> Oops, I meant to say $2\mathbb{Z}$, rather than $\mathbb{Z}/2$.

Oh, I see. Maybe I should have guessed.

> For _free_ modules, which are spanned by $n$ linearly independent generators, it seems like we could still retain the terminology of "dimension" to describe the number of linearly independent generators comprising a basis.

We could. But we don't: we say "rank". Part of succeeding in life is just knowing how to talk like other people do, regardless of whether it makes sense. Sometimes it pays to fight, but one should choose ones battle's wisely.

Oh, I see. Maybe I should have guessed.

> For _free_ modules, which are spanned by $n$ linearly independent generators, it seems like we could still retain the terminology of "dimension" to describe the number of linearly independent generators comprising a basis.

We could. But we don't: we say "rank". Part of succeeding in life is just knowing how to talk like other people do, regardless of whether it makes sense. Sometimes it pays to fight, but one should choose ones battle's wisely.