Examples
Any two isomorphic rings are Morita equivalent.
The ring of n-by-n matrices with elements in R, denoted Mn(R), is Morita-equivalent to R for any n > 0. Notice that this generalizes the classification of simple artinian rings given by Artin–Wedderburn theory. To see the equivalence, notice that if M is a left R-module then Mn is an Mn(R)-module where the module structure is given by matrix multiplication on the left of column vectors from M. This allows the definition of a functor from the category of left R-modules to the category of left Mn(R)-modules. The inverse functor is defined by realizing that for any Mn(R)-module there is a left R-module V and a positive integer n such that the Mn(R)-module is obtained from V as described above.
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