mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the irrelevant ideal is the ideal of a

graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, but ...

generated by the homogeneous elements of degree greater than zero. It corresponds to the origin in the

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

, which cannot be mapped to a point in the

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal. The terminology arises from the connection with

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

. If ''R'' = ''k'' 'x''₀, ..., ''x_n''(a multivariate polynomial ring in ''n''+1 variables over an

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

''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 ideal Radical (from Latin: ', root) may refer to: Politics and ideology Politics *Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century *Radical politics ...

s of ''R'' not equal to the irrelevant ideal; this is known as the projective Nullstellensatz. More generally, for an arbitrary graded ring ''R'', the

Proj construction In algebraic geometry, Proj is a construction analogous to the spectrum of a ring, spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective variety, projective varie ...

disregards all irrelevant ideals of ''R''.

Notes

References

*Sections 1.5 and 1.8 of * * Commutative algebra Algebraic geometry {{algebraic-geometry-stub