In the following, note that the free algebra F itself is a valid ideal (the ideal containing 1).
Given ideals I and J belonging to the same algebra F, return the sum of I and J, which is the ideal generated by the union of the generators of I and J.
Given ideals I and J belonging to the same algebra A, return the product of I and J, which is the ideal generated by the products of the generators of I with those of J.
Given an algebra F over a field and an ideal J of F, return the fp-algebra F/J (see below).
Given an ideal I of a generic algebra A, return A.
Given two ideals I and J belonging to the same algebra F, return whether I and J are equal.
Given two ideals I and J belonging to the same algebra F, return whether I and J are not equal.
Given two ideals I and J belonging to the same algebra F return whether I is not contained in J.
Given two ideals I and J belonging to the same algebra F return whether I is contained in J.
Given an ideal I of the algebra F, return whether I is the zero ideal (contains zero alone).
Given a polynomial f from an algebra F, together with an ideal I of F, return whether f is in I.
Given a polynomial f from an algebra F, together with an ideal I of F, return the unique normal form of f with respect to (the Gröbner basis of) I. The normal form of f is zero if and only if f is in I.
Given a polynomial f from an algebra F, together with a set or sequence S of polynomials from F, return a normal form of f with respect to S. This is not unique in general. If the normal form of f is zero then f is in the ideal generated by S, but the converse is false in general. In fact, the normal form is unique if and only if S forms a Groëbner basis.
Given a polynomial f from an algebra F, together with an ideal I of F, return whether f is not in I.
> F<x,y,z> := FreeAlgebra(RationalField(), 3);
> I := ideal<F | (x + y)^3, (y - z)^2, y^2*z + z>;
> NormalForm(y^2*z + z, I);
0
> NormalForm(x^3, I);
-x^2*y - x*y*x - x*y*z - x*z*y + x*z^2 - y*x^2 - y*x*y - y*z*x -
y*z*y - z*y*x - z*y*z + z^2*x + z^3
> NormalForm(z^4 + y^2, I);
z^4 + y*z + z*y - z^2
> x + y in I;
false