The multiplicativity formula for degrees of a tower of fields is well-known. I wonder if the same formula still holds if we consider division rings instead of fields, namely:
Let $A \subseteq B \subseteq C$ be three division algebras. From the comments to this question follows that the notion of the rank (as modules) is well-defined.
Denote the rank of $B$ as an $A$-module by $r_A(B)$, etc.
Is it true that $r_A(C)=r_B(C) r_A(B)$? I think (but may be wrong) that the answer is yes, and the proof is similar to the commutative case (taking bases etc.).
Thank you very much for any help. I apologize if this question is trivial.