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单词 HomogeneousElementsOfAGradedRing
释义

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

R+=ω0Rω.

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:

(R)=ωRω.

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

References

  • 1 Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.
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更新时间:2025/5/4 10:58:27