I have a (D,F)-bimodule $A$ where $D,F$ are division rings and $A = aF$ for some $a\in A$.
then for every $d\in D$, $da\in A$ so there exists a unique $f_d\in F$ such that $da = af_d$. Then I define the map $\phi:D\to F:d\mapsto f_d$. How do we show that this is an isomorphism? In particular, How do we show surjectivity?