# 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*[*x*_{0}, ..., *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

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