In the question linear algebra over a division ring vs. over a field is discussed the relationship between linear algebra over a field and linear algebra over a division ring. Roughly speaking the quation is: what happens if the scalars do not commute?
This led me to think about the following: what happens if the scalars do not commute and are also not associative?
Are there any theorems that hold for a module over an associative division ring but do not hold for a module over a nonassociative division ring?
Remark: Let $D$ be a nonassociative division ring. A left $D$-module $M$ consists of an abelian group $(M, +)$ and an operation $D × M → M$ such that for all $r, s \in D$ and $x, y \in M$, we have:
r(x+y) = rx + ry
(r+s)x = rx + sx
1.x = x.
A right $D$-module is defined similarly.