Inmathematics, theirrelevant ideal is theideal of agraded ring generated by thehomogeneous elements of degree greater than zero. It corresponds to the origin in theaffine space, which cannot be mapped to a point in theprojective space. More generally, ahomogeneous ideal of a graded ring is called anirrelevant ideal if itsradical contains the irrelevant ideal.^[1]

The terminology arises from the connection withalgebraic geometry. IfR = k[x₀, ..., x_n] (amultivariate polynomial ring inn+1 variables over analgebraically closed fieldk) is graded with respect todegree, there is abijective correspondence betweenprojective algebraic sets inprojectiven-space overk and homogeneous,radical ideals ofR not equal to the irrelevant ideal; this is known as theprojective Nullstellensatz.^[2] More generally, for an arbitrary graded ringR, theProj construction disregards all irrelevant ideals ofR.^[3]

• Sections 1.5 and 1.8 ofEisenbud, 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
• Hartshorne, Robin (1977),Algebraic Geometry,Graduate Texts in Mathematics, vol. 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, vol. 29 (Reprint of the 1960 ed.), Berlin, New York:Springer-Verlag,ISBN 978-0-387-90171-8,MR 0389876

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