Proposition 14

Let A and B be $k$-algebras. Then A and B are Morita equivalent if and only if there exist an integer $n\geqslant 1$ and a square matrix $e= (e_{ij})$ in $\mathbf{M}_n(B)$ satisfying the following conditions:

(i) We have $e^2=e$.

(ii) The two-sided ideal of B generated by the elements $e_{ij}$ is equal to B.

(iii) The $k$-algebra A is isomorphic to $e\mathbf{M}_n(B)e$.

If conditions (i) and (ii) are satisfied, then the $(e\mathbf{M}_n(B)e,B)_k$-bimodule $eB^n_d$ is invertible.

In view of Theorem 1 (VIII, p. 101), the $k$-algebra A is Morita equivalent to B if and only if it is isomorphic to the endomorphism algebra of a projective, finitely generated, and generating right B-module. The proposition therefore follows from Lemmas 3 and 4 below.