Discrete valuation ring
Encyclopedia
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:
to the numerator and denominator of r to write r as 2kp/q, where p, q, and k are integers with p and q odd. In this case, we define ν(r)=k.
Then Z(2) is the discrete valuation ring corresponding to ν. The maximal ideal of Z(2) is the principal ideal generated by 2, and the "unique" irreducible element (up to units) is 2.
Note that Z(2) is the localization
of the Dedekind domain
Z at the prime ideal
generated by 2. Any localization of a Dedekind domain 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 Z(p) for any prime
p in complete analogy.
For an example more geometrical in nature, take the ring R = { f/g : f, g polynomial
s in R[X] and g(0) ≠ 0}, considered as a subring
of the field of rational function
s R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined 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.
Another important example of a DVR is the ring of formal power series
R = KT
in one variable T over some field K. The "unique" irreducible element is T, the maximal ideal of R is the principal ideal generated by T, and the valuation ν assigns to each power series the index of the first non-zero coefficient.
If we restrict ourselves to real
or complex
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 also a discrete valuation ring.
Finally, the ring Zp of p-adic integers is a DVR, for any prime p. Here p is an irreducible element; the valuation assigns to each p-adic integer x the largest integer k such that pk divides x.
If we fix a uniformizing parameter, 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 different, and so are the powers of M. Every non-zero element x of R can be written in the form αt k with α a unit in R and k≥0, both uniquely determined by x. The valuation is given by ν(x) = k. 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 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
its valuation ν(z) is large.
A DVR is compact
if and only if it is complete
and its residue field
R/M is a finite field
.
Examples of complete 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 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 functions to power series.
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 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 Z(p) (which can be seen as the set of all rational numbers that are p-adic integers) is the ring of all p-adic integers Zp.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a discrete valuation ring (DVR) is a principal ideal domain
Principal ideal domain
In abstract algebra, 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 principal, although some authors refer to PIDs as...
(PID) with exactly one non-zero maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
.
This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:
- R is a localLocal ringIn abstract algebra, more particularly in 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 number fields examined at a particular place, or...
principal ideal domainPrincipal ideal domainIn abstract algebra, 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 principal, although some authors refer to PIDs as...
, and not a fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. - R is a valuation ringValuation ringIn 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....
with a value group isomorphic to the integers under addition. - R is a localLocal ringIn abstract algebra, more particularly in 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 number fields examined at a particular place, or...
Dedekind domainDedekind domainIn 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 unique up to the order of the factors...
and not a field. - R is a noetherianNoetherianIn mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...
local ringLocal ringIn abstract algebra, more particularly in 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 number fields examined at a particular place, or...
with Krull dimensionKrull dimensionIn commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
one, and the maximal idealIdeal (ring theory)In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
of R is principal. - R is an integrally closedIntegrally closedIn mathematics, more specifically in abstract algebra, the concept of integrally closed has two meanings, one for groups and one for rings. -Commutative rings:...
noetherianNoetherianIn mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...
local ringLocal ringIn abstract algebra, more particularly in 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 number fields examined at a particular place, or...
with Krull dimensionKrull dimensionIn commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
one. - R is a principal ideal domain with a unique non-zero prime idealPrime idealIn algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
. - R is a principal ideal domain with a unique irreducible elementIrreducible elementIn abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.Irreducible elements should not be confused with prime elements...
(up toUp toIn mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
multiplication by unitUnit (ring theory)In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
s). - R is a unique factorization domainUnique factorization domainIn mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
with a unique irreducible element (up to multiplication by units). - R is not a fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, and every nonzero fractional idealFractional idealIn 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 domain are like ideals where denominators are allowed...
of R is irreducibleIrreducible idealIn mathematics, an ideal of a commutative ring is said to be irreducible if it cannot be written as a finite intersection of ideals properly containing it....
in the sense that it cannot be written as finite intersection of fractional ideals properly containing it. - There is some Dedekind valuation ν on the field of fractionsField of fractionsIn abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
K of R, such that R={x : x in K, ν(x) ≥ 0}.
Examples
Let Z(2)={ p/q : p, q in Z, q odd }. Then the field of fractions of Z(2) is Q. Now, for any nonzero element r of Q, we can apply unique factorizationFundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
to the numerator and denominator of r to write r as 2kp/q, where p, q, and k are integers with p and q odd. In this case, we define ν(r)=k.
Then Z(2) is the discrete valuation ring corresponding to ν. The maximal ideal of Z(2) is the principal ideal generated by 2, and the "unique" irreducible element (up to units) is 2.
Note that Z(2) is the localization
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
of the Dedekind domain
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 unique up to the order of the factors...
Z at the prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
generated by 2. Any localization of a Dedekind domain 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 Z(p) for any prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p in complete analogy.
For an example more geometrical in nature, take the ring R = { f/g : f, g polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s in R[X] and g(0) ≠ 0}, considered as a subring
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
of the field of rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined 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.
Another important example of a DVR is the ring of formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
R = K
T
T is the 20th letter in the basic modern Latin alphabet. It is the most commonly used consonant and the second most common letter in the English language.- History :Taw was the last letter of the Western Semitic and Hebrew alphabets...
If we restrict ourselves to real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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 also a discrete valuation ring.
Finally, the ring Zp of p-adic integers is a DVR, for any prime p. Here p is an irreducible element; the valuation assigns to each p-adic integer x the largest integer k such that pk divides x.
Uniformizing parameter
Given a DVR R, then 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, or a uniformizer).If we fix a uniformizing parameter, 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 different, and so are the powers of M. Every non-zero element x of R can be written in the form αt k with α a unit in R and k≥0, both uniquely determined by x. The valuation is given by ν(x) = k. 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.
Topology
Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ringTopological ring
In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...
. 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
IFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
its valuation ν(z) is large.
A DVR is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
if and only if it is complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
and its residue field
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...
R/M is a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
.
Examples of complete 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
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
, a complete 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 functions to power series.
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 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 Z(p) (which can be seen as the set of all rational numbers that are p-adic integers) is the ring of all p-adic integers Zp.