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homogeneous elements of a graded ring
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(Definition)
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Let $k$ be a field, and let $R$ be a connected commutative $k$ -algebra graded by $\mb{N}^m$ . Then via the grading, we can decompose $R$ into a direct sum of vector
spaces: $R=\coprod_{\omega\in\mb{N}^m} R_\omega$ , where $R_0=k$ .
For an arbitrary ring element $x\in R$ , we define the homogeneous degree of $x$ to be the value $\omega$ such that $x\in R_\omega$ , and we denote this by $\deg(x)=\omega$ . (See also homogeneous ideal)
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 $\mathcal{H}(R_+)$ .
- 1
- Richard P. Stanley, Combinatorics and Commutative Algebra, Second edition, Birkhauser Press. Boston, MA. 1986.
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See Also: homogeneous ideal
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homogeneous element, homogeneous degree, irrelevant ideal, homogeneous union |
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Cross-references: homogeneous system of parameters, homogeneous ideal, ring, vector spaces, direct sum, grading, commutative, connected, field
There is 1 reference to this entry.
This is version 3 of homogeneous elements of a graded ring, born on 2004-03-12, modified 2004-04-30.
Object id is 5694, canonical name is HomogeneousElementsOfAGradedRing.
Accessed 6825 times total.
Classification:
| AMS MSC: | 13A02 (Commutative rings and algebras :: General commutative ring theory :: Graded rings) |
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Pending Errata and Addenda
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