Corollary 2

Let A and B be $k$-algebras and P an $(A,B)_k$-bimodule. The following properties are equivalent:

(i) The A-module P is generating, and the mapping $b\mapsto b_P$ from B to End$_A(P)$ is bijective.

(ii) The right B-module P is projective, finitely generated, faithful, and balanced, and the mapping $a\mapsto a_P$ from A to End$_B(P)$ is bijective.

The implication (i) $\Rightarrow$ (ii) follows from Proposition 4, a) and Remark 1 (VIII, p. 97). Assume that (ii) holds; then the A-module P is generating (Proposition 4, b) applied to the $(B^o,A^o)_k$-bimodule P). Since the B-module P is faithful and balanced, the second assertion of (i) is also satisfied (VIII, p. 97, Remark 1).