Field of fractions

Given an integral domain, $R$ , we may informally define the field of fractions of $R$ (also called the fraction field or the quotient field), denoted by $\text{Frac}(R)$ , as the set $\left\{\frac{a}{b} \mid a,b\in R, b\neq 0\right\}$ . This is analogous to the construction of the rational numbers $\mathbb{Q}$ from the integers, $\mathbb{Z}$ , and can be viewed as a way turning $R$ into a field.

Formal Definition

While the above definition makes sense intuitively, it is not entirely satisfactory. In general, division in $R$ is undefined, so an expression like $\frac{a}{b}$ is meaningless.

To get around this, we consider the set of ordered pairs $S=R\times (R-\{0\}) = \{(a,b) | a,b\in R, b\neq 0\}$ and define an equivalence relation, $\sim$ , on $S$ by $(a,b)\sim (c,d)$ if $ad = bc$ . Then we can define $\text{Frac}(R)$ as the set of equivalence classes of $S$ under $\sim$ .

Intuitively we can think of each ordered pair $(a,b)$ as representing the fraction $\frac{a}{b}$ , and our definition of $\sim$ is equivalent to the statement $\frac{a}{b}=\frac{c}{d}\Leftrightarrow ad=bc.$

We can now define addition and multiplication on $\text{Frac}(R)$ in the 'obvious way':

$(a,b) + (c,d) = (ad+bc,ad)$
$(a,b)(c,d) = (ac,bd)$

Notice that we have actually defined these operations on $S$ , not on $\text{Frac}(R)$ . However it is now easy to verify that if $(a_1,a_2)\sim (b_1,b_2)$ and $(c_1,c_2)\sim (d_1,d_2)$ for $(a_1,a_2),(b_1,b_2),(c_1,c_2),(d_1,d_2)\in S$ then $(a_1,a_2)+(c_1,c_2)=(b_1,b_2)+(d_1,d_2)$ and $(a_1,a_2)(c_1,c_2)=(b_1,b_2)(d_1,d_2)$ , so we can view these operations as operations on $\text{Frac}(R)$ . It is now a simple matter to verify that $\text{Frac}(R)$ is indeed a field under these operations.

We can view $R$ as a subring of $\text{Frac}(R)$ via the embedding $r\mapsto (r,1)$ . We can now think of $\text{Frac}(R)$ as the 'smallest' field which contains $R$ .

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Field of fractions

Formal Definition