# PlanetPhysics/Morita Equivalence Lemma for Arbitrary Algebras

Let us consider first an example of Morita equivalence; thus, for an integer ${\displaystyle n\geq 1}$, let ${\displaystyle Mat_{n}(A)}$ be the algebra of ${\displaystyle n\times n}$-matrices with entries in an algebra ${\displaystyle A}$. The following is a typical example of Morita equivalence that involves noncommutative algebras.
For any algebra ${\displaystyle A}$ and any integer ${\displaystyle n\geq 1}$, the algebras ${\displaystyle A}$ and ${\displaystyle Mat_{n}(A)}$ are Morita equivalent. \end{theorem}
• Even if ${\displaystyle A}$ is a commutative algebra, the algebra ${\displaystyle Mat_{n}(A)}$ is of course not commutative for any ${\displaystyle n>1}$ because the matrix multiplication is generally non-commutative.
• In general, the algebra ${\displaystyle A}$ cannot be recovered from its corresponding abelian category ${\displaystyle A}$-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.