Here is an illustration of the ripple effects of discarding the division of scalars.

With vector spaces, we have the following:

* Every maximal set of linearly independent vectors is a basis
* Every minimal spanning set of vectors is a basis

But not so for general modules.

Take the ring of integers \$\mathbb{Z}\$, which itself is a one-dimensional \$\mathbb{Z}\$-module. There are two bases for \$\mathbb{Z}\$: \$\{1\}\$, and \$\{-1\}\$.

But:

* The set \$\{2\}\$ is a maximal independent set, yet it does not span the whole space
* The set \$\{2,3\}\$ is a minimal spanning set, yet it is linearly dependent