Editing Valuation ring

{{Short description|Concept in algebra}}
In [[abstract algebra]], a '''valuation ring''' is an [[integral domain]] ''D'' such that for every non-zero element ''x'' of its [[field of fractions]] ''F'', at least one of ''x'' or ''x''<sup>&minus;1</sup> belongs to ''D''.

Given a [[field (mathematics)|field]] ''F'', if ''D'' is a [[subring]] of ''F'' such that either ''x'' or ''x''<sup>&minus;1</sup> belongs to
''D'' for every nonzero ''x'' in ''F'', then ''D'' is said to be '''a valuation ring for the field ''F''''' or a '''place''' of ''F''. Since ''F'' in this case is indeed the field of fractions of ''D'', a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field ''F'' is that valuation rings ''D'' of ''F'' have ''F'' as their field of fractions, and their [[ideal (ring theory)|ideals]] are [[totally ordered]] by [[subset|inclusion]]; or equivalently their [[principal ideal]]s are totally ordered by inclusion. In particular, every valuation ring is a [[local ring]].

The valuation rings of a field are the maximal elements of the set of the local subrings in the field [[partial order|partially ordered]] by '''dominance''' or '''refinement''',{{sfn|Hartshorne|1977|loc=Theorem I.6.1A}} where
:<math>(A,\mathfrak{m}_A)</math> dominates <math>(B,\mathfrak{m}_B)</math> if <math>A \supseteq B</math> and <math>\mathfrak{m}_A \cap B = \mathfrak{m}_B</math>.{{sfn|Efrat|2006|p=55}}

Every local ring in a field ''K'' is dominated by some valuation ring of ''K''.

An integral domain whose [[localization of a ring|localization]] at any [[prime ideal]] is a valuation ring is called a [[Prüfer domain]].

== Definitions ==
There are several equivalent definitions of valuation ring (see below for the characterization in terms of dominance). For an [[integral domain]] ''D'' and its [[field of fractions]] ''K'', the following are equivalent:
# For every non-zero ''x'' in ''K'', at least one of ''x'' or ''x''<sup>−1</sup> is in ''D''.
# The ideals of ''D'' are [[totally ordered]] by inclusion.
# The principal ideals of ''D'' are [[totally ordered]] by inclusion (i.e. the elements in ''D'' are, [[up to]] [[unit (ring theory)|units]], totally ordered by [[divisibility (ring theory)|divisibility]].)
# There is a [[totally ordered group|totally ordered]] [[abelian group]] Γ (called the '''value group''') and a '''[[valuation (algebra)|valuation]]''' ν: ''K'' → Γ ∪ {∞} with ''D'' = { ''x'' &isin; ''K'' | ν(''x'') ≥ 0 }.

The equivalence of the first three definitions follows easily. A theorem of {{harv|Krull|1939}} states that any [[ring (mathematics)|ring]] satisfying the first three conditions satisfies the fourth: take Γ to be the [[quotient group|quotient]] ''K''<sup>×</sup>/''D''<sup>×</sup> of the [[unit group]] of ''K'' by the unit group of ''D'', and take ν to be the natural projection. We can turn Γ into a [[totally ordered group]] by declaring the residue classes of elements of ''D'' as "positive".{{efn|More precisely, Γ is totally ordered by defining <math>[x] \geq [y]</math> [[if and only if]] <math>x y^{-1} \in D</math> where [''x''] and [''y''] are equivalence classes in Γ. cf. {{harvp|Efrat|2006|p=39}}}}

Even further, given any totally ordered abelian group Γ, there is a valuation ring ''D'' with value group Γ (see [[Hahn series]]).

From the fact that the ideals of a valuation ring are totally ordered, one can conclude that a valuation ring is a local domain, and that every finitely generated ideal of a valuation ring is principal (i.e., a valuation ring is a [[Bézout domain]]). In fact, it is a theorem of Krull that an integral domain is a valuation ring if and only if it is a local Bézout domain.{{sfn|Cohn|1968|loc=Proposition 1.5}} It also follows from this that a valuation ring is [[Noetherian ring|Noetherian]] if and only if it is a [[principal ideal domain]]. In this case, it is either a field or it has exactly one non-zero prime ideal; in the latter case it is called a [[discrete valuation ring]]. (By convention, a field is not a discrete valuation ring.)

