finitely generated projective moduleFinitelyGeneratedProjectiveModule2003-02-26 20:30:352003-11-18 09:24:55Definition\usepackage{amssymb}
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\newcommand*{\defn}[1]{\textbf{#1}}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
\[
\langle u,v \rangle = \sum_{i=1}^n u_i^* v_i, \quad u,v \in \hilbmod.
\]