Let $A \subseteq B$ be two division $k$-algebras, where $k$ is a field of characteristic zero. I am not sure if I wish to further assume that $B$ is affine over $A$, namely, if $B$ is finitely generated as an $A$-algebra.
If $A$ and $B$ happen to be commutative, namely fields, then the dimension of the field extension $A \subseteq B$ is naturally defined as the dimension of $B$ as a vectore space over $A$. (Actually, it is enough that $A$ is a field, and $B$ can be a non-commutative division ring).
I wonder if there is a 'natural' way to define the dimension of $A \subseteq B$, when both $A$ and $B$ are non-commutative?
Any ideas are welcome. Thanks!