Proposition 9

Tag 04NI, permanent. Cite it as [Bourbaki, Tag 04NI].

Commutative Algebra, chapter V, § 1. Notion of an integral element

Let $A$ be a ring, $(R_i)_{1 \leq i \leq n}$ a finite family of $A$ -algebras and $A'_i$ the integral closure of $A$ in $R_i$ , $(1 \leq i \leq n)$ . Then the integral closure of $A$ in $R = \prod_{i=1}^n R_i$ , is equal to $\prod_{i=1}^n A'_i$ .

This is an immediate consequence of no. 1, Proposition 3.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

no. 1, Proposition 3