A value group is called ''discrete'' if it is [[group isomorphism|isomorphic]] to the additive group of the [[Integer#Algebraic_properties|integers]], and a valuation ring has a discrete valuation group if and only if it is a [[discrete valuation ring]].{{sfn|Efrat|2006|p=43}}

Very rarely, ''valuation ring'' may refer to a ring that satisfies the second or third condition but is not necessarily a domain.  A more common term for this type of ring is ''[[serial module|uniserial ring]]''.

== Examples ==
* Any field <math>\mathbb{F}</math> is a valuation ring. For example, the field of rational functions <math>\mathbb{F}(X)</math> on an [[algebraic variety]] <math>X</math>.<ref>[https://math.stackexchange.com/q/946933 The role of valuation rings in algebraic geometry]</ref><ref>[https://mathoverflow.net/q/28595 Does there exist a Riemann surface corresponding to every field extension? Any other hypothesis needed?]</ref>
* A simple non-example is the integral domain <math>\Complex[X]</math> since the inverse of a generic <math>f/g \in \Complex(X)</math> is <math>g/f \not\in \Complex[X]</math>.
* The field of [[power series]]:
::<math>\mathbb{F}((X)) =\left\{ f(X) =\! \sum_{i>-\infty}^\infty a_iX^i \, :\ a_i \in \mathbb{F} \right\}</math>
:has the [[valuation (algebra)|valuation]] <math>v(f) = \inf\nolimits_{a_n \neq 0} n</math>. The subring <math>\mathbb{F}[[X]]</math> is a valuation ring as well.

* <math>\Z_{(p)},</math> the [[localization of a ring|localization]] of the integers <math>\Z</math> at the prime ideal (''p''), consisting of ratios where the numerator is any integer and the denominator is not divisible by ''p''. The field of fractions is the field of rational numbers <math>\Q.</math>
* The ring of [[meromorphic function]]s on the entire [[complex plane]] which have a [[Maclaurin series]] ([[Taylor series]] expansion at zero) is a valuation ring. The field of fractions are the functions meromorphic on the whole plane. If ''f'' does not have a Maclaurin series then 1/''f'' does.
* Any ring of [[p-adic integer|''p''-adic integers]] <math>\Z_p</math> for a given [[prime number|prime]] ''p'' is a [[local ring]], with field of fractions the [[p-adic number|''p''-adic numbers]] <math>\Q_p</math>. The [[integral closure]] <math>\Z_p^{\text{cl}}</math> of the ''p''-adic integers is also a local ring, with field of fractions <math>\Q_p^{\text{cl}}</math> (the [[algebraic closure]] of the ''p''-adic numbers). Both <math>\Z_p</math> and <math>\Z_p^{\text{cl}}</math> are valuation rings.
* Let '''k''' be an [[ordered field]]. An element of '''k''' is called finite if it lies between two integers ''n'' < ''x'' < ''m''; otherwise it is called infinite. The set ''D'' of finite elements of '''k''' is a valuation ring. The set of elements ''x'' such that ''x'' ∈ ''D'' and ''x''<sup>−1</sup> ∉ ''D'' is the set of [[infinitesimal]] elements; and an element ''x'' such that ''x'' ∉ ''D'' and ''x''<sup>−1</sup> ∈ ''D'' is called infinite.
* The ring '''F''' of finite elements of a [[hyperreal number|hyperreal field]] *'''R''' (an ordered field containing the [[real number]]s) is a valuation ring of *'''R'''. '''F''' consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, which is equivalent to saying a hyperreal number ''x'' such that −''n'' < ''x'' < ''n'' for some standard integer ''n''. The [[residue field]], finite hyperreal numbers modulo the ideal of infinitesimal hyperreal numbers, is isomorphic to the real numbers.
* A common geometric example comes from [[algebraic plane curve]]s.  Consider the [[polynomial ring]] <math>\Complex[x, y]</math> and an [[irreducible polynomial]] <math>f</math> in that ring. Then the ring <math>\Complex[x, y] / (f)</math> is the ring of polynomial functions on the curve <math>\{(x, y) : f(x, y) = 0\}</math>.  Choose a point <math>P = (P_x, P_y) \in \Complex ^2</math> such that <math>f(P) = 0</math> and it is a [[regular point]] on the curve; i.e., the local ring ''R'' at the point is a [[regular local ring]] of [[Krull dimension]] one or a [[discrete valuation ring]].<!-- by assumption, it is a valuation ring Then the [[localization of a ring|localization]] <math>\Complex[x, y]_{(x - P_x, y - P_y)} / (f)</math> is a discrete valuation ring. This ring represents [[germ (mathematics)|germs]] of polynomial functions at the point <math>P</math>.  The fraction field is the ring of germs of rational functions at <math>P</math>, and the valuation ring property means that if a rational function has a pole at <math>P</math>, then its reciprocal does not. The valuation of a polynomial <math>g</math> can be computed as the length of the [[Artinian ring]] <math>\Complex[x, y]_{(x-P_x, y-P_y)} / (f, g)</math> (assuming that ''g'' is in a general position) and the valuation of a quotient <math>g / h</math> can be computed as the difference of valuations.-->
* For example, consider the inclusion <math>(\mathbb{C}[[X^2]],(X^2)) \hookrightarrow (\mathbb{C}[[X]],(X))</math>. These are all subrings in the field of bounded-below power series <math>\mathbb{C}((X))</math>.

