13A02-HomogeneousElementsOfAGradedRing.tex

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\pmtitle{homogeneous elements of a graded ring}
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\pmtype{Definition}
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\pmrelated{HomogeneousIdeal}
\pmdefines{homogeneous element}
\pmdefines{homogeneous degree}
\pmdefines{irrelevant ideal}
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\begin{document}
Let $k$ be a field, and let $R$ be a connected commutative $k$-algebra \PMlinkname{graded}{GradedAlgebra} by $\mb{N}^m$.  Then via the grading, we can decompose $R$ into a direct sum of vector spaces:  $R=\coprod_{\omega\in\mb{N}^m} R_\omega$, where $R_0=k$.

For an arbitrary ring element $x\in R$, we define the \emph{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 \emph{irrelevant ideal} of $R$, denoted by $R^+$, and given by

\begin{align*}
R_+=\coprod_{\omega\neq 0}R_\omega.
\end{align*}

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 \emph{homogeneous union} of $R$:

\begin{align*}
\mathcal{H}(R)=\bigcup_\omega R_\omega.
\end{align*}

In particular, in defining a homogeneous system of parameters, we are looking at elements of $\mathcal{H}(R_+)$.
\begin{thebibliography}{9}
\bibitem{Stan} Richard P. Stanley, {\em Combinatorics and Commutative Algebra}, Second edition, Birkhauser Press.  Boston, MA.  1986.
\end{thebibliography}
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