I think that $k[x,y,z]/(z-1, x^2-y)$ can be identified as a subset of $k[x,y]$ with all polynomials whose $x$ terms are only degree one.
Therefore I conclude that $k[x,y,z]/(z-1, x^2-y)$ is integral domain and $(z-1, x^2-y)$ is a prime ideal. Could anyone tell me if this is a correct logic?
Here $k$ is a field.