Let $K$ be a field, assume $A,B$ are $M_n(K)$ modules with $\dim_{\mathbb{C}}A=\dim_{\mathbb{C}}B<\infty$. I am supposed to show that $A=B$ and that $n$ divides $\dim_{\mathbb{C}}A$.
If I had that $\dim_KA=\dim_KB$ I can easily settle it since $M_n(K)$ is simple and thus it has a unique simple module $U$. Then $A\simeq U^a$ and $B\simeq U^b$ and $a=b$ by dimension comparison. Is there any way to show this or is it a typo perhaps?