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
Version 2.1.8p2
