Editing Valuation ring (section)

== Ideals in valuation rings ==
We may describe the ideals in the valuation ring by means of its value group.

Let Γ be a [[totally ordered group|totally ordered]] [[abelian group]]. A subset Δ of Γ is called a ''segment'' if it is nonempty and, for any α in Δ, any element between −α and α is also in Δ (end points included). A subgroup of Γ is called an ''isolated subgroup'' if it is a segment and is a proper subgroup.

Let ''D'' be a valuation ring with valuation ''v'' and value group Γ. For any subset ''A'' of ''D'', we let <math>\Gamma_A</math> be the complement of the union of <math>v(A - 0)</math> and <math>-v(A - 0)</math> in <math>\Gamma</math>. If ''I'' is a proper ideal, then <math>\Gamma_I</math> is a segment of <math>\Gamma</math>. In fact, the mapping <math>I \mapsto \Gamma_I</math> defines an inclusion-reversing bijection between the set of proper ideals of ''D'' and the set of segments of <math>\Gamma</math>.{{sfn|Zariski|Samuel|1975|loc=Ch. VI, Theorem 15}} Under this correspondence, the nonzero prime ideals of ''D'' correspond bijectively to the isolated subgroups of Γ.

Example: The ring of ''p''-adic integers <math>\Z_p</math> is a valuation ring with value group <math>\Z</math>. The zero subgroup of <math>\Z</math> corresponds to the unique maximal ideal <math>(p) \subseteq \Z_p</math> and the whole group to the [[zero ideal]]. The maximal ideal is the only isolated subgroup of <math>\Z</math>.

The set of isolated subgroups is totally ordered by inclusion. The '''height''' or '''rank''' ''r''(Γ) of Γ is defined to be the [[cardinality]] of the set of isolated subgroups of Γ. Since the nonzero prime ideals are totally ordered and they correspond to isolated subgroups of Γ, the height of Γ is equal to the [[Krull dimension]] of the valuation ring ''D'' associated with Γ.

The most important special case is height one, which is equivalent to Γ being a subgroup of the [[real number]]s <math>\mathbb{R}</math> under addition (or equivalently, of the [[positive real numbers]] <math>\mathbb{R}^{+}</math> under multiplication.) A valuation ring with a valuation of height one has a corresponding [[absolute value (algebra)|absolute value]] defining an [[ultrametric]] [[place (mathematics)|place]]. A special case of this are the [[discrete valuation ring]]s mentioned earlier.

The '''rational rank''' ''rr''(Γ) is defined as the rank of the value group as an abelian group,
:<math>\mathrm{dim}_\Q(\Gamma \otimes_\Z \Q).</math>