# complete set of primitive orthogonal idempotents

Let $A$ be a unital algebra over a field $k$. Recall that $e\in A$ is an idempotent^{} iff ${e}^{2}=e$. If ${e}_{1},{e}_{2}\in A$ are idempotents, then we will say that they are orthogonal^{} iff ${e}_{1}{e}_{2}={e}_{2}{e}_{1}=0$. Furthermore an idempotent $e\in A$ is called primitive^{} iff $e$ cannot be written as a sum $e={e}_{1}+{e}_{2}$ where both ${e}_{1},{e}_{2}\in A$ are nonzero idempotents. An idempotent is called trivial iff it is either $0$ or $1$.

Now assume that $A$ is an algebra^{} such that

$$A={M}_{1}\oplus {M}_{2}$$ |

as right modules and $1={m}_{1}+{m}_{2}$ for some ${m}_{1}\in {M}_{1}$, ${m}_{2}\in {M}_{2}$. Then ${m}_{1}$, ${m}_{2}$ are orthogonal idempotents in $A$ and ${M}_{1}={m}_{1}A$, ${M}_{2}={m}_{2}A$. Furthermore ${M}_{i}$ is indecomposable^{} (as a right module) if and only if ${m}_{i}$ is primitive. This can be easily generalized to any number (but finite) of summands.

If $A$ is additionally finite-dimensional, then

$$A={P}_{1}\oplus \mathrm{\cdots}\oplus {P}_{n}$$ |

for some (unique up to isomorphism^{}) right (ideals) indecomposable modules ${P}_{i}$. It follows from the preceding that

$${P}_{i}={e}_{i}A$$ |

for some ${e}_{i}\in A$ and $\{{e}_{1},\mathrm{\dots},{e}_{n}\}$ is a set of pairwise orthogonal, primitive idempotents. This set is called the complete set of primitive orthogonal idempotents of $A$.

Title | complete set of primitive orthogonal idempotents |
---|---|

Canonical name | CompleteSetOfPrimitiveOrthogonalIdempotents |

Date of creation | 2013-03-22 19:17:38 |

Last modified on | 2013-03-22 19:17:38 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 16S99 |

Classification | msc 20C99 |

Classification | msc 13B99 |