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

## Synopsis

• 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 .