TITLE: Left-ratio and right-ratio in (not necessarily commutative) field
QUESTION [2 upvotes]: I am reading Birkhoff–Neumann (1936) about (not necessarily commutative) field $F$. 
The authors use terms right-ratio and left-ratio in section 13.
Right-ratio is denoted as $[x_1, x_2, ... x_n]_r\,$  $x_i \in F$ where not all elements $x_i=0$. Similary left-ratio is $[y_1, y_2, ... y_n]_l$.
Two right-ratios $[x_1, ... x_n]_r\, ,[\xi_1, ... \xi_n]_r\,$  are "equal" iff $\exists z\in F:\xi_i=x_iz,\ i=1, ... n$
But in section 14, they also use term ennuples: not right or left ratios. We can define vector-operations and inner-product for ennuples. 
I don't understand to this term (ennuples) in not commutative field and their connection to ratios. Any example (of left/right-ratios with ennuples) will be helpful.

Birkhoff, Garrett; von Neumann, John
“The logic of quantum mechanics.”
Annals of Mathemathics (2nd series) 37 (1936), no. 4, 823–843.
MR1503312
JFM62.1061.04
DOI:10.2307/1968621

REPLY [1 votes]: An "ennuple" just sounds like an element of $F^n$, and "right-ratio" just sounds like an equivalence class of ennuples under the equivalence relation $x\sim y$ iff $\exists z\in F^\ast$ such that $xz=y$. (Here I'm using the $^\ast$ to denote the nonzero elements of the ring $F$).
If we're talking about a division ring $D$, then an ennuple $x\in D^n$ would be equivalent to the members of the set of ennuples $xD^\ast$, and so together they all form a single right-ratio.
For commutative fields, both of these things would just boil down to the same thing: the subspace spanned by $x$. But in general I guess there is no guarantee they are the same subspace.
The "ratio" aspect probably is a reference to the following example. If you think about pairs of nonzero integers (from $\Bbb Z^\ast$), then you can have ennuples with $n=2$: $\Bbb Z^\ast\times\Bbb Z^\ast$, and their left/right ratios would be a model of $\Bbb Q^\ast$. (For example, $2(1,2)=(2,4)$ evidences that $\frac{1}{2}=\frac{2}{4}$ as ratios. Actually I suddenly realized that the relation in this case would not be symmetric! But I still think it gives a little idea of the distinction between the ennuples and equivalence classes.
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