== Dominance and integral closure ==
The [[unit (ring theory)|units]], or invertible elements, of a valuation ring are the elements ''x'' in ''D'' such that ''x''<sup>&nbsp;−1</sup> is also a member of ''D''. The other elements of ''D'' – called nonunits – do not have an inverse in ''D'', and they form an [[ideal (ring theory)|ideal]] ''M''. This ideal is maximal among the (totally ordered) ideals of D. Since ''M'' is a [[maximal ideal]], the [[quotient ring]] ''D''/''M'' is a field, called the '''residue field''' of ''D''.

In general, we say a [[local ring]] <math>(S,\mathfrak{m}_S)</math> dominates a local ring <math>(R,\mathfrak{m}_R)</math> if <math>S \supseteq R</math> and <math>\mathfrak{m}_S \cap R = \mathfrak{m}_R</math>; in other words, the inclusion <math>R \subseteq S</math> is a [[local ring homomorphism]]. Every local ring <math>(A, \mathfrak{p})</math> in a field ''K'' is dominated by some valuation ring of ''K''. Indeed, the set consisting of all subrings ''R'' of ''K'' containing ''A'' and <math>1 \not\in \mathfrak{p}R</math> is [[empty set|nonempty]] and is inductive; thus, has a maximal element <math>R</math> by [[Zorn's lemma]]. We claim ''R'' is a valuation ring. ''R'' is a local ring with maximal ideal containing <math>\mathfrak{p}R</math> by maximality. Again by maximality it is also integrally closed. Now, if <math>x \not\in R</math>, then, by maximality, <math>\mathfrak{p}R[x] = R[x]</math> and thus we can write:
:<math>1 = r_0 + r_1 x + \cdots + r_n x^n, \quad r_i \in \mathfrak{p}R</math>.

Since <math>1 - r_0</math> is a unit element, this implies that <math>x^{-1}</math> is integral over ''R''; thus is in ''R''. This proves ''R'' is a valuation ring. (''R'' dominates ''A'' since its maximal ideal contains <math>\mathfrak{p}</math> by construction.)

