## Take Flight Snohomish County

Groups and subgroups in vector spaces over ordered fields

I have taken two first-year calculus courses, so I am not particularly well-versed in abstract algebra. Nevertheless, I am not clear on some of my readings about modules over ordered fields (it is worth noting that I am not even clear on what a positive element is; for instance, we were working over a field with the usual ordering).
Given an ordered field $F$ and a vector space $V$, an element $v\in V$ is said to be positive if for all $w\in V$, $w-v$ is nonnegative. Then, given $U$ and $V$ subspaces of $V$, it is true that $U$ is positive if and only if $V/U$ is a positive vector space. However, $U$ is positive if and only if the induced map $V/U\to V$ is injective. So why are these equivalent?

A:

Let $F$ be an ordered field, and let $V$ be a vector space over $F$. A vector $v\in V$ is said to be positive if for every positive $u\in V$, $u+v$ is positive.
Here is a quick proof, that assumes $V$ has a basis, $v$ is non-zero, and that $F$ is positive.
Lemma: Let $F$ be an ordered field, and let $U$ be a subspace of $V$. Then $U$ is positive iff the induced map $V/U\to V$ is injective.
Proof: If $v\in V$ is positive, then $V-\{0\}$ is positive. But $U-\{0\}$ is positive, and $U\subseteq V-\{0\}$ is a subspace, so $V/U$ is positive. Thus if $U$ is positive, the induced

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