# Two Categories mod-$R$ and mod-$M_n(R)$

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..

An equivalence between categories is naturally isomorphic to an isomorphism of categories iff the cardinalities of all the corresponding isomorphism classes of objects in the two categories are equal. That is, all you need to know to promote an equivalence $F$ into an isomorphism is that for each object $X$ in the domain category, there are just as many objects isomorphic to $X$ as there are objects isomorphic to $F(X)$. See this answer of mine for a sketch of the proof and some related discussion.
In this case every object in a module category has a proper class of other objects isomorphic to it and this proper class can in fact be put in bijection with the class $V$ of all sets. So any equivalence between two module categories is naturally isomorphic to an isomorphism.
• Thank you very much! I didn't expect "extending" or "modifying" $F$ to be so easy! Apr 8 '17 at 5:15