# 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 $\deg(x)=\omega$. (See also homogeneous ideal)

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

 $\displaystyle R_{+}=\coprod_{\omega\neq 0}R_{\omega}.$

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 $R$:

 $\displaystyle\mathcal{H}(R)=\bigcup_{\omega}R_{\omega}.$

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

## References

• 1 Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.
Title homogeneous elements of a graded ring HomogeneousElementsOfAGradedRing 2013-03-22 14:14:52 2013-03-22 14:14:52 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 13A02 HomogeneousIdeal homogeneous element homogeneous degree irrelevant ideal homogeneous union