Oops, I meant to say $2 \mathbb{Z}$, rather than $\mathbb{Z}_2$.

I just applied this fix to messages 25 and 26, and also changed the terms "subspaces" to "submodules."

So the sentence you referred to now reads:

> Observe that $2 \mathbb{Z}$ is a _proper_ one-dimensional submodule of the one-dimensional module $\mathbb{Z}$.

For a _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.

In any case, that is what I meant by calling the ideals $k \mathbb{Z}$ one-dimensional submodules -- being that they are generated by the linearly independent set $\{k\}$.

I just applied this fix to messages 25 and 26, and also changed the terms "subspaces" to "submodules."

So the sentence you referred to now reads:

> Observe that $2 \mathbb{Z}$ is a _proper_ one-dimensional submodule of the one-dimensional module $\mathbb{Z}$.

For a _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.

In any case, that is what I meant by calling the ideals $k \mathbb{Z}$ one-dimensional submodules -- being that they are generated by the linearly independent set $\{k\}$.