Let $D$, $E$ be division rings and $_ D M$ and $_ E N$ finite-dimensional vector spaces. Show that $End_D(M) \cong End_E(N)$, if and only if $D \overset{\psi}{\cong} E$ and $\dim_D(M) = \dim_E(N)$.
($\Leftarrow$) if $\dim_D(M) = \dim_E(N)$, then $M \overset{\varphi}{\cong} N$, and each $f: {_D} M \rightarrow {_D} M$ homomorphism goes to $g: {_E} N \rightarrow {_E} N$ with $\varphi(m) \in N$ for each $m \in M$ and $\psi(d) \in E$ for each $d \in D$ on the formula of $f$.
($\Rightarrow$) However, i have only the result that if $End_D(M) \cong End_E(N)$ then $End_D(M)=S$ is semi-simple as regular module and all simple $S$-module are isomorphic. How could i prove that $N \cong M$ and $D \cong E$?