# 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 $k[X_{1},\ldots,X_{n}]$ 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