Is it true that by equivalence between category of $R$-modules and $S$-modules we conclude that $R$ and $S$ are isomorphic?($R$ and $S$ are unital rings.)
2026-05-06 00:53:53.1778028833
Equivalence between category of $R$-modules and $S$-modules
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If $A$ is any ring, then $A$ and $M_n(A)$ have equivalent categories of modules, and usually $A$ and $M_n(A)$ are not isomorphic.
This is the simplest example of a Morita equivalence.