# Which one of the following ideals is prime?

Consider the polynomial ring $\mathbb R[x,y,z]$. Which one of the following ideals is prime?

(a) $\langle x^{2},y+z\rangle$; (b) $\langle x^{2}yz,z\rangle$; (c) $\langle xy,xz+yz\rangle$.

Since $xy\in\langle xy,xz+yz\rangle$ but $x$ and $y$ do not belong to $\langle xy,xz+yz\rangle$, so $\langle xy,xz+yz\rangle$ is not prime. There is a similar proof for (a) which is not prime. But (b)?

For (b) observe that $x^2yz\in \langle z \rangle \Rightarrow ...$
Compare the ideal of (b) with the ideal $\langle z \rangle$.