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.^[1] 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.

Formal definition

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

1. ${\displaystyle A_{\mathfrak {p))}$ is a discrete valuation ring for all ${\displaystyle {\mathfrak {p))\in P}$,
2. ${\displaystyle A}$ is the intersection of these discrete valuation rings (considered as subrings of the quotient field of ${\displaystyle A}$),
3. any nonzero element of ${\displaystyle A}$ is contained in only a finite number of height 1 prime ideals.

It is also possible to characterize Krull rings by mean of valuations only:^[2]

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:

1. for any ${\displaystyle x\in K\setminus \{0\))$ and all ${\displaystyle i}$, except possibly a finite number of them, ${\displaystyle v_{i}(x)=0}$,
2. for any ${\displaystyle x\in K\setminus \{0\))$, ${\displaystyle x}$ belongs to ${\displaystyle A}$ if and only if ${\displaystyle v_{i}(x)\geq 0}$ for all ${\displaystyle i\in I}$.

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 ${\displaystyle A_{\mathfrak {p))}$.^[3] 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.

Properties

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.

• An element ${\displaystyle x\in K}$ belongs to ${\displaystyle U}$ if, and only if, ${\displaystyle v_{\mathfrak {p))(x)=0}$ for every ${\displaystyle {\mathfrak {p))\in P}$. Indeed, in this case, ${\displaystyle x\not \in A_{\mathfrak {p)){\mathfrak {p))}$ for every ${\displaystyle {\mathfrak {p))\in P}$, hence ${\displaystyle x^{-1}\in A_{\mathfrak {p))}$; by the intersection property, ${\displaystyle x^{-1}\in A}$. Conversely, if ${\displaystyle x}$ and ${\displaystyle x^{-1))$ are in ${\displaystyle A}$, then ${\displaystyle v_{\mathfrak {p))(xx^{-1})=v_{\mathfrak {p))(1)=0=v_{\mathfrak {p))(x)+v_{\mathfrak {p))(x^{-1})}$, hence ${\displaystyle v_{\mathfrak {p))(x)=v_{\mathfrak {p))(x^{-1})=0}$, since both numbers must be ${\displaystyle \geq 0}$.
• An element ${\displaystyle x\in A}$ is uniquely determined, up to a unit of ${\displaystyle A}$, by the values ${\displaystyle v_{\mathfrak {p))(x)}$, ${\displaystyle {\mathfrak {p))\in P}$. Indeed, if ${\displaystyle v_{\mathfrak {p))(x)=v_{\mathfrak {p))(y)}$ for every ${\displaystyle {\mathfrak {p))\in P}$, then ${\displaystyle v_{\mathfrak {p))(xy^{-1})=0}$, hence ${\displaystyle xy^{-1}\in U}$ by the above property (q.e.d). This shows that the application ${\displaystyle x\ {\rm {mod))\ U\mapsto \left(v_{\mathfrak {p))(x)\right)_((\mathfrak {p))\in P))$ is well defined, and since ${\displaystyle v_{\mathfrak {p))(x)\not =0}$ for only finitely many ${\displaystyle {\mathfrak {p))}$, it is an embedding of ${\displaystyle A^{\times }/U}$ into the free Abelian group generated by the elements of ${\displaystyle P}$. Thus, using the multiplicative notation "${\displaystyle \cdot }$" for the later group, there holds, for every ${\displaystyle x\in A^{\times ))$, ${\displaystyle x=1\cdot {\mathfrak {p))_{1}^{\alpha _{1))\cdot {\mathfrak {p))_{2}^{\alpha _{2))\cdots {\mathfrak {p))_{n}^{\alpha _{n))\ {\rm {mod))\ U}$, where the ${\displaystyle {\mathfrak {p))_{i))$ are the elements of ${\displaystyle P}$ containing ${\displaystyle x}$, and ${\displaystyle \alpha _{i}=v_((\mathfrak {p))_{i))(x)}$.
• The valuations ${\displaystyle v_{\mathfrak {p))}$ are pairwise independent.^[4] As a consequence, there holds the so-called weak approximation theorem,^[5] an homologue of the Chinese remainder theorem: if ${\displaystyle {\mathfrak {p))_{1},\ldots {\mathfrak {p))_{n))$ are distinct elements of ${\displaystyle P}$, ${\displaystyle x_{1},\ldots x_{n))$ belong to ${\displaystyle K}$ (resp. ${\displaystyle A_{\mathfrak {p))}$), and ${\displaystyle a_{1},\ldots a_{n))$ are ${\displaystyle n}$ natural numbers, then there exist ${\displaystyle x\in K}$ (resp. ${\displaystyle x\in A_{\mathfrak {p))}$) such that ${\displaystyle v_((\mathfrak {p))_{i))(x-x_{i})=n_{i))$ for every ${\displaystyle i}$.
• A consequence of the weak approximation theorem is a characterization of when Krull rings are noetherian; namely, a Krull ring ${\displaystyle A}$ is noetherian if and only if all of its quotients ${\displaystyle A/{\mathfrak {p))}$ by height-1 primes are noetherian.
• Two elements ${\displaystyle x}$ and ${\displaystyle y}$ of ${\displaystyle A}$ are coprime if ${\displaystyle v_{\mathfrak {p))(x)}$ and ${\displaystyle v_{\mathfrak {p))(y)}$ are not both ${\displaystyle >0}$ for every ${\displaystyle {\mathfrak {p))\in P}$. The basic properties of valuations imply that a good theory of coprimality holds in ${\displaystyle A}$.
• Every prime ideal of ${\displaystyle A}$ contains an element of ${\displaystyle P}$.^[6]
• Any finite intersection of Krull domains whose quotient fields are the same is again a Krull domain.^[7]
• If ${\displaystyle L}$ is a subfield of ${\displaystyle K}$, then ${\displaystyle A\cap L}$ is a Krull domain.^[8]
• If ${\displaystyle S\subset A}$ is a multiplicatively closed set not containing 0, the ring of quotients ${\displaystyle S^{-1}A}$ is again a Krull domain. In fact, the essential valuations of ${\displaystyle S^{-1}A}$ are those valuation ${\displaystyle v_{\mathfrak {p))}$ (of ${\displaystyle K}$) for which ${\displaystyle {\mathfrak {p))\cap S=\emptyset }$.^[9]
• If ${\displaystyle L}$ is a finite algebraic extension of ${\displaystyle K}$, and ${\displaystyle B}$ is the integral closure of ${\displaystyle A}$ in ${\displaystyle L}$, then ${\displaystyle B}$ is a Krull domain.^[10]

