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.

Theorem 1.1 Morita equivalence Lemma for arbitrary algebras

For any algebra and any integer $n \geq 1$ , the algebras and are Morita equivalent.

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”.

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