TITLE: Regular Ring is Integrally Closed?
QUESTION [6 upvotes]: Studying some topics in Algebraic Geometry I've bumped into the following question:

Let $A$ be a regular ring. Is $A$ integrally closed?

Someone said me that with the hypothesis $A$ local regular ring it is true (Can anyone give me a proof of this assert or give me some references?), but what about the general case? 
There are some counterexamples?
Thank you

REPLY [2 votes]: If you do know that local regular rings are integrally closed, you know that a regular ring has all its localizations at prime ideals integrally closed. Therefore, the only piece of knowledge you need to reach the conclusion you want is that

an integral domain all of whose localizations at prime ideals is integrally closed is integrally closed

which is part of the content of Proposition 5.13 in the book by Atiyah and MacDonald.
You can replace prime by maximal throughout.