Are mod-$R$ and mod-$M_n(R)$ isomorphic for $n>1$?
Let $R$ be a ring (with 1) and $M_n(R)$ be the ring of $n$ by $n$ matrices with entries in $R$. Let mod-$R$ be the category of right $R$ modules and mod-$M_n(R)$ be the category of right $M_n(R)$ modules.
In Jacobson's Basic Algebra 2, (page 31) it is proved that the above two categories are equivalent. One of the exercises in that section asks whether the two categories are isomorphic.
For $n=1$, they are obviously isomorphic. For $n>1$, I believe that they cannot be isomorphic but I don't see a way to explain this. The examples of isomorphic categories I'm aware of are just the trivial ones, and I'm having a problem showing that the two categories are not isomorphic..
Please Enlighten me.