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A [[Commutative ring|commutative ring]] with a unit that has a unique maximal ideal. If  $  A $
is a local ring with maximal ideal  $  \mathfrak m $, 
then the quotient ring  $  A / \mathfrak m $
is a field, called the residue field of  $  A $.

Examples of local rings. Any field or valuation ring is local. The ring of formal power series  $  k [ [ X _ {1} \dots X _ {n} ] ] $
over a field  $  k $
or over any local ring is local. On the other hand, the polynomial ring  $  k [ X _ {1} \dots X _ {n} ] $
with  $  n \geq  1 $
is not local. Let  $  X $
be a topological space (or a differentiable manifold, an analytic space or an algebraic variety) and let  $  x $
be a point of  $  X $. 
Let  $  A $
be the ring of germs at  $  x $
of continuous functions (respectively, differentiable, analytic or regular functions); then  $  A $
is a local ring whose maximal ideal consists of the germs of functions that vanish at  $  x $.

Some general ring-theoretical constructions lead to local rings, the most important of which is localization (cf. [[Localization in a commutative algebra|Localization in a commutative algebra]]). Let  $  A $
be a commutative ring and let  $  \mathfrak p $
be a prime ideal of  $  A $. 
The ring  $  A _ {\mathfrak p }  $, 
which consists of fractions of the form  $  a / s $, 
where  $  a \in A $, 
$  s \in A \setminus  \mathfrak p $, 
is local and is called the localization of the ring  $  A $
at  $  \mathfrak p $. 
The maximal ideal of  $  A _ {\mathfrak p }  $
is  $  \mathfrak p A _ {\mathfrak p }  $, 
and the residue field of  $  A _ {\mathfrak p }  $
is identified with the field of fractions of the integral quotient ring  $  A / \mathfrak p $. 
Other constructions that lead to local rings are Henselization (cf. [[Hensel ring|Hensel ring]]) or [[Completion|completion]] of a ring with respect to a maximal ideal. Any quotient ring of a local ring is also local.

A property of a ring  $  A $(
or an  $  A $-
module  $  M $, 
or an  $  A $-
algebra  $  B $) 
is called a local property if its validity for  $  A $(
or  $  M $, 
or  $  B $) 
is equivalent to its validity for the rings  $  A _ {\mathfrak p }  $(
respectively, modules  $  M \otimes _ {A} A _ {\mathfrak p }  $
or algebras  $  B \otimes _ {A} A _ {\mathfrak p }  $) 
for all prime ideals  $  \mathfrak p $
of  $  A $(
see [[Local property|Local property]]).

The powers  $  \mathfrak m  ^ {n} $
of the maximal ideal  $  \mathfrak m $
of a local ring  $  A $
determine a basis of neighbourhoods of zero of the so-called local-ring topology (or  $  \mathfrak m $-
adic topology). For a Noetherian local ring this topology is separated (Krull's theorem), and any ideal of it is closed.

From now on only Noetherian local rings are considered (cf. also [[Noetherian ring|Noetherian ring]]). A local ring is called a complete local ring if it is complete with respect to the  $  \mathfrak m $-
adic topology; in this case  $  A = \lim\limits _ {\leftarrow n }   A / \mathfrak m  ^ {n} $. 
In a complete local ring the  $  \mathfrak m $-
adic topology is weaker than any other separated topology (Chevalley's theorem). Any complete local ring can be represented as the quotient ring of the ring  $  S [ [ X _ {1} \dots X _ {n} ] ] $
of formal power series, where  $  S $
is a field (in the case of equal characteristic) or of a complete discrete valuation ring (in the case of different characteristic). This theorem makes it possible to prove that complete local rings have a number of specific properties that are absent in arbitrary Noetherian local rings (see [[#References|[5]]]); for example, a complete local ring is an [[Excellent ring|excellent ring]].

A finer quantitative investigation of a local ring  $  A $
is connected with the application of the concept of the adjoint graded ring  $   \mathop{\rm Gr} ( A) = \oplus _ {n \geq  0 }  ( \mathfrak m  ^ {n} / \mathfrak m  ^ {n+} 1 ) $. 
Let  $  H _ {A} ( n) $
be the dimension of the vector space  $  \mathfrak m  ^ {n} / \mathfrak m  ^ {n+} 1 $
over the residue field  $  A / \mathfrak m $; 
as a function of the integer argument  $  n $
it is called the Hilbert–Samuel function (or characteristic function) of the local ring  $  A $. 
For large  $  n $
this function coincides with a certain polynomial  $  \overline{H}\; _ {A} ( n) $
in  $  n $, 
which is called the Hilbert–Samuel polynomial of the local ring  $  A $(
see also [[Hilbert polynomial|Hilbert polynomial]]). This fact can be expressed in terms of a Poincaré series: The formal series

