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

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