For an arbitrary ring $R$ and a positive integer $n >1$, are the category of $R$-modules and the category of $M_n(R)$-modules isomorphic?
Here, $M_n(R)$ denotes the $n\times n$ matrices over the ring $R$.
I know these two categories are equivalent, and I guess they are not necessarily isomorphic, but I don't know how to prove it...
Many thanks :)