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.


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