Formal Definition
Two rings R and S (associative, with 1) are said to be Morita equivalent (or equivalent) if there is an equivalence of the category of (left) modules over R, R-Mod, and the category of (left) modules over S, S-Mod. Under the equivalence functors, each R module corresponds to an S module, and vice versa.
It can be shown that the left module categories R-Mod and S-Mod are equivalent if and only if the right module categories Mod-R and Mod-S are equivalent. This means that the notion of Morita equivalence does not depend on whether you are talking about left or right modules.
Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically additive.
Read more about this topic: Morita Equivalence
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