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
Canonical name	FinitelyGeneratedProjectiveModule
Date of creation	2013-03-22 13:29:37
Last modified on	2013-03-22 13:29:37
Owner	mhale (572)
Last modified by	mhale (572)
Numerical id	6
Author	mhale (572)
Entry type	Definition
Classification	msc 16D40
Synonym	finite projective module
Related topic	HilbertModule