1 Morita equivalence
This entry presents both the definition of Morita equivalent algebras and the Morita equivalence
theorem, with a brief proof included.
Definition 1.1. Let A and B be two associative, but not necessarily commutative, algebras.
Such algebras A and B are called Morita equivalent, if there is an equivalence of categories
between A-mod and B-mod.
Theorem 1.1. Morita Equivalence Theorem Commutative algebras A and B are Morita
equivalent if and only if they are isomorphic.
Proof. Following the above definition, isomorphic algebras are Morita equivalent. Let us assume
that A and B are any two such Morita equivalent associative algebras. It follows then
that
, and thus one also has that
If A and B are both commutative, then by the Associative Algebra Lemma one also has that
A = ZA and B = ZB.
