Quotient field of a domain Let $A$ be a commutative domain and $K=Quot(A)$, its field of fractions (quotient field). Prove that $K$ is a f.g. $A$-module if and only if $A=K$.

Let $\\{\frac{1}{a_1},\ldots,\frac{1}{a_n}\\}$ generate $K$ as an $A$ module, with each $a_j\in A$, and let $x$ be an element of $K$. Then there exist $b_1,\ldots,b_n$ in $A$ such that $\frac{x}{a_1a_2\cdots a_n}=\sum\limits_{j=1}^n\frac{b_j}{a_j}$. Multiplying both sides of the equation by $a_1a_2\cdots a_n$ shows that $x$ is in $A$.