Example 1

Let B be a $k$-algebra, $n$ an integer $\geqslant 1$, and A the $k$-algebra $\mathbf{M}_n(B)$. The right B-module $P = B^n_d$ is projective, finitely generated, and generating, and A can be identified with the endomorphism algebra of P (II, §10, No. 7, p. 349). By Theorem 1, the $(A,B)_k$-bimodule P is invertible. The algebras B and $\mathbf{M}_n(B)$ are therefore Morita equivalent.