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.
In this article, a ring is commutative and has unity.
be the set of all prime ideals of
of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal.
is a Krull ring if It is also possible to characterize Krull rings by mean of valuations only:[2] An integral domain
is a Krull ring if there exists a family
of discrete valuations on the field of fractions
are called essential valuations of
, one can associate a unique normalized valuation
satisfies the conditions of the equivalent definition.
is the minimal set of normalized valuations satisfying the equivalent definition.
denote the set of units of
is a height 1 prime ideal of
The set of prime divisors of
is a formal integral linear combination of prime divisors.
They form an Abelian group, noted
, is called a principal divisor.
is a Krull domain containing
As usual, we say that a prime ideal
lies above a prime ideal
Denote the ramification index of
the set of prime divisors of
is contained in at most finitely many elements of
[19] For example, the following generalizes a theorem of Gauss: The application
is a unique factorization domain, then so is
[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] A Cartier divisor of a Krull ring is a locally principal (Weil) divisor.
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.