Proposition 3

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

(i) The A-module P is generating.

(ii) The image of the mapping Λ from $P\otimes_BP^*$ to $_sA_d$ (VIII, p. 95) contains the unit element.

(iii) There exist a $(B,A)_k$-bimodule Q and a surjective homomorphism of $(A,A)_k$-bimodules from $P\otimes_BQ$ to $_sA_d$.

If, moreover, the mapping $b\mapsto b_P$ from B to End$_A(P)$ is bijective, then the assertions are equivalent to the following statement:

(iv) The mapping Λ is an isomorphism from $P\otimes_BP^*$ to $_sA_d$.

(i) $\Leftrightarrow$ (ii): The image of Λ is the trace ideal $\tau (P)$ (VIII, p. 80). The equivalence of (i) and (ii) therefore follows from Theorem 1 of VIII, p. 80.

(ii) $\Rightarrow$ (iii): It suffices to take $Q = P^*$.

(iii) $\Rightarrow$ (ii): Let Q be a $(B,A)_k$-bimodule and $\psi$ a homomorphism of $(A,A)_k$-bimodules from $P\otimes_BQ$ to $_sA_d$. There exists a homomorphism of $(B,A)_k$-bimodules Ψ from Q to $P^*$ such that $\psi (x\otimes y) =\langle x,\Psi (y)\rangle$ for $x\in P$ and $y\in Q$, and we have the equality $\psi = \Lambda \circ (1_P\otimes \Psi )$. If $\psi$ is surjective, then so is Λ.

It is clear that (iv) implies (ii). Conversely, suppose that property (ii) holds and that the mapping $b\mapsto b_P$ from B to End$_A(P)$ is bijective. Let $e$ be an element of $P\otimes_BP^*$ such that $\Lambda (e) = 1$. By relation (7) of VIII, p. 96, we have $u= \Lambda (u)e$ for every $u$ in $P\otimes_BP^*$; the injectivity of Λ follows.