The main result for Morita equivalent algebras is provided by the following proposition.
Let
and
be two arbitrary rings, and also let
be an additive, right exact functor. Then, there is a
-bimodule
, which is unique up to isomorphism, so that
is isomorphic to the functor
given by
There are also two important and fairly straightforward corollaries of the Morita (uniqueness) theorem.
Two rings,
and
, are Morita equivalent if and only if there
is an
-bimodule
and a
-bimodule
so that
as
as
. Also
Proof. All equivalences of categories are exact functors, and therefore they preserve projective objects as required by Corollary 1.
which takes
Proof.
Let
and
be the bimodules already defined in Corollary 1.
For proposition (i), one utilizes the functors
and
to prove the equivalence of the two categories.
For the second proposition (ii), one needs to employ the functor
to prove the natural equivalence of the latter two categories.
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