Editing Local ring

{{Short description|(Mathematical) ring with a unique maximal ideal}}
In [[mathematics]], more specifically in [[ring theory]], '''local rings''' are certain [[ring (mathematics)|rings]] that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on [[algebraic varieties]] or [[manifold]]s, or of [[algebraic number fields]] examined at a particular [[place (mathematics)|place]], or prime. '''Local algebra''' is the branch of [[commutative algebra]] that studies [[commutative ring|commutative]] local rings and their [[module (mathematics)|modules]].

In practice, a commutative local ring often arises as the result of the [[localization of a ring]] at a [[prime ideal]].

The concept of local rings was introduced by [[Wolfgang Krull]] in 1938 under the name ''Stellenringe''.<ref name="Krull">
{{cite journal
  | last = Krull
  | first = Wolfgang
  | author-link = Wolfgang Krull
  | title = Dimensionstheorie in Stellenringen
  | journal = J. Reine Angew. Math.
  | volume = 1938
  | issue = 179
 | page = 204
  | year = 1938
  | language = de
  | doi = 10.1515/crll.1938.179.204
| s2cid = 115691729
 }}</ref>  The English term ''local ring'' is due to [[Zariski]].<ref name = "Zariski">
{{cite journal
 | last = Zariski
 | first = Oscar
 | author-link = Oscar Zariski
 |date=May 1943
 | title = Foundations of a General Theory of Birational Correspondences
 | journal = Trans. Amer. Math. Soc.
 | volume = 53
 | issue = 3
 | doi = 10.2307/1990215
 | jstor = 1990215
 | publisher = American Mathematical Society
 | pages = 490–542 [497]
| url = http://www.ams.org/tran/1943-053-03/S0002-9947-1943-0008468-9/S0002-9947-1943-0008468-9.pdf
 | doi-access = free
 }}</ref>

== Definition and first consequences ==

A [[ring (mathematics)|ring]] ''R'' is a '''local ring''' if it has any one of the following equivalent properties:
* ''R'' has a unique [[maximal ideal|maximal]] left [[ring ideal|ideal]].
* ''R'' has a unique maximal right ideal.
* 1 ≠ 0 and the sum of any two non-[[unit (algebra)|unit]]s in ''R'' is a non-unit.
* 1 ≠ 0 and if ''x'' is any element of ''R'', then ''x'' or {{nowrap|1 &minus; ''x''}} is a unit.
* If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0).

If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's [[Jacobson radical]]. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal,<ref>Lam (2001), p. 295, Thm. 19.1.</ref> necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring ''R'' is local if and only if there do not exist two [[coprime]] proper ([[principal ideal|principal]]) (left) ideals, where two ideals ''I''<sub>1</sub>, ''I''<sub>2</sub> are called ''coprime'' if {{nowrap|1=''R'' = ''I''<sub>1</sub> + ''I''<sub>2</sub>}}.

<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Local Ring |url=https://mathworld.wolfram.com/LocalRing.html |access-date=2024-08-26 |website=mathworld.wolfram.com |language=en}}</ref> In the case of [[commutative ring]]s, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal.
Before about 1960 many authors required that a local ring be (left and right) [[Noetherian ring|Noetherian]], and (possibly non-Noetherian) local rings were called '''quasi-local rings'''. In this article this requirement is not imposed.

A local ring that is an [[integral domain]] is called a '''local domain'''.

== Examples ==
*All [[field (mathematics)|field]]s (and [[skew field]]s) are local rings, since {0} is the only maximal ideal in these rings.
*The ring <math>\mathbb{Z}/p^n\mathbb{Z}</math> is a local ring ({{mvar|p}} prime, {{math|''n'' ≥ 1}}). The unique maximal ideal consists of all multiples of {{mvar|p}}.
*More generally, a nonzero ring in which every element is either a unit or [[nilpotent]] is a local ring.
*An important class of local rings are [[discrete valuation ring]]s, which are local [[principal ideal domain]]s that are not fields.
*The ring <math>\mathbb{C}[[x]]</math>, whose elements are infinite series <math display="inline">\sum_{i=0}^\infty a_ix^i </math> where multiplications are given by <math display="inline">(\sum_{i=0}^\infty a_ix^i)(\sum_{i=0}^\infty b_ix^i)=\sum_{i=0}^\infty c_ix^i</math> such that <math display="inline">c_n=\sum_{i+j=n}a_ib_j</math>, is local. Its unique maximal ideal consists of all elements that are not invertible. In other words, it consists of all elements with constant term zero. 
*More generally, every ring of [[formal power series]] over a local ring is local; the maximal ideal consists of those power series with [[constant term]] in the maximal ideal of the base ring.
*Similarly, the [[algebra over a field|algebra]] of [[dual numbers]] over any field is local. More generally, if ''F'' is a local ring and ''n'' is a positive integer, then the [[quotient ring]] ''F''[''X'']/(''X''<sup>''n''</sup>) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of ''F'', since one can use a [[geometric series]] to invert all other polynomials [[Ideal (ring theory)|modulo]] ''X''<sup>''n''</sup>. If ''F'' is a field, then elements of ''F''[''X'']/(''X''<sup>''n''</sup>) are either [[nilpotent]] or [[invertible]]. (The dual numbers over ''F'' correspond to the case {{nowrap|1=''n'' = 2}}.)
*Nonzero quotient rings of local rings are local.
*The ring of [[rational number]]s with [[odd number|odd]] denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. It is the integers [[localization of a ring|localized]] at 2. 
*More generally, given any [[commutative ring]] ''R'' and any [[prime ideal]] ''P'' of ''R'', the [[localization of a ring|localization]] of ''R'' at ''P'' is local; the maximal ideal is the ideal generated by ''P'' in this localization; that is, the maximal ideal consists of all elements ''a''/''s'' with ''a'' ∈ ''P'' and ''s'' ∈ ''R'' - ''P''.

