Irrelevant ideal
Encyclopedia
In mathematics
, the irrelevant ideal is the ideal
of a graded ring consisting of all 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.
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. More generally, for an arbitrary graded ring R, the Proj construction
disregards all irrelevant ideals of R.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the irrelevant ideal is the ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
of a graded ring consisting of all homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical
Radical of an ideal
In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical...
contains the irrelevant ideal.
The terminology arises from the connection with algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. If R = k[x0, ..., xn] (a multivariate polynomial ring in n+1 variables over an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
k) graded with respect to degree
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
, there is a bijective correspondence
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
between projective algebraic sets in projective n-space over k
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
and homogeneous, radical ideals of R not equal to the irrelevant ideal. More generally, for an arbitrary graded ring R, the Proj construction
Proj construction
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties...
disregards all irrelevant ideals of R.