Prove that $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ is an integral domain

Question

Let $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ be an analytic algebra. I am trying to prove that $R$ is an integral domain.

Basically I know that if $\langle x^2 - yz\rangle$ is a prime ideal in $K\langle x,y,z\rangle$ it follows, that $R$ is an integral domain. But I'm having problems proving that.

Answer 1

Assuming you mean that $K$ is a field and so on, then yes, showing that it is irreducible should suffice.

You could also try writing out a multiplication of two non-zero elements and see when the product is zero.

Also, technically you don't know if the zero ideal is prime, since that is the same as being an integral domain.

Answer 2

$x^2-yz$ is irreducible by Eisenstein. In a factorial ring, any irreducible element generates a prime ideal.

Note that you should be careful about notions of irreducible. Irreducible ideals and irreducible elements are not the same notion. For example the ideal $(x^2)$ irreducible, but $x^2$ is of course not irreducible as an element of the polynomial ring.