Examples

1. Any unique factorization domain is a Krull domain. Conversely, a Krull domain is a unique factorization domain if (and only if) every prime ideal of height one is principal.^[11][12]
2. Every integrally closed noetherian domain is a Krull domain.^[13] In particular, Dedekind domains are Krull domains. Conversely, Krull domains are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed.
3. If ${\displaystyle A}$ is a Krull domain then so is the polynomial ring ${\displaystyle A[x]}$ and the formal power series ring ${\displaystyle A[[x]]}$.^[14]
4. The polynomial ring ${\displaystyle R[x_{1},x_{2},x_{3},\ldots ]}$ in infinitely many variables over a unique factorization domain ${\displaystyle R}$ is a Krull domain which is not noetherian.
5. Let ${\displaystyle A}$ be a Noetherian domain with quotient field ${\displaystyle K}$, and ${\displaystyle L}$ be a finite algebraic extension of ${\displaystyle K}$. Then the integral closure of ${\displaystyle A}$ in ${\displaystyle L}$ is a Krull domain (Mori–Nagata theorem).^[15]
6. Let ${\displaystyle A}$ be a Zariski ring (e.g., a local noetherian ring). If the completion ${\displaystyle {\widehat {A))}$ is a Krull domain, then ${\displaystyle A}$ is a Krull domain (Mori).^[16][17]
7. Let ${\displaystyle A}$ be a Krull domain, and ${\displaystyle V}$ be the multiplicatively closed set consisting in the powers of a prime element ${\displaystyle p\in A}$. Then ${\displaystyle S^{-1}A}$ is a Krull domain (Nagata).^[18]

