In an algebraic integer ring, two ideals whose intersection contains 3 may be comaximal

Let K be an algebraic number field with ring of integers \mathcal{O}, and let A,B \subseteq \mathcal{O} be nontrivial ideals. If 3 \in A \cap B, must it be the case that (A,B) \neq (1)?


Consider K = \mathbb{Q}(\sqrt{-2}), so that \mathcal{O} = \mathbb{Z}[\sqrt{-2}]. Consider A = (1+\sqrt{-2}) and B = (1-\sqrt{-2}). Now AB = (3), so that 3 \in A \cap B. However, note that 1 = (1+\sqrt{-2})(1-\sqrt{-2}) - (1+\sqrt{-2}) - (1-\sqrt{-2}) = (A,B), so that (A,B) = (1).

Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: