# homogeneous ideal

Let $R={\oplus}_{g\in G}{R}_{g}$ be a graded ring^{}. Then an element $r$ of $R$ is said to be *homogeneous* if it is an element of some ${R}_{g}$. An ideal $I$ of $R$ is said to be homogeneous if it can be generated by a set of homogeneous elements, or equivalently if it is the ideal generated by the set of elements ${\bigcup}_{g\in G}I\cap {R}_{g}$.

One observes that if $I$ is a homogeneous ideal and $r={\sum}_{i}{r}_{{g}_{i}}$ is the sum of homogeneous elements ${r}_{{g}_{i}}$ for distinct ${g}_{i}$, then each ${r}_{{g}_{i}}$ must be in $I$.

To see some examples, let $k$ be a field, and take $R=k[{X}_{1},{X}_{2},{X}_{3}]$ with the usual grading by total degree. Then the ideal generated by ${X}_{1}^{n}+{X}_{2}^{n}-{X}_{3}^{n}$ is a homogeneous ideal. It is also a radical ideal. One reason homogeneous ideals in $k[{X}_{1},\mathrm{\dots},{X}_{n}]$ are of interest is because (if they are radical^{}) they define projective varieties; in this case the projective variety is the Fermat (http://planetmath.org/FermatsLastTheorem) curve. For contrast, the ideal generated by ${X}_{1}+{X}_{2}^{2}$ is not homogeneous.

Title | homogeneous ideal |

Canonical name | HomogeneousIdeal |

Date of creation | 2013-03-22 11:45:00 |

Last modified on | 2013-03-22 11:45:00 |

Owner | archibal (4430) |

Last modified by | archibal (4430) |

Numerical id | 11 |

Author | archibal (4430) |

Entry type | Definition |

Classification | msc 13A15 |

Classification | msc 33C75 |

Classification | msc 33E05 |

Classification | msc 86A30 |

Classification | msc 14H52 |

Classification | msc 14J27 |

Related topic | GradedRing |

Related topic | ProjectiveVariety |

Related topic | HomogeneousElementsOfAGradedRing |

Related topic | HomogeneousPolynomial |

Defines | homogeneous |

Defines | homogeneous element |