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 -> 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.