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In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. It corresponds to the origin in the affine space, which cannot be mapped to a point in the projective space. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal.cite-ref-1[1]

The terminology arises from the connection with algebraic geometry. If R = k[x0, ..., xn] (a multivariate polynomial ring in n+1 variables over an algebraically closed field k) is 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; this is known as the projective Nullstellensatz.cite-ref-2[2] More generally, for an arbitrary graded ring R, the Proj construction disregards all irrelevant ideals of R.cite-ref-3[3]

• Notes

• References

cite-note-11. ↑ Zariski & Samuel 1975, §VII.2, p. 154

cite-note-22. ↑ Hartshorne 1977, Exercise I.2.4

cite-note-33. ↑ Hartshorne 1977, §II.2

• Sections 1.5 and 1.8 of citerefeisenbud1995Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94269-8, MR 1322960

• citerefhartshorne1977Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

• citerefzariskisamuel1975Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra volume II, Graduate Texts in Mathematics, vol. 29 (Reprint of the 1960 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876