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.
|Date of creation||2013-03-22 11:45:00|
|Last modified on||2013-03-22 11:45:00|
|Last modified by||archibal (4430)|