This section describes operations on affine algebras. Most of the operations are very similar to those for multivariate polynomial rings; such operations are done by mapping the computation to the preimage ideal and then by mapping the result back into the affine algebra. See the corresponding functions for the multivariate polynomial rings for details.
Given an affine algebra Q, return the i-th indeterminate of Q as an element of Q.
Return the coefficient ring of the affine algebra Q.
Return the rank of the affine algebra Q (the number of indeterminates of Q).
Given an ideal I of an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the ideal J.
Given an ideal I of an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the ideal I' of P such that the image of I' under the natural epimorphism P -> Q is I.
Given an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the polynomial ring P.
Return the generic polynomial ring P such that Q is P/J for some ideal J of P.
Given two ideals I and J of the same affine algebra Q, return true if and only if I and J are equal.
Given two ideals I and J of the same affine algebra Q, return true if and only if I is contained in J.
Given two ideals I and J of the same affine algebra Q, return the sum I + J.
Given two ideals I and J of the same affine algebra Q, return the product I * J.
Given an ideal I of an affine algebra Q and an integer n, return the power In.
Given two ideals I and J of the same affine algebra Q, return the intersection I∩J.
Given an ideal I of the affine algebra Q, return whether I is proper; that is, whether I is strictly contained in Q.
Given an ideal I of the affine algebra Q, return whether I is the zero ideal. Note that this is equivalent to whether the preimage ideal of I is the divisor ideal of Q.
Given an ideal I of the affine algebra Q, return whether I is a prime ideal.
Given an ideal I of the affine algebra Q, return whether I is a primary ideal.
Given an ideal I of the affine algebra Q, return whether I is a radical ideal.
Given an ideal I of the affine algebra Q, return the primary decomposition of I, together with the associated primes.
Given an ideal I of the affine algebra Q, return the (prime) decomposition of the radical of I.
> Q := RationalField();
> A<x,y,z> := AffineAlgebra<Q,x,y,z | x^2 - y + 1, y^3 + z - 1>;
> A;
Affine Algebra of rank 3 over Rational Field
Lexicographical Order
Variables: x, y, z
Quotient relations:
[
x^2 - y + 1,
y^3 + z - 1
]
> I := ideal<A | x^3*y*z^2>;
> IsRadical(I);
false
> Radical(I);
Affine Algebra of rank 3 over Rational Field
Lexicographical Order
Variables: x, y, z
Quotient relations:
[
x^2 - y + 1,
y^3 + z - 1
]
Generating basis:
[
x*y^2 + x*y - x*z + x,
y*z,
z^2 - z
]
> PQ, PP := PrimaryDecomposition(I);
> #PQ;
3
> PQ[1];
Affine Algebra of rank 3 over Rational Field
Lexicographical Order
Variables: x, y, z
Quotient relations:
[
x^2 - y + 1,
y^3 + z - 1
]
Generating basis:
[
y + 5/81*z^3 + 1/9*z^2 + 1/3*z - 1,
x*z^3,
y + 5/81*z^3 + 1/9*z^2 + 1/3*z - 1,
z^4
]
> PP[1];
Affine Algebra of rank 3 over Rational Field
Lexicographical Order
Variables: x, y, z
Quotient relations:
[
x^2 - y + 1,
y^3 + z - 1
]
Generating basis:
[
x,
y - 1,
z
]