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[parent] homogeneous ideal (Definition)

Let $ R = \oplus_{g\in G} R_g$ be a graded ring. Then an element $ r$ of $ R$ is said to be homogeneous if it is an element of some $ R_g$. An ideal $ I$ of $ R$ 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 $ \bigcup_{g\in G} I\cap R_g$.

One observes that if $ I$ is a homogeneous ideal and $ r=\sum_i r_{g_i}$ is the sum of homogeneous elements $ r_{g_i}$ for distinct $ g_i$, then each $ r_{g_i}$ must be in $ I$.

To see some examples, let $ k$ be a field, and take $ R=k[X_1,X_2,X_3]$ with the usual grading by total degree. Then the ideal generated by $ X_1^n+X_2^n-X_3^n$ is a homogeneous ideal. It is also a radical ideal. One reason homogeneous ideals in $ k[X_1,\ldots,X_n]$ are of interest is because (if they are radical) they define projective varieties; in this case the projective variety is the Fermat curve. For contrast, the ideal generated by $ X_1+X_2^2$ is not homogeneous.



"homogeneous ideal" is owned by archibal. [ full author list (2) | owner history (1) ]
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See Also: graded ring, projective variety, homogeneous elements of a graded ring, homogeneous polynomial

Also defines:  homogeneous, homogeneous element
Keywords:  commutative algebra, algebraic geometry

Pronunciation (guide):
 homogeneous: /hoh-moh-gee''-nee-uhs/

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Cross-references: curve, projective varieties, radical, radical ideal, degree, grading, field, sum, ideal generated by, generated by, ideal, graded ring
There are 13 references to this entry.

This is version 6 of homogeneous ideal, born on 2001-10-15, modified 2004-02-16.
Object id is 190, canonical name is HomogeneousIdeal.
Accessed 7414 times total.

Classification:
AMS MSC: 13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)
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