# 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, $\mathcal{E}=p{A}^{n}$ is a finitely generated projective right $A$-module.
Further, $\mathcal{E}$ is a pre-Hilbert $A$-module with ($A$-valued) inner product

$$\u27e8u,v\u27e9=\sum _{i=1}^{n}{u}_{i}^{*}{v}_{i},u,v\in \mathcal{E}.$$ |

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 |