# homogeneous elements of a graded ring

Let $k$ be a field, and let $R$ be a connected commutative^{} $k$-algebra^{} graded (http://planetmath.org/GradedAlgebra) by ${\mathbb{N}}^{m}$. Then via the grading, we can decompose $R$ into a direct sum^{} of vector spaces: $R={\coprod}_{\omega \in {\mathbb{N}}^{m}}{R}_{\omega}$, where ${R}_{0}=k$.

For an arbitrary ring element $x\in R$, we define the *homogeneous degree* of $x$ to be the value $\omega $ such that $x\in {R}_{\omega}$, and we denote this by $\mathrm{deg}(x)=\omega $. (See also homogeneous ideal^{})

A set of some importance (ironically), is the *irrelevant ideal* of $R$, denoted by ${R}^{+}$, and given by

${R}_{+}={\displaystyle \coprod _{\omega \ne 0}}{R}_{\omega}.$ |

Finally, we often need to consider the elements of such a ring $R$ without using the grading, and we do this by looking at the *homogeneous union* of $R$:

$\mathscr{H}(R)={\displaystyle \bigcup _{\omega}}{R}_{\omega}.$ |

In particular, in defining a homogeneous system of parameters, we are looking at elements of $\mathscr{H}({R}_{+})$.

## References

- 1 Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.

Title | homogeneous elements^{} of a graded ring |
---|---|

Canonical name | HomogeneousElementsOfAGradedRing |

Date of creation | 2013-03-22 14:14:52 |

Last modified on | 2013-03-22 14:14:52 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 13A02 |

Related topic | HomogeneousIdeal |

Defines | homogeneous element |

Defines | homogeneous degree |

Defines | irrelevant ideal |

Defines | homogeneous union |