In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, a discrete valuation ring (DVR) is a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...
(PID) with exactly one non-zero
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
.
This means a DVR is 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 ...
''R'' which satisfies any one of the following equivalent conditions:
# ''R'' is a
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...
, and not a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
# ''R'' is a
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' such ...
with a value group isomorphic to the integers under addition.
# ''R'' is a
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
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 necessari ...
and not a field.
# ''R'' is a
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 leng ...
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
whose maximal
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
is principal, and not a field.
[https://mathoverflow.net/a/155639/114772]
# ''R'' is an
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 leng ...
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
with
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generall ...
one.
# ''R'' is a principal ideal domain with a unique non-zero
prime ideal.
# ''R'' is a principal ideal domain with a unique
irreducible element
In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.
Relationship with prime elements
Irreducible elements should not be confus ...
(
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
multiplication by
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
s).
# ''R'' 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 ...
with a unique irreducible element (up to multiplication by units).
# ''R'' is Noetherian, not a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, and every nonzero
fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral dom ...
of ''R'' is
irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it.
# There is some
discrete valuation ν on the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
''K'' of ''R'' such that ''R'' =
.
Examples
Algebraic
Localization of Dedekind rings
Let
. Then, the field of fractions of
is
. For any nonzero element
of
, we can apply
unique factorization
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 an ...
to the numerator and denominator of ''r'' to write ''r'' as where ''z'', ''n'', and ''k'' are integers with ''z'' and ''n'' odd. In this case, we define ν(''r'')=''k''.
Then
is the discrete valuation ring corresponding to ν. The maximal ideal of
is the principal ideal generated by 2, i.e.
, and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter). Note that
is the
localization of the
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 necessari ...
at the
prime ideal generated by 2.
More generally, any
localization of a
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 necessari ...
at a non-zero
prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
:
for any
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
''p'' in complete analogy.
p-adic integers
The
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of
''p''-adic integers is a DVR, for any
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
. Here
is an
irreducible element
In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.
Relationship with prime elements
Irreducible elements should not be confus ...
; the
valuation assigns to each
-adic integer
the largest
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
such that
divides
.
Formal power series
Another important example of a DVR is the
ring of formal power series
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 ...
in one variable
over some field
. The "unique" irreducible element is
, the maximal ideal of
is the principal ideal generated by
, and the valuation
assigns to each power series the index (i.e. degree) of the first non-zero coefficient.
If we restrict ourselves to
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (201 ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
coefficients, we can consider the ring of power series in one variable that ''converge'' in a neighborhood of 0 (with the neighborhood depending on the power series). This is a discrete valuation ring. This is useful for building intuition with the
Valuative criterion of properness In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ...
.
Ring in function field
For an example more geometrical in nature, take the ring ''R'' = , considered as a
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of the field of
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s R(''X'') in the variable ''X''. ''R'' can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a
neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is ''X'' and the valuation assigns to each function ''f'' the order (possibly 0) of the zero of ''f'' at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line.
Scheme-theoretic
Henselian trait
For a DVR
it is common to write the fraction field as
and
the residue field. These correspond to the
generic
Generic or generics may refer to:
In business
* Generic term, a common name used for a range or class of similar things not protected by trademark
* Generic brand, a brand for a product that does not have an associated brand or trademark, other ...
and closed points of
For example, the closed point of
is
and the generic point is
. Sometimes this is denoted as
:
where
is the generic point and
is the closed point .
Localization of a point on a curve
Given an
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
, the
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
at a smooth point
is a discrete valuation ring, because it is a principal valuation ring. Note because the point
is smooth, the
completion of the
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
is
isomorphic to the
completion of the
localization of
at some point
.
Uniformizing parameter
Given a DVR ''R'', any irreducible element of ''R'' is a generator for the unique maximal ideal of ''R'' and vice versa. Such an element is also called a uniformizing parameter of ''R'' (or a uniformizing element, a uniformizer, or a prime element).
If we fix a uniformizing parameter ''t'', then ''M''=(''t'') is the unique maximal ideal of ''R'', and every other non-zero ideal is a power of ''M'', i.e. has the form (''t''
''k'') for some ''k''≥0. All the powers of ''t'' are distinct, and so are the powers of ''M''. Every non-zero element ''x'' of ''R'' can be written in the form α''t''
''k'' with α a unit in ''R'' and ''k''≥0, both uniquely determined by ''x''. The valuation is given by ''ν''(''x'') = ''kv''(''t''). So to understand the ring completely, one needs to know the group of units of ''R'' and how the units interact additively with the powers of ''t''.
The function ''v'' also makes any discrete valuation ring into a
Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. ...
.
Topology
Every discrete valuation ring, being a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
, carries a natural topology and is a
topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps:
R \times R \to R
where R \times R carries the product topology. That means R is an additive t ...
. We can also give it a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
structure where the distance between two elements ''x'' and ''y'' can be measured as follows:
:
(or with any other fixed real number > 1 in place of 2). Intuitively: an element ''z'' is "small" and "close to 0"
iff
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
its
valuation ν(''z'') is large. The function , x-y, , supplemented by , 0, =0, is the restriction of an
absolute value defined on the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of the discrete valuation ring.
A DVR is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
if and only if it is
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...
and its
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is ...
''R''/''M'' is a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
.
Examples of
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...
DVRs include
* the ring of ''p''-adic integers and
* the ring of formal power series over any field
For a given DVR, one often passes to its
completion, a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...
DVR containing the given ring that is often easier to study. This
completion procedure can be thought of in a geometrical way as passing from
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s to
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
, or from
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s to the
reals.
Returning to our examples: the ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined (i.e. finite) in a neighborhood of 0 on the real line; it is also the completion of the ring of all real power series that converge near 0. The completion of
(which can be seen as the set of all rational numbers that are ''p''-adic integers) is the ring of all ''p''-adic integers Z
''p''.
See also
*
:Localization (mathematics)
*
Local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
*
Ramification of local fields In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups.
In thi ...
*
Cohen ring
In algebra, a Cohen ring is a field or a complete discrete valuation ring of mixed characteristic (0, p) whose maximal ideal is generated by ''p''. Cohen rings are used in the Cohen structure theorem for complete
Complete may refer to:
Logi ...
*
Valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' such ...
References
*
* {{Citation , last1=Dummit , first1=David S. , last2=Foote , first2=Richard M. , title=Abstract algebra , publisher=
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, i ...
, location=New York , edition=3rd , isbn=978-0-471-43334-7 , mr=2286236 , year=2004
Discrete valuation ring The ''
Encyclopaedia of Mathematics''.
Commutative algebra
Localization (mathematics)