$$ 
P _ {A} ( t)  =  \sum _ {n \geq  0 } H _ {A} ( n) \cdot t  ^ {n}
$$

is a rational function of the form  $  f ( t) ( 1 - t )  ^ {-} d( A) $, 
where  $  f ( t) \in \mathbf Z [ t] $
is a polynomial and  $  d ( A) - 1 $
is the degree of  $  \overline{H}\; _ {A} $. 
The integer  $  d ( A) $
is the (Krull) dimension  $   \mathop{\rm dim}  A $
of the ring  $  A $
and is one of the most important invariants of a ring. Moreover,  $  d ( A) $
is equal to the least number of elements  $  a _ {1} \dots a _ {d} \in A $
for which the quotient ring  $  A / ( a _ {1} \dots a _ {d} ) $
is Artinian (cf. [[Artinian ring|Artinian ring]]). If these elements can be chosen in such a way that they generate the maximal ideal  $  \mathfrak m $, 
then  $  A $
is called a regular local ring. The regularity of  $  A $
is equivalent to the fact that  $   \mathop{\rm dim} ( \mathfrak m / \mathfrak m  ^ {2} ) =  \mathop{\rm dim}  A $. 
For a  $  d $-
dimensional regular ring  $  A $,

$$ 
H _ {A} ( n)  = \ 
\left ( \begin{array}{c}
n + d - 1 \\
 d - 1 
\end{array}
 \right )
$$

and  $  P _ {A} ( t) = ( 1 - t )  ^ {-} d $. 
Geometrically, regularity means that the corresponding point of the (analytic or algebraic) variety is non-singular.

Besides the characteristic function  $  H _ {A} $
and the dimension and multiplicity connected with it, a local ring has various invariants of a homological kind. The main one of these is the depth  $   \mathop{\rm depth}  A $(
see [[Depth of a module|Depth of a module]]); the condition  $   \mathop{\rm depth}  A =  \mathop{\rm dim}  A $
distinguishes among local rings the so-called Cohen–Macaulay rings (cf. [[Cohen–Macaulay ring|Cohen–Macaulay ring]]). It is not known (1989) whether there is a module  $  M $
with  $   \mathop{\rm depth}  M =  \mathop{\rm dim}  A $
for an arbitrary or a complete local ring  $  A $. 
Other homological invariants are the so-called Betti numbers  $  b _ {i} ( A) $
of a local ring  $  A $, 
that is, the dimensions of the  $  k $-
spaces  $   \mathop{\rm Tor} _ {i}  ^ {A} ( k , k ) $, 
where  $  k $
is the residue field of  $  A $. 
The question of the rationality of the Poincaré series  $  \sum _ {n \geq  0 }  b _ {n} ( A) t  ^ {n} $
is open, although for many classes of rings an affirmative answer is known. There are also invariants of an algebraic-geometrical nature; for their definition one uses resolution of the singularity corresponding to the local ring.

A similar theory has been constructed for semi-local rings; that is, rings that have finitely many maximal ideals. The role of a maximal ideal for them is played by the [[Jacobson radical|Jacobson radical]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Krull,   "Dimensionstheorie in Stellenringen"  ''J. Reine Angew. Math.'' , '''179'''  (1939)  pp. 204–226</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Chevalley,   "On the theory of local rings"  ''Ann. of Math. (2)'' , '''44'''  (1943)  pp. 690–708</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.S. Cohen,   "On the structure and ideal theory of complete local rings"  ''Trans. Amer. Math. Soc.'' , '''59'''  (1946)  pp. 54–106</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Samuel,   "Algèbre locale" , Gauthier-Villars  (1953)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Nagata,   "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  O. Zariski,   P. Samuel,   "Commutative algebra" , '''2''' , Springer  (1975)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.-P. Serre,   "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer  (1965)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M.F. Atiyah,   I.G. Macdonald,   "Introduction to commutative algebra" , Addison-Wesley  (1969)</TD></TR></table>

====Comments====
For the notion of Krull dimension see [[Dimension|Dimension]] of an associative ring.

A counter-example to the question of the rationality of the Poincaré series was given by D. Anick [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Anick,   "Construction d'espaces de lacets et d'anneaux locaux à séries de Poincaré–Betti non rationelles"  ''C.R. Acad. Soc. Paris'' , '''290'''  (1980)  pp. 1729–1732  (English abstract)</TD></TR></table>