Let R be a domain. On the set of pairs (t,n) from R with n 0, we define an equivalence relation eaq (equal as quotient):

(t,n) eaq (t',n')   if and only if   tn' = t'n.

We call t the numerator and n the denominator of the pair (t,n). Denote the equivalence class containing (t,n) by t/n, and the set of equivalence classes by Q(R). Addition and multiplication on these classes are defined as follows:

• addition: n/t + m/s = (ns + tm)/ts;
• zero element: 0/1;
• multiplication: (n/t) * (m/s) = nm/ts;
• unit element: 1/1.

It is readily checked that these operations are well defined and that Q(R) is a ring. Even more is true:

The field Q(R) is called the field of fractions of R.

The map

R -> Q(R),     x -> x/1

is an injective morphism of rings. Thus, R may be viewed as a subring of Q(R).