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# quadraticClosure -- Creates the subideal generated by quadratic elements of a given ideal

- Usage:
    `quadraticClosure I`
- Inputs:
  - I, an instance of the type NCIdeal,
- Outputs:
  - an instance of the type NCIdeal, the quadratic closure of I

## Description

The quadratic closure of an NCIdeal in an NCPolynomialRing is the NCIdeal generated by the elements of degree at most 2. Commonly used with Link to homogDual in the case where the ideal generators are homogeneous of degree greater than 1.

If the input is an NCQuotientRing, the method is applied to the defining ideal of the quotient ring and the corresponding quotient ring is returned. At the moment, quotients of quotients is not implemented, and the ambient ring of the input NCQuotientRing is assumed to be an NCPolynomialRing.

This method is commonly used in conjunction with homogDual.

    i1 : A = QQ{x,y,z}

    o1 = A

    o1 : NCPolynomialRing

    i2 : I = ncIdeal{x*z-z*x, y*z, x*y^2-y^2*x, x^3*y-y*x^3}

                                        2    2     3  3
    o2 = Two-sided ideal {-zx+xz, yz, -y x+xy , -yx +x y}

    o2 : NCIdeal

    i3 : J = quadraticClosure I

    o3 = Two-sided ideal {-zx+xz, yz}

    o3 : NCIdeal

## See also

- homogDual -- Computes the dual of a pure homogeneous ideal

## Ways to use quadraticClosure:

- `quadraticClosure(NCIdeal)`
- `quadraticClosure(NCQuotientRing)`

## For the programmer

The object quadraticClosure is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NCAlgebra/NCAlgebraDoc.m2:2753:0.