Is there another terminology to designate this?

Question

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 $F$?

Answer 1

If you look careful at $M=t(M)\oplus F$ can notice that $F\simeq M/t(M)$. But finitely generated torsion-free modules over a PID are free, so $M/t(M)$ is free and thus has a (finite) rank. That's why Lang calls this the rank of $M$ which seems a reasonable choice. (Bourbaki also calls this the rank of $M$.) But as you can see there is no standard terminology. (Betti number comes from group theory rather than from modules which have assigned Betti numbers but these have a different meaning.)

Answer 2

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)