In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.

In this article, a ring is commutative and has unity.

Let ${\displaystyle A}$ be an integral domain and let ${\displaystyle P}$ be the set of all prime ideals of ${\displaystyle A}$ of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then ${\displaystyle A}$ is a Krull ring if

It is also possible to characterize Krull rings by mean of valuations only:

An integral domain ${\displaystyle A}$ is a Krull ring if there exists a family ${\displaystyle \{v_{i}\}_{i\in I}}$ of discrete valuations on the field of fractions ${\displaystyle K}$ of ${\displaystyle A}$ such that:

The valuations ${\displaystyle v_{i}}$ are called essential valuations of ${\displaystyle A}$.

The link between the two definitions is as follows: for every ${\displaystyle {\mathfrak {p}}\in P}$, one can associate a unique normalized valuation ${\displaystyle v_{\mathfrak {p}}}$ of ${\displaystyle K}$ whose valuation ring is ${\displaystyle A_{\mathfrak {p}}}$. Then the set ${\displaystyle {\mathcal {V}}=\{v_{\mathfrak {p}}\}}$ satisfies the conditions of the equivalent definition. Conversely, if the set ${\displaystyle {\mathcal {V}}'=\{v_{i}\}}$ is as above, and the ${\displaystyle v_{i}}$ have been normalized, then ${\displaystyle {\mathcal {V}}'}$ may be bigger than ${\displaystyle {\mathcal {V}}}$, but it must contain ${\displaystyle {\mathcal {V}}}$. In other words, ${\displaystyle {\mathcal {V}}}$ is the minimal set of normalized valuations satisfying the equivalent definition.

With the notations above, let ${\displaystyle v_{\mathfrak {p}}}$ denote the normalized valuation corresponding to the valuation ring ${\displaystyle A_{\mathfrak {p}}}$, ${\displaystyle U}$ denote the set of units of ${\displaystyle A}$, and ${\displaystyle K}$ its quotient field.