The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-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 non-zero divisors.
When defining addition and multiplication of fractions, all that is required of the denominators is that they be multiplicatively
closed, i.e., if 
,
then 
,
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(1)
|
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(2)
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Given a multiplicatively closed set 
in a ring 
, the ring of fractions is all elements
of the form 
with 
and 
.
Of course, it is required that 
and that fractions of the form 
and 
be considered equivalent. With the above definitions of
addition and multiplication, this set forms a ring.
The original ring may not embed in this ring of fractions 
if it is not an integral
domain. For instance, if 
for some 
, then 
in the ring of fractions.
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 
.
