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# getMaxIdeal -- computes a maximal ideal containing a given ideal in a polynomial ring

- Usage:
    `M = getMaxIdeal I`
    `M = getMaxIdeal(I,L)`
- Inputs:
  - I, an ideal, an ideal of a polynomial ring over QQ, ZZ, or ZZ/p for p a prime integer.
  - L, a list, a subset of the variables of the ring. This list must contain the variables that appear in the ideal I. By default, L is assumed to be the list of variables in the ring.
- Optional inputs:
  - Verbose (missing documentation) => an integer, default value 0, which controls the level of output of the method (0, 1, 2, 3, or 4).
- Outputs:
  - M, an ideal, maximal with respect to the variables in L.

## Description

In absence of an input list, getMaxIdeal yields a maximal ideal containing the input ideal I.

    i1 : R = ZZ/3[x,y]

    o1 = R

    o1 : PolynomialRing

    i2 : I = ideal(x*(x-1)*(x-2)*y*(y-1)*(y-2)+1)

                3 3    3       3
    o2 = ideal(x y  - x y - x*y  + x*y + 1)

    o2 : Ideal of R

    i3 : J = getMaxIdeal I

                        2
    o3 = ideal (x - y, y  + 1)

    o3 : Ideal of R

    i4 : isSubset(I,J)

    o4 = true

The function isSubset shows that I is contained in our new ideal. To see that J is indeed maximal, consider the codimension and the minimal primes.

    i5 : codim J

    o5 = 2

    i6 : P = minimalPrimes J

                         2
    o6 = {ideal (x - y, y  + 1)}

    o6 : List

    i7 : J == P_0

    o7 = true

The optional list argument allows us to restrict our maximal ideal to a polynomial ring defined by a subset of the variables of the ambient ring. Note that the list must contain the variables that appear in the generators of I.

    i8 : R = ZZ[x,y,z,a,b,c]

    o8 = R

    o8 : PolynomialRing

    i9 : I = ideal(27,x^2+1)

                     2
    o9 = ideal (27, x  + 1)

    o9 : Ideal of R

    i10 : J = getMaxIdeal(I,{x,y,z})

                        2
    o10 = ideal (z, y, x  + 1, 3)

    o10 : Ideal of R

    i11 : isSubset(I,J)

    o11 = true

## Ways to use getMaxIdeal:

- `getMaxIdeal(Ideal)`
- `getMaxIdeal(Ideal,List)`

## For the programmer

The object getMaxIdeal is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/QuillenSuslin.m2:2163:0.