=== Non-examples ===
{{Expand section|date=January 2022}}
*The [[Polynomial ring|ring of polynomials]] <math>K[x]</math> over a field <math>K</math> is not local, since <math>x</math> and <math>1 - x</math> are non-units, but their sum is a unit.
*The ring of integers <math>\Z</math> is not local since it has a maximal ideal <math>(p)</math> for every prime <math>p</math>.
*<math>\Z</math>/(''pq'')<math>\Z</math>, where ''p'' and ''q'' are distinct prime numbers. Both (''p'') and (''q'') are maximal ideals here.

=== Ring of germs ===

{{main|Germ (mathematics)}}

To motivate the name "local" for these rings, we consider real-valued [[continuous function]]s defined on some [[interval (mathematics)|open interval]] around 0 of the [[real line]]. We are only interested in the behavior of these functions near 0 (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an [[equivalence relation]], and the [[equivalence class]]es are what are called the "[[germ (mathematics)|germs]] of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring.

To see that this ring of germs is local, we need to characterize its invertible elements. A germ ''f'' is invertible if and only if {{nowrap|''f''(0) ≠ 0}}. The reason: if {{nowrap|''f''(0) ≠ 0}}, then by continuity there is an open interval around 0 where ''f'' is non-zero, and we can form the function {{nowrap|1=''g''(''x'') = 1/''f''(''x'')}} on this interval. The function ''g'' gives rise to a germ, and the product of ''fg'' is equal to 1. (Conversely, if ''f'' is invertible, then there is some ''g'' such that ''f''(0)''g''(0) = 1, hence {{nowrap|''f''(0) ≠ 0}}.)

With this characterization, it is clear that the sum of any two non-invertible germs is again non-invertible, and we have a commutative local ring. The maximal ideal of this ring consists precisely of those germs ''f'' with {{nowrap|1=''f''(0) = 0}}.

Exactly the same arguments work for the ring of germs of continuous real-valued functions on any [[topological space]] at a given point, or the ring of germs of [[differentiable]] functions on any [[differentiable manifold]] at a given point, or the ring of germs of [[rational function]]s on any [[algebraic variety]] at a given point. All these rings are therefore local. These examples help to explain why [[scheme (mathematics)|scheme]]s, the generalizations of varieties, are defined as special [[locally ringed space]]s.

=== Valuation theory ===
{{main|Valuation (algebra)}}

Local rings play a major role in valuation theory. By definition, a [[valuation ring]] of a field ''K'' is a subring ''R'' such that for every non-zero element ''x'' of ''K'', at least one of ''x'' and ''x''<sup>&minus;1</sup> is in ''R''. Any such subring will be a local ring. For example, the ring of [[rational number]]s with [[odd number|odd]] denominator (mentioned above) is a valuation ring in  <math>\mathbb{Q}</math>.

Given a field ''K'', which may or may not be a  [[Function field of an algebraic variety|function field]], we may look for local rings in it. If ''K'' were indeed the function field of an [[algebraic variety]] ''V'', then for each point ''P'' of ''V'' we could try to define a valuation ring ''R'' of functions "defined at" ''P''. In cases where ''V'' has dimension 2 or more there is a difficulty that is seen this way: if ''F'' and ''G'' are rational functions on ''V'' with

:''F''(''P'') = ''G''(''P'') = 0,

the function

:''F''/''G''

is an [[indeterminate form]] at ''P''. Considering a simple example, such as

:''Y''/''X'',

approached along a line

:''Y'' = ''tX'',

one sees that the ''value at'' ''P'' is a concept without a simple definition. It is replaced by using valuations.

=== Non-commutative ===

Non-commutative local rings arise naturally as [[endomorphism ring]]s in the study of [[Direct sum of modules|direct sum]] decompositions of [[module (mathematics)|modules]] over some other rings. Specifically, if the endomorphism ring of the module ''M'' is local, then ''M'' is [[indecomposable module|indecomposable]]; conversely, if the module ''M'' has finite [[length of a module|length]] and is indecomposable, then its endomorphism ring is local.

If ''k'' is a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] {{nowrap|''p'' > 0}} and ''G'' is a finite [[p-group|''p''-group]], then the [[group ring|group algebra]] ''kG'' is local.

