Definition 3

A n integral domain $A$ is called a Krull domain if there exists a family $(v_i)_{i \in I}$ of valuations on the field of fractions $K$ of $A$ with the following properties: (AK$_I$) the valuations $v_i$ are discrete; (AK$_{II}$) the intersection of the rings of the $v_i$ is $A$; (AK$_{III}$) for all $x \in K^*$, the set of indices $t \in I$ such that $v_t(x) \neq 0$ is finite.

It obviously suffices to verify condition (AK$_{III}$) for the elements $x$ of $A - (0)$.

Examples (1) Every discrete valuation ring is a Krull domain. (2) More generally, every principal ideal domain $A$ is a Krull domain. For let $(p_i)_{i \in I}$ be a representative system of extremal elements of $A$ and let $v_i$ be the valuation on the field of fractions of $A$ defined by $p_i$ (Chapter VI, § 3, no. 3, Example 4). It is immediately seen that the family $(v_i)_{i \in I}$ satisfies properties (AK$_I$), (AK$_II$) and (AK$_{III}$). (3) Let $F$ be a field and $(R_i)_{1 \leq i \leq n}$ a finite family of subrings of $F$ which are Krull domains. Then their intersection $S = \bigcap_{j=1}^n R_j$, is a Krull domain. For $1 \leq j \leq n$ let $(v_{j,t})_{t \in I_j}$ be a family of valuations on the field of fractions of $R_j$, satisfying (AK$_I$), (AK$_{II}$), (AK$_{III}$) (where $A$ is replaced by $R_j$). Let $w_{j,t}$ denote the restriction of $v_{j,t}$ to the field of fractionsof $S$. Then the family $(v_{j,t})_{1 \leq j \leq n, t \in I_j}$ obviously satisfies (AK$_{II}$) (where $A$ is replaced by $S$) and also (AK$_{III}$) since the set of indices$j$ is finite. The valuations $w_{j,t}$ are either discrete or improper. By retaining only those which are discrete, a family is obtained which obviously satisfies (AK$_I$), (AK$_{II}$) and (AK$_{,,}$) (where $A$ is replaced by $S$). Hence $S$ is certainly a Krull domain. (4) In particular, if $A$ is a Krull domain and $K'$ a subfield of the field of fractions $K$ of $A$, $K' \cap A$ is a Krull domain.