In the following, note that since ideals of a full polynomial ring P are regarded as subrings of P, the ring P itself is a valid ideal as well (the ideal containing 1).
Given ideals I and J of the same polynomial ring P, return the sum of I and J, which is the ideal generated by the generators of I and those of J.
Given ideals I and J of the same polynomial ring P, return the product of I and J, which is the ideal generated by the products of the generators of I and those of J.
Given an ideal I of the polynomial ring P, and an integer k, return the k-th power of I.
Given an ideal I of a local polynomial ring R over a field K, return the dimension of P/I as a K-vector space. Note that this is quite different from the function Dimension below (which returns the Krull dimension of an ideal).
Given an ideal I of a generic local polynomial ring R, return R.
Given an ideal I, return the leading monomial ideal of I; that is, the ideal generated by all the leading monomials of I.
Given ideals I and J of the same polynomial ring P, return the intersection of I and J.
Given a set or sequence S of ideals of the same local polynomial ring R, return the intersection of all the ideals of S.
Given two ideals I and J of the same polynomial ring P, return whether I and J are equal.
Given two ideals I and J of the same polynomial ring P, return whether I and J are not equal.
Given two ideals I and J in the same polynomial ring P return whether I is not contained in J.
Given two ideals I and J in the same polynomial ring P return whether I is contained in J.
Given an ideal I of the local polynomial ring R, return whether I is the zero ideal (contains zero alone).
Given an ideal I of the local polynomial ring R, return whether I is proper; that is, whether I is strictly contained in R (or whether the standard basis of I does not contain 1 alone).
Given an ideal I of the local polynomial ring R, return whether I is zero-dimensional (so the quotient of P by I has non-zero finite dimension as a vector space over the coefficient field -- see the section on dimension for further details). Note that the ring R has dimension -1, so it is not zero-dimensional.
> R<x,y,z> := LocalPolynomialRing(RationalField(), 3);
> I := ideal<R | x*y - z, x^3*z^2 - y^2, x*z^3 - x - y>;
> J := ideal<R | x*y - z, x^2*z - y, x*z^3 - x - y>;
> A := I * J;
> _ := StandardBasis(A);
> A;
Ideal of Localization of Polynomial Ring of rank 3 over Rational Field
Order: Local Lexicographical
Variables: x, y, z
Inhomogeneous, Dimension 0
Standard basis:
[
x^2 - y^2 + 2*x^3*z,
x*y + y^2 - x^3*z,
y^3,
x*z + y*z,
y*z,
z^2
]
> M := I meet J;
> M;
Ideal of Localization of Polynomial Ring of rank 3 over Rational Field
Order: Local Lexicographical
Variables: x, y, z
Homogeneous
Basis:
[
x + y,
y^2,
z
]
> A eq M;
false
> A subset M;
true
Given a polynomial f from a local polynomial ring R, together with an ideal I of R, return whether f is in I.
Given a polynomial f from a local polynomial ring R, together with an ideal I of R, return a normal form of f with respect to (the standard basis of) I. The normal form of f is zero if and only if f is in I.
Given a polynomial f from a polynomial ring P, together with an ideal I of P, return whether f is not in I.
> R<x,y,z> := LocalPolynomialRing(RationalField(), 3); > I := ideal<R | (x + y)^3, (y - z)^2, y^2*z + z>; > NormalForm(y^2*z + z, I); 0 > NormalForm(x^3, I); -3*x^2*y > x + y in I; false