Morita equivalence

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

$\displaystyle A-mod \sim B-mod$

, and thus one also has that

$\displaystyle Z(A-mod) \simeq Z(B-mod).$

If $ A$ and $ B$ are both commutative, then by the Associative Algebra Lemma one also has that $ A = Z_A$ and $ B = Z_B.$



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As of this snapshot date, this entry was owned by bci1.