# 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 $\pi$ such that any non-zero ideal is generated by some power of the element $\pi$; 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 $u \pi ^ {n}$, where $u$ is an invertible element and $n \geq 0$ is an integer. Examples of discretely-normed rings include the ring $\mathbf Z _ {p}$ of $p$- adic integers, the ring $k [[ T ]]$ of formal power series in one variable $T$ over a field $k$, and the ring of Witt vectors (cf. Witt vector) $W ( k)$ for a perfect field $k$.

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 $\mathbf Z$.

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 $k [[ T ]]$, where $k$ is a finite field, or else is a finite extension of $\mathbf Z _ {p}$.

If $A \subset B$ is a local homomorphism of discretely-normed rings with uniformizing elements $\pi$ and $\Pi$, then $\pi = u \Pi ^ {e}$, where $u$ is an invertible element in $B$. The integer $e = e ( B / A )$ is the ramification index of the extension $A \subset B$, and

$$[ B / \Pi B : A / \pi A ] = f ( B / A )$$

is called the residue degree. This situation arises when one considers the integral closure $B$ of a discretely-normed ring $A$ with a field of fractions $K$ in a finite extension $L$ of $K$. In such a case $B$ is a semi-local principal ideal ring; if $\mathfrak n _ {1} \dots \mathfrak n _ {s}$ are its maximal ideals, then the localizations $B _ {i} = B _ {\mathfrak n _ {i} }$ are discretely-normed rings. If $L$ is a separable extension of $K$ of degree $n$, the formula

$$\sum _ {i = 1 } ^ { s } e ( B _ {i} / A ) f ( B _ {i} / A ) = n$$

is valid. If $L / K$ is a Galois extension, then all $e ( B _ {i} / A )$ and all $f ( B _ {i} / A )$ are equal, and $n = sef$. If $A$ is a complete discretely-normed ring, $B$ itself will be a discretely-normed ring and $e ( B / A ) f ( B / A ) = n$. On these assumptions the extension $A \subset B$( and also $L$ over $K$) is known as an unramified extension if $e ( B / A ) = 1$ and the field $B / \mathfrak n$ is separable over $A / \mathfrak m$; it is weakly ramified if $e ( B / A )$ is relatively prime with the characteristic of the field $A / \mathfrak m$ while $B / \mathfrak n$ is separable over $A / \mathfrak m$; it is totally ramified if $f ( B / A ) = 1$.

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

Let $A$ be a discretely-normed ring with uniformizing parameter $\pi$. The associated valuation is then defined by $\nu ( a) = n$ if $a = u \pi ^ {n}$, $u$ a unit of $A$. A corresponding norm on $A$ is defined by $| a | = c ^ {\nu ( a ) }$, $| 0 | = 0$, where $c$ is a real number between $0$ and $1$. This makes $A$ a normal ring. If the residue field $k = A ( \pi )$ of $A$ is finite it is customary to take $c = q ^ {-} 1$ where $q$ is the number of elements of $k$.