Great. I'm all in favor of using the standard terminology -- I just didn't know this one until reading it above. Thanks.

Now using these standard terms, I will restate the point that I was making above.

With vector spaces, you can never have a submodule of rank k that is properly contained in another submodule of rank k (proper inclusions always correspond to an increase of dimension).

But the $\mathbb{Z}$-module $\mathbb{Z}$ has an entire lattice of submodules, all of rank 1.

This is a consequence of not being able to perform division on the scalars.

So with modules, the lattice of submodules can have some interesting and rich structures, which cannot be present in the lattice of submodules of a vector space.