How to show that ideal is prime in $\mathbb{R}[x,y,z]$ modulo some other ideal

Question

Let $R:=\mathbb{R}[x,y,z]$ and $g:=x^2+y^2-z^2\in R$. I would like to know how to show that $(x,y-z)/(g)$ is a prime ideal in $R/(g)$, and whether it is maximal or not.

Thanks for the help!

Answer 1

Hint: Is $\mathbb R[x,y,z]/(x,y-z)$ an integral domain? Is it a field?

Answer 2

Hint: By the third isomorphism theorem $(R/(g))/((x,y-z)/(g))\simeq R/(x,y-z)$