# finitely generated projective module

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

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

 $\langle u,v\rangle=\sum_{i=1}^{n}u_{i}^{*}v_{i},\quad u,v\in\mathord{\mathcal{% E}}.$
Title finitely generated projective module FinitelyGeneratedProjectiveModule 2013-03-22 13:29:37 2013-03-22 13:29:37 mhale (572) mhale (572) 6 mhale (572) Definition msc 16D40 finite projective module HilbertModule