Ring with discrete valuation

In a few words, a ring with discrete valuation is a just Anneau in which all the elements are factorized in irreducible elements, provided of only one element first $T$. The theory of factorization in such a ring is then simple, the question which one installation is of knowing which power of $t$ divides an element of A. That gives a valuation $v\; then:\; With\; \backslash \; backslash\; \backslash \; \{0\; \backslash \}\; \backslash \; rightarrow\; \backslash \; mathbb\; \{NR\}$. The rings with discrete valuation are tools very used in Théorie of the numbers like in Géométrie.

Definition

That is to say (F, v) a Body valué, the ring of valuation of v is the subset: $A:=\; \backslash \; \{X\; \backslash \; in\; F\; |\; v\; (X)\; \backslash \; geq\; 0\; \backslash \}$ It is obvious that has is well a ring and any ring of this form is called ring with discrete valuation.
There is $v\; (x^\; \{-\; 1\})\; =\; -\; v\; (X)$ in $F^*$, and thus $x$ is a unit of $A$ if and only if $v\; (X)\; =\; 0$. Thus a ring with discrete valuation is a Local ring of maximum ideal: $m\; =\; \backslash \; \{X\; \backslash \; in\; F\; |\; v\; (X)\; >\; 0\; \backslash \}$. That is to say $t$ an element of $F$ such as $v\; (T)\; =\; 1$; then there is $m\; =\; (T)$ and any ideal not no one of $A$ is form $(t^n)$. In particular $A$ is noethérien. A generator $t$ of $m$ is called a local parameter (or an element first, or one standardizing) ring $A$.

Criteria

One seeks criteria which make that a given ring is indeed local.
For example a local ring noethérien whose maximum ideal is principal $m\; =\; (T)$ for a $T$. One defines $v\; (X)\; =\; n$ where $N$ is the single entirety checking $X\; =\; t^n\; U$ for a unit $u$ of $a$ (it will be checked that it is well defined). There is then a valuation on the Corps of the fractions of $A$ given by $v\; (x/y)\; =\; v\; (X)\; -\; v\; (there)$.

When has is a unit commutative ring, the following assertions are equivalent:

1. has is a ring of discrete valuation;

2. has is a principal ring of valuation;
3. has is a ring of valuation noethérien;
4. has is a local ring, noethérien, of principal maximum ideal not nilpotent;
5. has is a local ring, noethérien, completely closed and of dimension of Krull 1;
6. has is a local ring, noethérien, just and of invertible maximum ideal.

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