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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)?

Please help me.

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@reme: If you want to mark this as a duplicate you should use the "close" button, not unilaterally add a duplicate banner. In fact your question came later, so your action is doubly absurd. – Rahul Jan 26 '13 at 9:21
up vote 1 down vote accepted

For (b) observe that $x^2yz\in \langle z \rangle \Rightarrow ...$

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Compare the ideal of (b) with the ideal $\langle z \rangle$.

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