== Some facts and definitions ==

=== Commutative case===

We also write {{nowrap|(''R'', ''m'')}} for a commutative local ring ''R'' with maximal ideal ''m''. Every such ring becomes a [[topological ring]] in a natural way if one takes the powers of ''m'' as a [[neighborhood base]] of 0. This is the [[I-adic topology|''m''-adic topology]] on ''R''. If {{nowrap|(''R'', ''m'')}} is a commutative [[Noetherian ring|Noetherian]] local ring, then

:<math>\bigcap_{i=1}^\infty m^i = \{0\}</math>
('''Krull's intersection theorem'''), and it follows that ''R'' with the ''m''-adic topology is a [[Hausdorff space]]. The theorem is a consequence of the [[Artin–Rees lemma]] together with [[Nakayama's lemma]], and, as such, the "Noetherian" assumption is crucial. Indeed, let ''R'' be the ring of germs of infinitely differentiable functions at 0 in the real line and ''m'' be the maximal ideal <math>(x)</math>. Then a nonzero function <math>e^{-{1 \over x^2}}</math> belongs to <math>m^n</math> for any ''n'', since that function divided by <math>x^n</math> is still smooth.

As for any topological ring, one can ask whether {{nowrap|(''R'', ''m'')}} is [[Complete uniform space|complete]] (as a [[uniform space]]); if it is not, one considers its [[Completion (ring theory)|completion]], again a local ring. Complete Noetherian local rings are classified by the [[Cohen structure theorem]].

In [[algebraic geometry]], especially when ''R'' is the local ring of a scheme at some point ''P'', {{nowrap|''R'' / ''m''}} is called the ''[[residue field]]'' of the local ring or residue field of the point ''P''.

If {{nowrap|(''R'', ''m'')}} and {{nowrap|(''S'', ''n'')}} are local rings, then a '''local ring homomorphism''' from ''R'' to ''S'' is a [[ring homomorphism]] {{nowrap|''f'' : ''R'' → ''S''}} with the property {{nowrap|''f''(''m'') ⊆ ''n''}}.<ref>{{Cite web|url=http://stacks.math.columbia.edu/tag/07BI|title=Tag 07BI}}</ref> These are precisely the ring homomorphisms that are continuous with respect to the given topologies on ''R'' and ''S''. For example, consider the ring morphism <math>\mathbb{C}[x]/(x^3) \to \mathbb{C}[x,y]/(x^3,x^2y,y^4)</math> sending <math>x \mapsto x</math>. The preimage of <math>(x,y)</math> is <math>(x)</math>. Another example of a local ring morphism is given by <math>\mathbb{C}[x]/(x^3) \to \mathbb{C}[x]/(x^2)</math>.

=== General case===

The [[Jacobson radical]] ''m'' of a local ring ''R'' (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of ''R''. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.<ref>The 2 by 2 matrices over a field, for example, has unique maximal ideal {0}, but it has multiple maximal right and left ideals.</ref>

For an element ''x'' of the local ring ''R'', the following are equivalent:
* ''x'' has a left inverse
* ''x'' has a right inverse
* ''x'' is invertible
* ''x'' is not in ''m''.

If {{nowrap|(''R'', ''m'')}} is local, then the [[factor ring]] ''R''/''m'' is a [[skew field]]. If {{nowrap|''J'' ≠ ''R''}} is any two-sided ideal in ''R'', then the factor ring ''R''/''J'' is again local, with maximal ideal ''m''/''J''.

A [[Kaplansky's theorem on projective modules|deep theorem]] by [[Irving Kaplansky]] says that any [[projective module]] over a local ring is [[free module|free]], though the case where the module is finitely-generated is a simple corollary to [[Nakayama's lemma]]. This has an interesting consequence in terms of [[Morita equivalence]]. Namely, if ''P'' is a [[finitely generated module|finitely generated]] projective ''R'' module, then ''P'' is isomorphic to the free module ''R''<sup>''n''</sup>, and hence the ring of endomorphisms <math>\mathrm{End}_R(P)</math> is isomorphic to the full ring of matrices <math>\mathrm{M}_n(R)</math>. Since every ring Morita equivalent to the local ring ''R'' is of the form <math>\mathrm{End}_R(P)</math> for such a ''P'', the conclusion is that the only rings Morita equivalent to a local ring ''R'' are (isomorphic to) the matrix rings over ''R''.

==Notes==
<references/>

== References ==
* {{Cite book| last=Lam| first=T.Y.| author-link=T.Y. Lam| year=2001| title= A first course in noncommutative rings| edition=2nd| series= Graduate Texts in Mathematics| publisher=Springer-Verlag| isbn = 0-387-95183-0}}
* {{Cite book| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| volume = 2 | publisher=Dover| isbn = 978-0-486-47187-7}}

==See also==

* [[Discrete valuation ring]]
* [[Semi-local ring]]
* [[Gorenstein local ring]]
* [[Regular local ring]]

== External links ==
*[https://mathoverflow.net/q/255511 The philosophy behind local rings]

[[Category:Ring theory]]
[[Category:Localization (mathematics)]]