Simple module over matrix rings

I’m trying to prove that if $M$ is a simple module over $M_n(D)$ where $D$ is a division algebra, then $M\cong D^n$. I know that if $M$ is a simple module over $R$ then it is isomorphic to $Rv=\{rv|r\in R\}$ for all $v\in M$. I’m trying to find a $M_n(D)$ module isomorphism $f:M_n(D)v\rightarrow D^n$. I’ve been told to try the map that maps $Xv\mapsto Xe_1$, where $e_1$ is the column vector with 1st entry 1 and the rest 0. I can prove that this is a homomorphism but I’m stuck on the injectivity. Any help would be well appreciated!

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