Proposition 5

The following properties are equivalent:

(i) The B-module A is projective.

(ii) There exists an element $e$ of the $(A,A)$-bimodule B satisfying the following two conditions: $\varepsilon (e) = 1$ and $ae=ea$ for every $a\in A$.

The mapping $\varepsilon : B\rightarrow A$ is surjective because we have $\varepsilon (a\otimes 1) =a$ for every $a\in A$; it is a homomorphism of $(A$, A)-modules, hence a B-linear mapping. If the B-module A is projective, then there exists a section $s$ of $\varepsilon$ (II, §2, No. 2, p. 231, Proposition 4); it is a homomorphism of $(A$, A)-bimodules from A to B. If we set $e=s$(1), then we have $\varepsilon (e) =\varepsilon (s(1)) = 1$ and $ae=s(a) =ea$ for every $a\in A$. So (i) implies (ii).

Conversely, let $e$ be an element of B satisfying the conditions in item (ii). Define a mapping $s$ from A to B by the formula

It is a homomorphism of $(A$, A)-modules, and we have $\varepsilon \circ s= 1_A$; in other words, $s$ is a B-linear section of the surjective mapping $\varepsilon$. Consequently, the B-module A is isomorphic to the direct factor submodule $s(A)$ of $B_s$ (II, §1, No. 9, p. 211, Proposition 15) and is therefore projective (II, §2, No. 2, p. 231, Proposition 4). This proves that (ii) implies (i).