I came across this problem and get stuck for quite sometime.
Problem: Let $R$ be a commutative domain that is not a field. Let $F$ be its field of fractions. Show that $F$ is an indecomposable $R$-module which is not finitely generated.
Attempt: I did a similar problem before showing that $\mathbb{Q}$ is a $\mathbb{Z}$-module that is not decomposable and not finitely generated. I think this problem feels like a generalization of what's happening in the rational number case. However, I don't know where to start.
Any help will be appreciated.