=2x^2y-5xy^3 \in D$.

Then $\Bbb C[x,y,z]/\langle 2x^2y-5xy^3+z\rangle = D[z]\langle a+z\rangle \cong D$ is a UFD.

Answer 2

The ideal is the kernel of the (clearly surjective) ring homomorphism $$\Bbb{C}[x,y,z]\ \longrightarrow\ \Bbb{C}[x,y]:\ f\ \longmapsto\ f(x,y,5xy^3-2x^2y),$$ so the quotient is isomorphic to $\Bbb{C}[x,y]$, which is a UFD.