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

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

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 .