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on page $38$ of the book "Ring theory" by Louis Rowen it is shown that $R-\mathbf{Mod}$ and $M_{n}(R)-\mathbf{Mod}$ are equivalent categories ($R$ a ring). Some lines below the author states that these categories are not isomorphic, but why is this?

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I don't think that is correct. (Modulo set theory, I guess...) the only obstruction to two equivalent categories to being isomorphic is that they have different number objects in each isomorphism class, but categories of modules are too large for this to happen; with a sufficiently good set-theoretic foundation —universes, or something— one should be able to construct an iso from every equivalence. What is true is that none of the equivalences we usually deal with are isomorphisms. – Mariano Suárez-Alvarez Feb 28 '12 at 19:42
Should it be $R$-Mod and $M_n(R)$-Mod? – Arturo Magidin Feb 28 '12 at 19:56
@Arturo Magidin: Oh, sorry! yes $M_{n}(R)-Mod$. – New Feb 28 '12 at 20:21
@Mariano: We'd need a sufficiently large axiom of choice, but then again, we need that to prove that full + faithful + essentially surjective on objects is enough to be (part of) an equivalence of categories. – Zhen Lin Feb 28 '12 at 22:17
@ZhenLin, ah, indeed! – Mariano Suárez-Alvarez Feb 28 '12 at 23:12

I guess the author might have wanted to write 'these rings are not isomorphic' instead of 'these categories are not isomorphic', and then to introduce Morita equivalence...

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