homogeneous ideal
Let be a graded ring. Then an element of is said to be homogeneous if it is an element of some . An ideal of 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 .
One observes that if is a homogeneous ideal and is the sum of homogeneous elements for distinct , then each must be in .
To see some examples, let be a field, and take with the usual grading by total degree. Then the ideal generated by is a homogeneous ideal. It is also a radical ideal. One reason homogeneous ideals in 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 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 |