Show that each element of $K$ can be written as $\frac{b}{a}$ where, $b\in B, a\in A$

Question

Let $A \subset B$ be integral domains and $B$ is a finitely generated $A$-module. Let $K$ be the fraction field of $B$. Show that each element of $K$ can be written as $\frac{b}{a}$ where, $b\in B, a\in A$.

Please give some hints.

Answer 1

Show that each non-zero element $b\in B$ satisfies some polynomial relation $p(b)=0$, where $p(X)\in A[X]$ has a non-zero constant term. Then write $p(X)=X\cdot p_1(X) + a_0$, and use that to manipulate a fraction of the form $b'/b$.

Answer 2

Hint: Every element in $B$ is integral over $A$.