# Singular Book 1.8.1 -- ideal membership

 i1 : A = QQ[x,y]; i2 : I = ideal(x^10+x^9*y^2, y^8-x^2*y^7); o2 : Ideal of A i3 : f = x^2*y^7+y^14; i4 : f % I 12 8 o4 = - x*y + y o4 : A
So this f is not in the ideal I.
 i5 : f = x*y^13+y^12; i6 : f % I o6 = 0 o6 : A
This f is in the ideal I.

Check inclusion and equality of ideals.

 i7 : K = ideal(f,x^2*y^7+y^14); o7 : Ideal of A i8 : (gens K) % I o8 = | 0 -xy12+y8 | 1 2 o8 : Matrix A <--- A
In Macaulay2, inclusion of ideals can be tested using isSubset(Ideal,Ideal) and equality can be checked using Ideal == Ideal. In both cases the necessary Groebner bases are computed, if they have not already been computed.
 i9 : isSubset(K,I) o9 = false i10 : K == I o10 = false
 i11 : K = ideal(f,y^14+x*y^12); o11 : Ideal of A i12 : (gens K) % I o12 = 0 1 2 o12 : Matrix A <--- A i13 : isSubset(K,I) o13 = true i14 : K == I o14 = false