Suppose that $R$ is an integral domain and $M$ is a finitely generated module. Does $M$ necessarily have finite rank?
DEFINITIONS: $M$ is finitely generated if there exists $x_1, \dots, x_n$ such that each element of $M$ can be written as a linear combination of $x_1, \dots, x_n$. We define the rank of $M$ to be $$ \sup_I |I| $$ where the supremum is taken over all $I\subseteq M$ where $I$ is a linearly independent set.
The result is true if we assume that $M$ is a free module. I am wondering if it also holds in this more general setting.