finitely generated projective module

Let $R$ be a unital ring.A finitely generated projective right $R$-module is of the form $e{R}^n$,$n\in \mathbb{N}$, where $e$ is an idempotent in ${End}_R(R^n)$.

Let $A$ be a unital $C^{*}$-algebra and $p$ be a projection in ${End}_A(A^n)$, $n\in \mathbb{N}$.Then, $ℰ=p{A}^n$ is a finitely generated projective right $A$-module.Further, $ℰ$ is a pre-Hilbert $A$-module with ($A$-valued) inner product

$$⟨u,v⟩=\sum _{i=1}^n{u}_i^{*}{v}_i,u,v\in ℰ.$$