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