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
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
in 1931. They are a higher-dimensional generalization of
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
s, which are exactly the Krull domains of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
at most 1.
In this article, a ring is commutative and has unity.
Formal definition
Let
be an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
and let
be the set of all
prime ideals of
of
height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then
is a Krull ring if
#
is a
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' i ...
for all
,
#
is the intersection of these discrete valuation rings (considered as subrings of the quotient field of
).
#Any nonzero element of
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:
An integral domain
is a Krull ring if there exists a family
of discrete valuations on the field of fractions
of
such that:
# for any
and all
, except possibly a finite number of them,
;
# for any
,
belongs to
if and only if
for all
.
The valuations
are called essential valuations of
.
The link between the two definitions is as follows: for every
, one can associate a unique normalized valuation
of
whose valuation ring is
. Then the set
satisfies the conditions of the equivalent definition. Conversely, if the set
is as above, and the
have been normalized, then
may be bigger than
, but it ''must'' contain
. In other words,
is the minimal set of normalized valuations satisfying the equivalent definition.
There are other ways to introduce and define Krull rings. The theory of Krull rings can be exposed in synergy with the theory of
divisorial ideals. One of the best references on the subject is ''Lecture on Unique Factorization Domains'' by P. Samuel.
Properties
With the notations above, let
denote the normalized valuation corresponding to the valuation ring
,
denote the set of units of
, and
its quotient field.
* ''An element
belongs to
if, and only if,
for every
.'' Indeed, in this case,
for every
, hence
; by the intersection property,
. Conversely, if
and
are in
, then
, hence
, since both numbers must be
.
* ''An element
is uniquely determined, up to a unit of
, by the values
,
.'' Indeed, if
for every
, then
, hence
by the above property (q.e.d). This shows that the application
is well defined, and since
for only finitely many
, it is an embedding of
into the free Abelian group generated by the elements of
. Thus, using the multiplicative notation "
" for the later group, there holds, for every
,
, where the
are the elements of
containing
, and
.
* The valuations
are pairwise independent. As a consequence, there holds the so-called ''weak approximation theorem'', an homologue of the Chinese remainder theorem: ''if
are distinct elements of
,
belong to
(resp.
), and
are
natural numbers, then there exist
(resp.
) such that
for every
.''
* Two elements
and
of
are ''coprime'' if
and
are not both
for every
. The basic properties of valuations imply that a good theory of coprimality holds in
.
* Every prime ideal of
contains an element of
.
* Any finite intersection of Krull domains whose quotient fields are the same is again a Krull domain.
* If
is a subfield of
, then
is a Krull domain.
* If
is a multiplicatively closed set not containing 0, the ring of quotients
is again a Krull domain. In fact, the essential valuations of
are those valuation
(of
) for which
.
* If
is a finite algebraic extension of
, and
is the integral closure of
in
, then
is a Krull domain.
Examples
#Any unique factorization domain is a Krull domain. Conversely, a Krull domain is a
unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
if (and only if) every prime ideal of height one is principal.
# Every
integrally closed noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
domain is a Krull domain. In particular,
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
s are Krull domains. Conversely, Krull domains are integrally closed, so a Noetherian domain is Krull if and only if it is integrally closed.
# If
is a Krull domain then so is the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
and the
formal power series ring
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
.
# The polynomial ring