Ideals and Quotient Rings

A differential ideal I⊂R of a differential ring R is an ideal of R that is closed under the derivation of R. However, we consider a differential ideal as an ideal of the underlying ring of R. More specifically, ideals of differential rings are restricted to those rings whose underlying rings are multivariate polynomial rings.

Defining Ideals and Quotient Rings

DifferentialIdeal(L) : [RngDiffElt] -> RngMPol

Given a sequence L with entries in a differential ring R, return the differential ideal generated by the entries of L as an ideal of the underlying ring of R. The underlying ring of R must be of type RngMPol. At first the elements of L may generate an ideal which is not closed under the derivation of R. By adding as many derivatives of the elements to the set of generators of the ideal as needed, one obtains a full set of generators for the calculated differential ideal.

QuotientRing(R, I) : RngDiff, RngMPol -> RngDiff, Map

Given a differential ring R and a differential ideal I, return the differential quotient ring Q=R/I. The derivation of Q is induced by the derivation of R. It maps Q.i to Q ! δ_R(R.i), for i=1, 2, ..., m where m is the number of generators of Q (or R). The induced quotient map from R to Q is also returned.

> P := PolynomialRing(Rationals(),1);
> f := map<P->P | a:->a*Derivative(a,1)>;
> R<T> := DifferentialRing(P, f, Rationals());
> L := [T^2+T-1];
> I := DifferentialIdeal(L);
> I;
Ideal of Polynomial ring of rank 1 over Rational Field
Lexicographical Order
Variables: T
Basis:
[
    T^2 + T - 1,
]
> Q<X>, toQ := QuotientRing(R,I);
> Q;
Differential Ring of Affine Algebra of rank 1 over Rational Field
Lexicographical Order
Variables: X
Quotient relations:
[
    X^2 + X - 1
]
with derivation given by Mapping from: Affine Algebra of rank 1 over Rational 
    Field to Affine Algebra of rank 1 over Rational Field given by a rule [no 
inverse]
> toQ(T);
X
> Derivative(T^2);
2*T^3
> Derivative(X^2);
X