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In this question I don't understand how distinct maximal ideals have non-empty intersection? I mean since $P, Q$ are distinct does not imply that $P\cap Q=\varnothing$? Or the intersection has another meaning here?

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Consider two arbitrary maximal ideals of $\mathbb{Z}$. What is their intersection?

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  • $\begingroup$ for example $\langle 3\rangle\cap\langle 2\rangle=0$ $\endgroup$ – Ronald Aug 11 '13 at 17:18
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    $\begingroup$ @Danial: No, in $\mathbb{Z}$ the intersection of these two maximal (prime) ideals is another principal ideal, $\langle 6 \rangle$. $\endgroup$ – hardmath Aug 11 '13 at 17:23
  • $\begingroup$ yes yes.. I understand it now. Thanks $\endgroup$ – Ronald Aug 11 '13 at 17:29
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In this context distinct means just that $P\ne Q$, note that any ideal must always contain 0.

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You are confusing "distinct" with "disjoint." $P\cap Q=\emptyset$ is "$P,Q$ disjoint."

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  • $\begingroup$ This is true :) $\endgroup$ – Ronald Aug 11 '13 at 17:30

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