Localization is an operation on rings and modules. Given a commutative unit
ring 
and a multiplicatively closed subset 
of 
with 
, the localization of 
at 
is the ring
 |
(1)
|
whose fractions represent equivalence classes of pairs 
.
Two fractions 
and 
are equal iff 
for
some 
.
The addition and multiplication
of fractions are defined by
 |
(2)
|
and
 |
(3)
|
The canonical ring homomorphism 
, 
, need not be an injection.
iff 
for some 
. Hence 
can be identified with a subring
of 
by this map iff multiplication by each
is injective
on 
(Stacks Project Authors 2026a).
The condition 
is sometimes imposed to exclude the trivial ring.
The construction also allows 
, in which case all fractions are equal and 
is the trivial ring (Stacks
Project Authors 2026a).
For an 
-module 
, the localization of 
at 
is the module tensor product 
. Its elements
can be represented by the elementary tensors
 |
(4)
|
where 
and 
,
also denoted 
(Stacks Project Authors 2026b).
The required properties of the subset 
hold in the following examples.
1. For a ring 
other than the trivial ring,
let 
be the set of nonzero elements
of 
that are not zero
divisors. In this case, 
is the ring of fractions
of 
.
2. Let 
be the complement 
of a prime ideal 
of 
. In this case, 
is denoted 
. It is the localization of 
at 
, a local ring with maximal
ideal 
.
3. For 
,
let 
.
The resulting localizations are denoted 
and 
. If 
is a nilpotent element,
then 
,
so 
is the trivial
ring and 
is the zero module (Stacks Project Authors 2026a).
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/Inline56.svg)
. The localization of 
at the maximal ideal generated
by the images 
of 
and 
of 
, denoted 
, describes 
at the origin. Its algebraic properties
provide clues to local geometric properties. For example, the localized ring
is not a regular local ring, 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 for 
.
For all other points 
of the 
-axis, with 
, the localized ring 
is a regular
local ring of Krull dimension 1, since 
generates the whole maximal
ideal 
.
Indeed, 
implies 
.
The same argument applies to the 
-axis. It follows that outside the origin,
the algebraic variety 
has no singular points.
