# How distinct maximal ideals have non-empty intersection?

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?

Consider two arbitrary maximal ideals of $\mathbb{Z}$. What is their intersection?
• for example $\langle 3\rangle\cap\langle 2\rangle=0$ – Ronald Aug 11 '13 at 17:18
• @Danial: No, in $\mathbb{Z}$ the intersection of these two maximal (prime) ideals is another principal ideal, $\langle 6 \rangle$. – hardmath Aug 11 '13 at 17:23
In this context distinct means just that $P\ne Q$, note that any ideal must always contain 0.
You are confusing "distinct" with "disjoint." $P\cap Q=\emptyset$ is "$P,Q$ disjoint."