The divisor class group of a Krull ring

Assume that ${\displaystyle A}$ is a Krull domain and ${\displaystyle K}$ is its quotient field. A prime divisor of ${\displaystyle A}$ is a height 1 prime ideal of ${\displaystyle A}$. The set of prime divisors of ${\displaystyle A}$ will be denoted ${\displaystyle P(A)}$ in the sequel. A (Weil) divisor of ${\displaystyle A}$ is a formal integral linear combination of prime divisors. They form an Abelian group, noted ${\displaystyle D(A)}$. A divisor of the form ${\displaystyle div(x)=\sum _{p\in P}v_{p}(x)\cdot p}$, for some non-zero ${\displaystyle x}$ in ${\displaystyle K}$, is called a principal divisor. The principal divisors of ${\displaystyle A}$ form a subgroup of the group of divisors (it has been shown above that this group is isomorphic to ${\displaystyle A^{\times }/U}$, where ${\displaystyle U}$ is the group of unities of ${\displaystyle A}$). The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of ${\displaystyle A}$; it is usually denoted ${\displaystyle C(A)}$.

Assume that ${\displaystyle B}$ is a Krull domain containing ${\displaystyle A}$. As usual, we say that a prime ideal ${\displaystyle {\mathfrak {P))}$ of ${\displaystyle B}$ lies above a prime ideal ${\displaystyle {\mathfrak {p))}$ of ${\displaystyle A}$ if ${\displaystyle {\mathfrak {P))\cap A={\mathfrak {p))}$; this is abbreviated in ${\displaystyle {\mathfrak {P))|{\mathfrak {p))}$.

Denote the ramification index of ${\displaystyle v_{\mathfrak {P))}$ over ${\displaystyle v_{\mathfrak {p))}$ by ${\displaystyle e({\mathfrak {P)),{\mathfrak {p)))}$, and by ${\displaystyle P(B)}$ the set of prime divisors of ${\displaystyle B}$. Define the application ${\displaystyle P(A)\to D(B)}$ by

${\displaystyle j({\mathfrak {p)))=\sum _((\mathfrak {P))|{\mathfrak {p)),\ {\mathfrak {P))\in P(B)}e({\mathfrak {P)),{\mathfrak {p))){\mathfrak {P))}$

(the above sum is finite since every ${\displaystyle x\in {\mathfrak {p))}$ is contained in at most finitely many elements of ${\displaystyle P(B)}$). Let extend the application ${\displaystyle j}$ by linearity to a linear application ${\displaystyle D(A)\to D(B)}$. One can now ask in what cases ${\displaystyle j}$ induces a morphism ${\displaystyle {\bar {j)):C(A)\to C(B)}$. This leads to several results.^[19] For example, the following generalizes a theorem of Gauss:

The application ${\displaystyle {\bar {j)):C(A)\to C(A[X])}$ is bijective. In particular, if ${\displaystyle A}$ is a unique factorization domain, then so is ${\displaystyle A[X]}$.^[20]

The divisor class group of a Krull rings are also used to set up powerful descent methods, and in particular the Galoisian descent.^[21]

Cartier divisor

A Cartier divisor of a Krull ring is a locally principal (Weil) divisor. The Cartier divisors form a subgroup of the group of divisors containing the principal divisors. The quotient of the Cartier divisors by the principal divisors is a subgroup of the divisor class group, isomorphic to the Picard group of invertible sheaves on Spec(A).

Example: in the ring k[x,y,z]/(xy–z²) the divisor class group has order 2, generated by the divisor y=z, but the Picard subgroup is the trivial group.^[22]