Editing Local ring (section)

== 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.