Operations on FP-Algebras

This section describes operations on fp-algebras. Most of the operations are very similar to those for noncommutative free algebras; such operations are done by mapping the computation to the preimage ideal and then by mapping the result back into the fp-algebra. See the corresponding functions for the noncommutative free algebras for details.

A . i: AlgFP, RngIntElt -> AlgFPElt

Given an fp-algebra \(A\), return the \(i\)-th indeterminate of \(A\) as an element of \(A\).

CoefficientRing(A): AlgFP -> Rng

Return the coefficient ring of the fp-algebra \(A\).

Rank(A): AlgFP -> RngIntElt

Return the rank of the fp-algebra \(A\) (the number of indeterminates of \(A\)).

DivisorIdeal(I): AlgFP -> AlgFr

Given an ideal \(I\) of an fp-algebra \(A\) which is the quotient ring \(F/J\), where \(F\) is a free algebra and \(J\) an ideal of \(F\), return the ideal \(J\).

PreimageIdeal(I): AlgFP -> AlgFr

Given an ideal \(I\) of an fp-algebra \(A\) which is the quotient ring \(F/J\), where \(F\) is a free algebra and \(J\) an ideal of \(F\), return the ideal \(I'\) of \(F\) such that the image of \(I'\) under the natural epimorphism \(F\rightarrow A\) is \(I\).

PreimageRing(A): AlgFP -> AlgFr

Given an fp-algebra \(A\) which is the quotient ring \(F/J\), where \(F\) is a free algebra and \(J\) an ideal of \(F\), return the free algebra \(F\).

OriginalRing(A): AlgFP -> Rng

Return the generic free algebra \(F\) such that \(A\) is \(F/J\) for some ideal \(J\) of \(F\).

IsCommutative(A): AlgFP -> BoolElt

Return whether the algebra \(A\) is commutative.

I eq J: AlgFP, AlgFP -> BoolElt

Given two ideals \(I\) and \(J\) of the same fp-algebra \(A\), return true if and only if \(I\) and \(J\) are equal.

I subset J: AlgFP, AlgFP -> BoolElt

Given two ideals \(I\) and \(J\) of the same fp-algebra \(A\), return true if and only if \(I\) is contained in \(J\).

I + J: AlgFP, AlgFP -> AlgFP

Given two ideals \(I\) and \(J\) of the same fp-algebra \(A\), return the sum \(I+J\).

I * J: AlgFP, AlgFP -> AlgFP

Given two ideals \(I\) and \(J\) of the same fp-algebra \(A\), return the product \(I*J\).

IsProper(I): AlgFP -> BoolElt

Given an ideal \(I\) of the fp-algebra \(A\), return whether \(I\) is proper; that is, whether \(I\) is strictly contained in \(A\).

IsZero(I): AlgFP -> BoolElt

Given an ideal \(I\) of the fp-algebra \(A\), return whether \(I\) is the zero ideal. Note that this is equivalent to whether the preimage ideal of \(I\) is the divisor ideal of \(A\).