Consider an ideal $I$ in $\mathbb{C}[x_1, x_2, x_3, x_4]$ such that $I$ is generated by $x_1x_3, x_2x_3, x_1x_4,$ and $x_2x_4$. I think this ideal cannot be generated by two elements, but can't give a rigorous proof.
Can anyone suggest a proof?
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1The ideal generated by $x_1$ and $x_2$ contains all the listed elements. – paw88789 Oct 16 '14 at 14:28
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Oh, I made a mistake. I will correct it. – Keith Oct 16 '14 at 14:35