Operations on Ideals

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).

Basic Operations

I + J: RngMPolLoc, RngMPolLoc -> RngMPolLoc: 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 \(J\).

I * J: RngMPolLoc, RngMPolLoc -> RngMPolLoc: 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 \(J\).

I ^ k: RngMPolLoc, RngIntElt -> RngMPolLoc: Given an ideal \(I\) of the polynomial ring \(P\), and an integer \(k\), return the \(k\)-th power of \(I\).

QuotientDimension(I): RngMPol -> RngIntElt: 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).

Generic(I): RngMPolLoc -> RngMPolLoc: Given an ideal \(I\) of a generic local polynomial ring \(R\), return \(R\).

LeadingMonomialIdeal(I): RngMPolLoc -> RngMPolLoc: Given an ideal \(I\), return the leading monomial ideal of \(I\); that is, the ideal generated by all the leading monomials of I.

I meet J: RngMPolLoc, RngMPolLoc -> RngMPolLoc: Given ideals \(I\) and \(J\) of the same polynomial ring \(P\), return the intersection of \(I\) and \(J\).

&meet S: [ RngMPolLoc ] -> RngMPolLoc: Given a set or sequence \(S\) of ideals of the same local polynomial ring \(R\), return the intersection of all the ideals of \(S\).

Ideal Predicates

I eq J: RngMPolLoc, RngMPolLoc -> BoolElt: Given two ideals \(I\) and \(J\) of the same polynomial ring \(P\), return whether \(I\) and \(J\) are equal.

I ne J: RngMPolLoc, RngMPolLoc -> BoolElt: Given two ideals \(I\) and \(J\) of the same polynomial ring \(P\), return whether \(I\) and \(J\) are not equal.

I notsubset J: RngMPolLoc, RngMPolLoc -> BoolElt: Given two ideals \(I\) and \(J\) in the same polynomial ring \(P\) return whether \(I\) is not contained in \(J\).

I subset J: RngMPolLoc, RngMPolLoc -> BoolElt: Given two ideals \(I\) and \(J\) in the same polynomial ring \(P\) return whether \(I\) is contained in \(J\).

IsZero(I): RngMPolLoc -> BoolElt: Given an ideal \(I\) of the local polynomial ring \(R\), return whether \(I\) is the zero ideal (contains zero alone).

IsProper(I): RngMPolLoc -> BoolElt: 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).

IsZeroDimensional(I): RngMPolLoc -> BoolElt: 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.

Example: Ideal Arithmetic in a Local Ring (ex-b95527)

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We construct some ideals in \({\mathbb{Q}}[x, y, z]\) and perform basic arithmetic on them.

> 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

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Operations on Elements of Ideals

f in I: RngMPolLocElt, RngMPolLoc -> BoolElt: Given a polynomial \(f\) from a local polynomial ring \(R\), together with an ideal \(I\) of \(R\), return whether \(f\) is in \(I\).

NormalForm(f, I): RngMPolLocElt, RngMPolLoc -> RngMPolLocElt: 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 \(I\).

f notin I: RngMPolLocElt, RngMPolLoc -> BoolElt: Given a polynomial \(f\) from a polynomial ring \(P\), together with an ideal \(I\) of \(P\), return whether \(f\) is not in \(I\).

Example: Element Operations with Local Ideals (ex-e2cb48)

We demonstrate the element operations with respect to an ideal of the localization of \({\mathbb{Q}}[x, y, z]\).

> 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

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Operations on Ideals#

Basic Operations#

Ideal Predicates#

Operations on Elements of Ideals#

Operations on Ideals

Basic Operations

Ideal Predicates

Operations on Elements of Ideals