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 $\mathscr{H}(R_{+})$.

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