homogeneous elements of a graded ring
Let be a field, and let be a connected commutative -algebra
graded (http://planetmath.org/GradedAlgebra) by . Then via the grading, we can decompose into a direct sum
of vector spaces: , where .
For an arbitrary ring element , we define the homogeneous degree of to be the value such that , and we denote this by . (See also homogeneous ideal)
A set of some importance (ironically), is the irrelevant ideal of , denoted by , and given by
Finally, we often need to consider the elements of such a ring without using the grading, and we do this by looking at the homogeneous union of :
In particular, in defining a homogeneous system of parameters, we are looking at elements of .
References
- 1 Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.