The ring of fractions of a commutative ring with multiplicative identity, other than the trivial ring, is the extension ring obtained by allowing division by all nonzero elements that are not zero divisors. The ring of fractions of an integral domain is always a field.
The term "ring of fractions" is sometimes used to denote any localization of a ring. The ring of fractions in the above meaning is then referred to as the total ring of fractions, and coincides with the localization with respect to the set of all nonzero elements that are not zero divisors.
For a commutative ring 
with multiplicative
identity, let 
contain 1 and be multiplicatively closed,
so that 
implies 
.
Addition and multiplication
of fractions are defined by
 |  |  |
(1)
|
 |  |  |
(2)
|
The elements of the localization 
are equivalence
classes of fractions 
with 
and 
. Two fractions 
and 
are equivalent iff there exists 
such that 
. This definition remains valid when 
has zero divisors. The condition
excludes the trivial
ring, but is not necessary for the localization
to be defined.
For a general localization, the natural ring homomorphism 
,
, need not be injective.
Its ring kernel consists of the elements 
for which 
for some 
(Stacks Project, Tag 00CM). For example, in 
, localizing at the powers of 2 sends the nonzero class
of 3 to zero. In contrast, the map from 
to its total ring of
fractions is always injective, because its denominators
are nonzero and are not zero
divisors.
When the complement of 
is an ideal 
, it must be a prime ideal because
is multiplicatively
closed. In this case, the ring of fractions is the localization
at 
.
When the ring is an integral domain, then the nonzero elements
are multiplicatively closed. Letting 
be the nonzero elements, then the ring of fractions is a field
called the field of fractions, or the total
ring of fractions. In this case one can also use the usual rule for division
of fractions, which is not normally available for more general 
.
