Discretely-normed ring
discrete valuation ring, discrete valuation domain
A ring with a discrete valuation, i.e. an integral domain with a unit element in which there exists an element such that any non-zero ideal is generated by some power of the element
; such an element is called a uniformizing parameter, and is defined up to multiplication by an invertible element. Each non-zero element of a discretely-normed ring can be uniquely written in the form
, where
is an invertible element and
is an integer. Examples of discretely-normed rings include the ring
of
-adic integers, the ring
of formal power series in one variable
over a field
, and the ring of Witt vectors (cf. Witt vector)
for a perfect field
.
A discretely-normed ring may also be defined as a local principal ideal ring; as a local one-dimensional Krull ring; as a local Noetherian ring with a principal maximal ideal; as a Noetherian valuation ring; or as a valuation ring with group of values .
The completion (in the topology of a local ring) of a discretely-normed ring is also a discretely-normed ring. A discretely-normed ring is compact if and only if it is complete and its residue field is finite; any such ring is either isomorphic to , where
is a finite field, or else is a finite extension of
.
If is a local homomorphism of discretely-normed rings with uniformizing elements
and
, then
, where
is an invertible element in
. The integer
is the ramification index of the extension
, and
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is called the residue degree. This situation arises when one considers the integral closure of a discretely-normed ring
with a field of fractions
in a finite extension
of
. In such a case
is a semi-local principal ideal ring; if
are its maximal ideals, then the localizations
are discretely-normed rings. If
is a separable extension of
of degree
, the formula
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is valid. If is a Galois extension, then all
and all
are equal, and
. If
is a complete discretely-normed ring,
itself will be a discretely-normed ring and
. On these assumptions the extension
(and also
over
) is known as an unramified extension if
and the field
is separable over
; it is weakly ramified if
is relatively prime with the characteristic of the field
while
is separable over
; it is totally ramified if
.
The theory of modules over a discretely-normed ring is very similar to the theory of Abelian groups [3]. Any module of finite type is a direct sum of cyclic modules; a torsion-free module is a flat module; any projective module or submodule of a free module is free. However, the direct product of an infinite number of free modules is not free. A torsion-free module of countable rank over a complete discretely-normed ring is a direct sum of modules of rank one.
References
[1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
[2] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) |
[3] | J. Kaplansky, "Modules over Dedekind rings and valuation rings" Trans. Amer. Math. Soc. , 72 (1952) pp. 327–340 |
Comments
Let be a discretely-normed ring with uniformizing parameter
. The associated valuation is then defined by
if
,
a unit of
. A corresponding norm on
is defined by
,
, where
is a real number between
and
. This makes
a normal ring. If the residue field
of
is finite it is customary to take
where
is the number of elements of
.
Discretely-normed ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discretely-normed_ring&oldid=46738