homogeneous ideal

Let R=gGRg be a graded ringMathworldPlanetmath. Then an element r of R is said to be homogeneous if it is an element of some Rg. 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 gGIRg.

One observes that if I is a homogeneous ideal and r=irgi is the sum of homogeneous elements rgi for distinct gi, then each rgi must be in I.

To see some examples, let k be a field, and take R=k[X1,X2,X3] with the usual grading by total degree. Then the ideal generated by X1n+X2n-X3n is a homogeneous ideal. It is also a radical ideal. One reason homogeneous ideals in k[X1,,Xn] are of interest is because (if they are radicalPlanetmathPlanetmathPlanetmath) they define projective varieties; in this case the projective variety is the Fermat (http://planetmath.org/FermatsLastTheorem) curve. For contrast, the ideal generated by X1+X22 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