PlanetPhysics/Morita Equivalence Lemma for Arbitrary Algebras

Morita equivalence lemma for arbitrary algebras
Let us consider first an example of Morita equivalence; thus, for an integer $$n \geq 1$$, let $$Mat_n(A)$$ be the algebra of $$n \times n$$-matrices with entries in an algebra $$A$$. 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 $$A$$ and any integer $$n \geq 1$$, the algebras $$A$$ and $$Mat_n(A)$$ are Morita equivalent. \end{theorem}

{\mathbf Important Notes:}


 * Even if $$A$$ is a commutative algebra, the algebra $$Mat_n(A)$$ is of course not commutative for any $$n > 1$$ because the matrix multiplication is generally non-commutative.
 * In general, the algebra $$A$$ cannot be recovered from its corresponding abelian category $$A$$-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.