Let $R$ be a unital ring. A finitely generated projective right $R$ -module is of the form $eR^n$ , $n \in \Nset$ , where $e$ is an idempotent in $\End_R(R^n)$ .

Let $A$ be a unital $C^*$ -algebra and $p$ be a projection in $\End_A(A^n)$ , $n \in \Nset$ . Then, $\hilbmod = pA^n$ is a finitely generated projective right $A$ -module. Further, $\hilbmod$ is a pre-Hilbert $A$ -module with ($A$ -valued) inner product$$ \langle u,v \rangle = \sum_{i=1}^n u_i^* v_i, \quad u,v \in \hilbmod.$$