Irrelevant ideal

In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal.^[1]

The terminology arises from the connection with algebraic geometry. If R = k[x₀, ..., x_n] (a multivariate polynomial ring in n+1 variables over an algebraically closed field k) graded with respect to degree, there is a bijective correspondence between projective algebraic sets in projective n-space over k and homogeneous, radical ideals of R not equal to the irrelevant ideal.^[2] More generally, for an arbitrary graded ring R, the Proj construction disregards all irrelevant ideals of R.^[3]

Notes

1. Zariski & Samuel 1975, §VII.2, p. 154
2. Hartshorne 1977, Exercise I.2.4
3. Hartshorne 1977, §II.2

References

• Sections 1.5 and 1.8 of Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94269-8, MR 1322960
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra volume II, Graduate Texts in Mathematics 29 (Reprint of the 1960 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876