Definition 5

An integral domain $A$ is called completely integrally closed if the following condition is satisfied: every element $x$ of the field of fractions $K$ of $A$ such that all the powers $x^n$ ($n \geqslant 0$) are contained in a finitely generated sub-A-module of $K$, belongs to $A$.

Note that the hypothesis that the $x^n$ are contained in a finitely generated sub-A-module of $K$ can also be expressed by saying that there exists a non-zero element $d \in A$ such that $dx^n \in A$ for all $n \geqslant 0$; for the latter condition means that $x^n \in Ad^{-1}$; and conversely, if $(b_i)_{1 \leqslant i \leqslant m}$ is a finite sequence of elements of $K$, there exists $d \in A$ such that $db_i \in A$ for $1 \leqslant i \leqslant m$, whence $dM \subset A$ for the sub-A-module $M$ of $K$ generated by the $b_i$.

Clearly a completely integrally closed domain is integrally closed; conversely, the Corollary to Proposition 1 of no. 1 shows that an integrally closed Noetherian domain is completely integrally closed. \* On the other hand, the ring of a valuation of height $\geqslant 2$ (Chapter VI, § 4, no. 4) is integrally closed but not completely integrally closed. \* If $(A_i)$ is a family of completely integrally closed domains with the same field of fractions $K$, $A = \bigcap_i A_i$ is completely integrally closed. For if $x \in K$ is such that for some non-zero $d$ in $A$, $dx^n$ belongs to $A$ for all $n > 0$, the hypothesis implies that $x \in A$, for all $i$ and hence $x \in A$.