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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 $n$ plus $n$ 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 :)

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4  
Why would you want to know? Just curiosity? Isomorphism of categories is usually too strong a property; that's why equivalence of categories is more prevalent, as it is normally just as useful but easier to handle. –  Arturo Magidin Mar 15 '11 at 21:55
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Thank you for taking time to comment. I want to know about this not just for curiosity. This a problem in Basic Algebra written by Nanthan Jacobson. –  ShinyaSakai May 1 '11 at 17:40
1  
Then say so in your question. It's called "giving context". –  Arturo Magidin May 1 '11 at 18:13

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