Irrelevant ideal

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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. 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, ..., xn] (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]


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