# Krull ring

(Redirected from Krull domain)

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 (1931). They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.

## 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.

## Properties

A Krull domain is a unique factorization domain if and only if every prime ideal of height one is principal.[1]

Let A be a Zariski ring (e.g., a local noetherian ring). If the completion ${\displaystyle {\widehat {A}}}$ is a Krull domain, then A is a Krull domain.[2]

## Examples

1. Every integrally closed noetherian domain is a Krull ring. In particular, Dedekind domains are Krull rings. Conversely Krull rings are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed.
2. If ${\displaystyle A}$ is a Krull ring then so is the polynomial ring ${\displaystyle A[x]}$ and the formal power series ring ${\displaystyle A[[x]]}$.
3. 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 ring which is not noetherian. In general, any unique factorization domain is a Krull ring.
4. 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 ring (Mori–Nagata theorem).[3]

## The divisor class group of a Krull ring

A (Weil) divisor of a Krull ring A is a formal integral linear combination of the height 1 prime ideals, and these form a group D(A). A divisor of the form div(x) for some non-zero x in A is called a principal divisor, and the principal divisors form a subgroup of the group of divisors. The quotient of the group of divisors by the subgroup of principal divisors is called the divisor class group of A.

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]/(xyz2) the divisor class group has order 2, generated by the divisor y=z, but the Picard subgroup is the trivial group.

## References

1. ^ "Krull ring - Encyclopedia of Mathematics". eom.springer.de. Retrieved 2016-04-14.
2. ^ Bourbaki, 7.1, no 10, Proposition 16.
3. ^ Huneke, Craig; Swanson, Irena (2006-10-12). Integral Closure of Ideals, Rings, and Modules. Cambridge University Press. ISBN 9780521688604.