A local ring ''R'' in a field ''K'' is a valuation ring if and only if it is a maximal element of the set of all local rings contained in ''K'' partially ordered by dominance. This easily follows from the above.{{efn|Proof: if ''R'' is a maximal element, then it is dominated by a valuation ring; thus, it itself must be a valuation ring. Conversely, let ''R'' be a valuation ring and ''S'' a local ring that dominates ''R'' but not ''R''. There is ''x'' that is in ''S'' but not in ''R''. Then <math>x^{-1}</math> is in ''R'' and in fact in the maximal ideal of ''R''. But then <math>x^{-1} \in \mathfrak{m}_S</math>, which is absurd. Hence, there cannot be such ''S''.}}

Let ''A'' be a subring of a field ''K'' and <math>f: A \to k</math> a [[ring homomorphism]] into an [[algebraically closed field]] ''k''. Then ''f'' extends to a ring homomorphism <math>g: D \to k</math>, ''D'' some valuation ring of ''K'' containing ''A''. (Proof: Let <math> g: R \to k </math> be a maximal extension, which clearly exists by Zorn's lemma. By maximality, ''R'' is a local ring with maximal ideal containing the [[kernel (algebra)|kernel]] of ''f''. If ''S'' is a local ring dominating ''R'', then ''S'' is algebraic over ''R''; if not, <math>S</math> contains a polynomial ring <math>R[x]</math> to which ''g'' extends, a contradiction to maximality. It follows <math>S/\mathfrak{m}_S</math> is an algebraic field extension of <math>R/\mathfrak{m}_R</math>. Thus, <math>S \to S/\mathfrak{m}_S \hookrightarrow k</math> extends ''g''; hence, ''S'' = ''R''.)

If a subring ''R'' of a field ''K'' contains a valuation ring ''D'' of ''K'', then, by checking Definition 1, ''R'' is also a valuation ring of ''K''. In particular, ''R'' is local and its maximal ideal contracts to some prime ideal of ''D'', say, <math>\mathfrak{p}</math>. Then <math>R = D_\mathfrak{p}</math> since <math>R</math> dominates <math>D_\mathfrak{p}</math>, which is a valuation ring since the ideals are totally ordered. This observation is subsumed to the following:{{sfn|Zariski|Samuel|1975|loc=Ch. VI, Theorem 3}} there is a [[bijection|bijective]] correspondence <math>\mathfrak{p} \mapsto D_\mathfrak{p}, \operatorname{Spec}(D) \to</math> the set of all subrings of ''K'' containing ''D''. In particular, ''D'' is integrally closed,{{sfn|Efrat|2006|p=38}}{{efn|To see more directly that valuation rings are integrally closed, suppose that ''x''<sup>''n''</sup>&nbsp;+&nbsp;''a''<sub>1</sub>''x''<sup>''n''−1</sup>&nbsp;+ ... +&nbsp;''a''<sub>0</sub>&nbsp;{{=}}&nbsp;0.  Then dividing by
''x''<sup>''n''−1</sup> gives us ''x'' {{=}} −''a''<sub>1</sub>&nbsp;−&nbsp;...&nbsp;−&nbsp;{{nowrap|''a''<sub>0</sub>''x''<sup>−''n''+1</sup>}}. If ''x'' were not in ''D'', then ''x''<sup>−1</sup> would be in ''D'' and this would express ''x'' as a finite sum of elements in ''D'', so that ''x'' would be in ''D'', a contradiction.}} and the [[Krull dimension]] of ''D'' is the number of proper subrings of ''K'' containing ''D''.

In fact, the [[integral closure]] of an integral domain ''A'' in the field of fractions ''K'' of ''A'' is the [[intersection (set theory)|intersection]] of all valuation rings of ''K'' containing ''A''.{{sfn|Matsumura|1989|loc=Theorem 10.4}} Indeed, the integral closure is contained in the intersection since the valuation rings are integrally closed. Conversely, let ''x'' be in ''K'' but not integral over ''A''. Since the ideal <math>x^{-1} A[x^{-1}]</math> is not <math>A[x^{-1}]</math>,{{efn|In general, <math>x^{-1}</math> is integral over ''A'' if and only if <math>xA[x] = A[x].</math>}} it is contained in a maximal ideal <math>\mathfrak{p}</math>. Then there is a valuation ring ''R'' that dominates the localization of <math>A[x^{-1}]</math> at <math>\mathfrak{p}</math>. Since <math>x^{-1} \in \mathfrak{m}_R</math>, <math>x \not\in R</math>.

