homogeneous elements of a graded ring

Let k be a field, and let R be a connected commutativePlanetmathPlanetmathPlanetmath k-algebraPlanetmathPlanetmath graded (http://planetmath.org/GradedAlgebra) by m. Then via the grading, we can decompose R into a direct sumPlanetmathPlanetmath of vector spaces: R=ωmRω, where R0=k.

For an arbitrary ring element xR, we define the homogeneous degree of x to be the value ω such that xRω, and we denote this by deg(x)=ω. (See also homogeneous idealMathworldPlanetmath)

A set of some importance (ironically), is the irrelevant ideal of R, denoted by R+, and given by


Finally, we often need to consider the elements of such a ring R without using the grading, and we do this by looking at the homogeneous union of R:


In particular, in defining a homogeneous system of parameters, we are looking at elements of (R+).


  • 1 Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.
Title homogeneous elementsPlanetmathPlanetmath of a graded ring
Canonical name HomogeneousElementsOfAGradedRing
Date of creation 2013-03-22 14:14:52
Last modified on 2013-03-22 14:14:52
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Definition
Classification msc 13A02
Related topic HomogeneousIdeal
Defines homogeneous element
Defines homogeneous degree
Defines irrelevant ideal
Defines homogeneous union