Krull ring

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 . 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 formula_1 be an integral domain and let formula_2 be the set of all prime ideals of formula_1 of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then formula_1 is a Krull ring if
Properties.
A Krull domain is a unique factorization domain if and only if every prime ideal of height one is principal.
Let "A" be a Zariski ring (e.g., a local noetherian ring). If the completion formula_10 is a Krull domain, then "A" is a Krull domain.
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 formula_22 for some non-zero "x" in "K", the fractional field of formula_23, 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"]/("xy"–"z"2) the divisor class group has order 2, generated by the divisor "y"="z", but the Picard subgroup is the trivial group.