How distinct maximal ideals have non-empty intersection?

Question

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?

score 3 · Answer 1 · answered Aug 11 '13 at 17:16

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

answered Aug 11 '13 at 17:16

Dune

7,397

for example $\langle 3\rangle\cap\langle 2\rangle=0$ – Ronald Aug 11 '13 at 17:18
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@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
yes yes.. I understand it now. Thanks – Ronald Aug 11 '13 at 17:29

score 3 · Answer 2 · answered Aug 11 '13 at 17:17

3

In this context distinct means just that $P\ne Q$, note that any ideal must always contain 0.

answered Aug 11 '13 at 17:17

Alex J Best

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score 1 · Answer 3 · answered Aug 11 '13 at 17:28

1

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

answered Aug 11 '13 at 17:28

Thomas Andrews

177,126

This is true :) – Ronald Aug 11 '13 at 17:30

How distinct maximal ideals have non-empty intersection?

3 Answers3