Example 2

Let A be a commutative $k$-algebra and P an A-module. We view P as an $(A,A)_k$-bimodule whose two laws of action are equal. If the $(A,A)_k$-bimodule P is invertible, then the A-module P is finitely generated (Theorem 1). By Theorem 3 of Comm. Alg., II, §5, No. 4, p. 114, the following properties are equivalent:

(i) The $(A,A)_k$-bimodule P is invertible.

(ii) There exists an A-module Q such that $P\otimes_AQ$ is isomorphic to A.

(iii) The A-module P is projective, finitely generated, and of rank $1.*$