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: Element Operations (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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