# PlanetPhysics/Morita Equivalence Lemma for Arbitrary Algebras

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## Morita equivalence lemma for arbitrary algebras

[edit | edit source]Let us consider first an example of Morita equivalence; thus, for an integer , let be the algebra of -matrices with entries in an algebra . The following is a typical example of Morita equivalence that involves noncommutative algebras.

\begin{theorem}{\mathbf Morita equivalence Lemma for arbitrary algebras}

For any algebra and any integer , the algebras and are Morita equivalent. \end{theorem}

{\mathbf Important Notes:}

- Even if is a commutative algebra, the algebra is of course not commutative for any because the matrix multiplication is generally non-commutative.
- In general, the algebra cannot be recovered from its corresponding abelian category -mod. Therefore, in order for a concept in noncommutative geometry to have or retain an intrinsic meaning, such a concept must be
*Morita invariant*that is, to remain within the same Morita equivalence class. This raises the important question: what properties of an algebra are Morita invariant ? The answer to this question is provided by the*Uniqueness Morita Theorem*.