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 nspace 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]}
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Projective Nullstellensatz
Transcription
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: SpringerVerlag, ISBN 9780387942698, MR 1322960
 Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: SpringerVerlag, ISBN 9780387902449, MR 0463157
 Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra volume II, Graduate Texts in Mathematics, 29 (Reprint of the 1960 ed.), Berlin, New York: SpringerVerlag, ISBN 9780387901718, MR 0389876