# extracting generators of an ideal

## obtain a single generator as an element

Once an ideal has been constructed it is possible to obtain individual elements using _. As always in Macaulay2, indexing starts at 0.
 i1 : R = ZZ[w,x,y,z]; i2 : I = ideal(z*w-2*x*y, 3*w^3-z^3,w*x^2-4*y*z^2,x); o2 : Ideal of R i3 : I_0 o3 = - 2x*y + w*z o3 : R i4 : I_3 o4 = x o4 : R

## the generators as a matrix or list of elements

Use generators or its abbreviation generators to get the generators of an ideal I as a matrix. Applying first entries to this matrix converts it to a list.
 i5 : gens I o5 = | -2xy+wz 3w3-z3 wx2-4yz2 x | 1 4 o5 : Matrix R <--- R i6 : first entries gens I 3 3 2 2 o6 = {- 2x*y + w*z, 3w - z , w*x - 4y*z , x} o6 : List

## number of generators

The command numgens gives the number of generators of an ideal I.
 i7 : numgens I o7 = 4

## minimal generating set

To obtain a minimal generating set of a homogeneous ideal use mingens to get the minimal generators as a matrix and use trim to get the minimal generators as an ideal.
 i8 : mingens I o8 = | x wz 4yz2 3w3-z3 | 1 4 o8 : Matrix R <--- R i9 : trim I 2 3 3 o9 = ideal (x, w*z, 4y*z , 3w - z ) o9 : Ideal of R
The function mingens is only well-defined for a homogeneous ideal or in a local ring. However, one can still try to get as small a generating set as possible and when it is implemented this function will be done by trim.

## obtaining the input form of an ideal

If the ideal was defined using a function like monomialCurveIdeal and the generators are desired in the usual format for input of an ideal, the function toString is very useful. (Note: We are changing rings because monomialCurveIdeal is not implemented for rings over ZZ.)
 i10 : R = QQ[a..d]; i11 : I = monomialCurveIdeal(R,{1,2,3}); o11 : Ideal of R i12 : toString I o12 = ideal(c^2-b*d,b*c-a*d,b^2-a*c)