Quotient ring is a UFD

Question

Prove that the ring $\mathbb R[x,y,z]/(x^2+y^2+z^2-1)$ is a unique factorization domain.

Answer 1

Hint. Let $R=\mathbb R[X,Y,Z]/(X^2+Y^2+Z^2-1)$. Note that $z-1$ is prime in $R$. Show that $R_{z-1}$ is a UFD and then use Nagata Criterion.