The dominance is used in [[algebraic geometry]]. Let ''X'' be an algebraic variety over a field ''k''. Then we say a valuation ring ''R'' in <math>k(X)</math> has "center ''x'' on ''X''" if <math>R</math> dominates the local ring <math>\mathcal{O}_{x, X}</math> of the structure sheaf at ''x''.{{sfn|Hartshorne|1977|loc=Ch II. Exercise 4.5}}

== 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>

==Places==
{{hatnote|This section is based on {{harvnb|Zariski|Samuel|1975}}.}}

=== General definition ===
A ''[[place (mathematics)|place]]'' of a field ''K'' is a ring homomorphism ''p'' from a valuation ring ''D'' of ''K'' to some field such that, for any <math>x \not\in D</math>, <math>p(1/x) = 0</math>. The image of a place is a field called the '''residue field''' of ''p''. For example, the canonical map <math>D \to D/\mathfrak{m}_D</math> is a place.

==== Example ====
Let ''A'' be a [[Dedekind domain]] and <math>\mathfrak{p}</math> a prime ideal. Then the canonical map <math>A_{\mathfrak{p}} \to k(\mathfrak{p})</math> is a place.

=== Specialization of places ===
We say a '''place ''p'' ''specializes to'' a place ''{{prime|p}}'',''' denoted by <math>p \rightsquigarrow p'</math>, if the valuation ring of ''p'' contains the valuation ring of ''p{{'}}''. In algebraic geometry, we say a prime ideal <math>\mathfrak{p}</math> specializes to <math>\mathfrak{p}'</math> if <math>\mathfrak{p} \subseteq \mathfrak{p}'</math>. The two notions coincide: <math>p \rightsquigarrow p'</math> if and only if a prime ideal corresponding to ''p'' specializes to a prime ideal corresponding to ''{{prime|p}}'' in some valuation ring (recall that if <math>D \supseteq D'</math> are valuation rings of the same field, then ''D'' corresponds to a prime ideal of <math>D'</math>.)

==== Example ====
For example, in the function field <math>\mathbb{F}(X)</math> of some algebraic variety <math>X</math> every prime ideal <math>\mathfrak{p} \in \text{Spec}(R)</math> contained in a maximal ideal <math>\mathfrak{m}</math> gives a specialization <math>\mathfrak{p} \rightsquigarrow \mathfrak{m}</math>.

==== Remarks ====
It can be shown: if <math>p \rightsquigarrow p'</math>, then <math>p' = q \circ p|_{D'}</math> for some place ''q'' of the residue field <math>k(p)</math> of ''p''. (Observe <math>p(D')</math> is a valuation ring of <math>k(p)</math> and let ''q'' be the corresponding place; the rest is mechanical.) If ''D'' is a valuation ring of ''p'', then its Krull dimension is the cardinarity of the specializations other than ''p'' to ''p''. Thus, for any place ''p'' with valuation ring ''D'' of a field ''K'' over a field ''k'', we have:

:<math> \operatorname{tr.deg}_k k(p) + \dim D \le \operatorname{tr.deg}_k K</math>.

If ''p'' is a place and ''A'' is a subring of the valuation ring of ''p'', then <math>\operatorname{ker}(p) \cap A</math> is called the ''center'' of ''p'' in ''A''.

=== Places at infinity ===
For the function field on an affine variety <math>X</math> there are valuations which are not associated to any of the primes of <math>X</math>. These valuations are called '''the places at infinity'''.[https://math.stackexchange.com/questions/3330320/place-at-infinity-in-function-fields] For example, the affine line <math>\mathbb{A}^1_k</math> has function field <math>k(x)</math>. The place associated to the localization of
:<math>k\left[\frac{1}{x}\right]</math>
at the maximal ideal
:<math>\mathfrak{m} = \left(\frac{1}{x}\right)</math>
is a place at infinity.

