Definition 4
Tag 04ND, permanent. Cite it as [Bourbaki, Tag 04ND].
Commutative Algebra, chapter V, § 1. Notion of an integral element
If A is an integral domain, the integral closure of A in its field of fractions is called the integral closure of A. An integral domain is called integrally closed if it is equal to its integral closure.
Note that an integrally closed domain is not necessarily closed in any ring containing it, as the example of a field which is not algebraically closed shows.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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