An operation on rings and modules. Given a commutative unit ring 
, and a subset 
of 
,
closed under multiplication, such that 
, and 
, the localization of 
at 
is the ring
 |
(1)
|
where the addition and the multiplication of the formal fractions 
are defined according to the natural rules,
 |
(2)
|
and
 |
(3)
|
The ring 
is a subring of 
via the identification 
.
For an 
-module
, the localization of 
at 
is defined as the tensor product 
, i.e., as the set of linear
combinations of the elementary tensors
 |
(4)
|
which are also denoted 
for short.
The properties required for the subset 
are fulfilled by
1. The set of non zero-divisors of 
; in this case 
is the ring of fractions of 
.
2. The complement 
of any prime ideal 
of 
: in this case the clumsy notation 
is replaced by 
. This ring is called the localization of 
at 
, and it is a local ring, with maximal ideal 
.
The name given to this operation derives from the geometric meaning it takes when applied to the rings associated with algebraic varieties.
The union 
of the coordinate axes of the real Cartesian plane is an algebraic variety given
by the equation
 |
(5)
|
and is associated with the quotient ring ![R=R[X,Y]/<XY>](/images/equations/Localization/Inline31.svg)
. The localization of 
at the maximal ideal generated by the residues 
of 
and 
of 
, denoted 
, describes 
at the origin; its algebraic properties provide clues to local
geometric properties. For example, the localized ring is nonregular since its Krull dimension is 1, whereas two elements are needed
to generate its maximal ideal 
. This is the algebraic counterpart
of the fact that the origin is a singular point (a knot) for 
. For all other points 
of the 
-axis, 
, the localized ring 
is regular of dimension 1, since 
generates the whole maximal ideal
:
being 
,
one has that 
.
The same argument applies to the 
-axis. It follows that outside the origin variety 
is regular in the geometric meaning of "smooth"
or "nonsingular."