== Notes ==
{{notelist}}

===Citations===
{{Reflist|30em}}

== Sources ==
{{refbegin|35em}}
*{{Cite book| title = Commutative Algebra | edition = First
 | last = Bourbaki | first = Nicolas | year = 1972
 | author-link = Nicolas Bourbaki
 | publisher = Addison-Wesley
 | series = Elements of Mathematics
 | isbn = 978-020100644-5
}}
*{{citation| title = Bezout rings and their subrings
 | last = Cohn | first = P. M. | authorlink = Paul Cohn|year = 1968
 | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]
 | volume = 64 | issue = 2 | pages = 251–264
 | url = http://www.lohar.com/researchpdf/bezout_rings_and_their_subrings.pdf
 | doi = 10.1017/s0305004100042791 | bibcode = 1968PCPS...64..251C | issn = 0008-1981 | mr = 0222065 | zbl = 0157.08401
| s2cid = 123667384 }}
*{{citation| title = Valuations, orderings, and Milnor ''K''-theory
 | last = Efrat | first = Ido | year = 2006
 | publisher = [[American Mathematical Society]] | location = Providence, RI
 | volume = 124 | series = Mathematical Surveys and Monographs
 | isbn = 0-8218-4041-X | zbl = 1103.12002
}}
*{{Citation| title = Modules over non-Noetherian domains
 | last1 = Fuchs | first1 = László
 | last2 = Salce | first2 = Luigi
 | year = 2001
 | publisher = [[American Mathematical Society]] | location = Providence, R.I.
 | volume = 84 | series = Mathematical Surveys and Monographs
 | isbn = 978-0-8218-1963-0 | mr = 1794715 | zbl = 0973.13001
}}
*{{Citation| title = [[Algebraic Geometry (book)|Algebraic Geometry]]
 | last = Hartshorne | first = Robin | year = 1977
 | author-link = Robin Hartshorne
 | publisher = Springer-Verlag | location = New York
 | volume = 52 | series = [[Graduate Texts in Mathematics]]
 | isbn = 978-0-387-90244-9 | mr = 0463157
}}
*{{Citation| title = Beiträge zur Arithmetik kommutativer Integritätsbereiche. VI. Der allgemeine Diskriminantensatz. Unverzweigte Ringerweiterungen
 | last = Krull | first = Wolfgang | year = 1939
 | author-link = Wolfgang Krull
 | journal = [[Mathematische Zeitschrift]]
 | volume = 45 | issue = 1 | pages = 1–19
 | doi = 10.1007/BF01580269 | issn = 0025-5874 | mr = 1545800 | zbl = 0020.34003
| s2cid = 121374449 }}
*{{citation| title = Commutative ring theory | edition = Second
 | last = Matsumura | first = Hideyuki | authorlink = Hideyuki Matsumura|year = 1989
 | others = Translated from the Japanese by [[Miles Reid]]
 | volume = 8 | series = Cambridge Studies in Advanced Mathematics
 | url = https://books.google.com/books?id=yJwNrABugDEC&q=%22valuation+ring%22
 | isbn = 0-521-36764-6 | zbl = 0666.13002
}}
*{{Citation| title = Commutative algebra. Vol. II
 | last1 = Zariski | first1 = Oscar
 | last2 = Samuel | first2 = Pierre
 | author1-link = Oscar Zariski
 | author2-link = Pierre Samuel
 | year = 1975
 | publisher = [[Springer-Verlag]] | location = Berlin, New York
 | isbn = 978-0-387-90171-8 | mr = 0389876
}}
{{refend}}

[[Category:Commutative algebra]]
[[Category:Field (mathematics)]]
[[Category:Localization (mathematics)]]
[[Category:Ring theory]]