Basic Operations on Ideals

In the following, note that the free algebra \(F\) itself is a valid ideal (the ideal containing 1).

Construction of New Ideals

I + J: AlgFr, AlgFr -> AlgFr

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 \(J\).

I * J: AlgFr, AlgFr -> AlgFr

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 \(J\).

F / J: AlgFr, AlgFr -> AlgFrRes

Given an algebra \(F\) over a field and an ideal \(J\) of \(F\), return the fp-algebra \(F/J\) (see below).

Generic(I): AlgFr -> AlgFr

Given an ideal \(I\) of a generic algebra \(A\), return \(A\).

Ideal Predicates

I eq J: AlgFr, AlgFr -> BoolElt

Given two ideals \(I\) and \(J\) belonging to the same algebra \(F\), return whether \(I\) and \(J\) are equal.

I ne J: AlgFr, AlgFr -> BoolElt

Given two ideals \(I\) and \(J\) belonging to the same algebra \(F\), return whether \(I\) and \(J\) are not equal.

I notsubset J: AlgFr, AlgFr -> BoolElt

Given two ideals \(I\) and \(J\) belonging to the same algebra \(F\) return whether \(I\) is not contained in \(J\).

I subset J: AlgFr, AlgFr -> BoolElt

Given two ideals \(I\) and \(J\) belonging to the same algebra \(F\) return whether \(I\) is contained in \(J\).

IsZero(I): AlgFr -> BoolElt

Given an ideal \(I\) of the algebra \(F\), return whether \(I\) is the zero ideal (contains zero alone).

Operations on Elements of Ideals

f in I: AlgFrElt, AlgFr -> BoolElt

Given a polynomial \(f\) from an algebra \(F\), together with an ideal \(I\) of \(F\), return whether \(f\) is in \(I\).

NormalForm(f, I): AlgFrElt, AlgFr -> AlgFrElt

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 \(I\).

NormalForm(f, S): AlgFrElt, [ AlgFrElt ] -> AlgFrElt

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.

f notin I: AlgFrElt, AlgFr -> BoolElt

Given a polynomial \(f\) from an algebra \(F\), together with an ideal \(I\) of \(F\), return whether \(f\) is not in \(I\).

Example: Normal Forms Modulo an Ideal (ex-3dd8c1)

We demonstrate the element operations with respect to an ideal of \({\mathbb{Q}}[x, y, z]\).

> 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

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