Is there another terminology to designate this? Let $R$ be a principal ideal domain.
Let $M$ be a finitely generated $R$-module. Then there exists a free $R$-submodule $F$ of $M$ such that $M=Tor(M)\oplus F$ and the ranks of such $F$'s are the same.
Lang, in his text, defines the rank of $F$ as the rank of $M$. But this contradicts the standard use of the term rank since usually the rank of a free module means the cardinality of a basis.
Is there another terminology to call the rank of $F$?
Dummit&Foote and Fraleigh calls this the Betti number or free rank of $M$, but I'm not sure these are standards.
What is the standard terminology of the rank of $F$?

I suspect what's happened is that Lang has used an odd choice of language and you've misread it. The definition is meant to be
rank of $M :=$ rank of any $F$ s.t. $M \cong Tor(M) \oplus F$
He isn't trying to define the rank of $F$, though I can see why you might read it that way.
Essentially, we're trying to extend the definition of rank for free modules to more general modules (since if $M$ is free then the definition just returns the rank of $M$ as a free module)
