Ideals of Orders

Construction of Ideals

lideal<O | E>: AlgAssVOrd, [AlgAssVOrdElt] -> AlgAssVOrdIdl
rideal<O | E>: AlgAssVOrd, [AlgAssVOrdElt] -> AlgAssVOrdIdl
ideal<O | E>: AlgAssVOrd, [AlgAssVOrdElt] -> AlgAssVOrdIdl: For an associative order \(O\), this constructs the left, right, or two sided \(O\)-ideal generated by the elements in the given sequence \(E\) (these elements should be coercible into \(O\)).

lideal<O | M>: AlgAssVOrd, PMat -> AlgAssVOrdIdl
lideal<O | M>: AlgAssVOrd, Mtrx -> AlgAssVOrdIdl
rideal<O | M>: AlgAssVOrd, PMat -> AlgAssVOrdIdl
rideal<O | M>: AlgAssVOrd, Mtrx -> AlgAssVOrdIdl
ideal<O | M>: AlgAssVOrd, PMat -> AlgAssVOrdIdl
ideal<O | M>: AlgAssVOrd, Mtrx -> AlgAssVOrdIdl: Constructs a left, right, or two sided ideal of the associative order \(O\) whose basis is given by \(M\), which may be either a matrix or a pseudo matrix.

O * e: AlgAssVOrd, RngElt -> AlgAssVOrdIdl
e * O: RngElt, AlgAssVOrd -> AlgAssVOrdIdl: The principal left (right) ideal of the associative order \(O\) generated by the element \(e\).

Attributes of Ideals

In this section, \(I\) is an ideal of some order in an associative algebra.

Order(I): AlgAssVOrdIdl -> AlgAssVOrd: For an ideal \(I\) created as an ideal of an order \(O\), this returns \(O\).

Algebra(I): AlgAssVOrdIdl -> AlgAssV: This returns Algebra(Order(I)).

LeftOrder(I): AlgAssVOrdIdl -> AlgAssVOrd
RightOrder(I): AlgAssVOrdIdl -> AlgAssVOrd: For an ideal \(I\), this returns the (full) order in Algebra(I) which stabilizes \(I\) under multiplication on the left or right respectively. Explicitly, the left order of \(I\) is defined as \(\{ x : x I \subseteq I \}\).

ArithmeticRadical(O, p): AlgAssVOrd[RngOrd], RngOrdIdl -> AlgAssVOrdIdl
ArithmeticRadical(O, p): AlgQuat, RngElt -> AlgQuatOrdIdl: Given an order \(O\) over a number ring \(R\) and a prime ideal \(p\) in \(R\), return the Arithmetic Radical of \(O\) over \(p\). This is the unique two-sided ideal \(J\) of \(O\) over \(pO\) such that \(J/pO\) is the Jacobson radical of \(O/pO\).

RadicalIdealizer(O, p): AlgAssVOrd[RngOrd], RngOrdIdl -> AlgAssVOrd

RadicalIdealizer(O, p): AlgQuat, RngElt -> AlgQuat

Side: MonStgElt                    Default: "Intersection"

Returns either the left or right order of the arithmetical radical of \(O\) at \(p\) or the intersection of these two orders, depending on whether Side is "Left", "Right", or "Intersection".

Neighbors(I, p): AlgAssVOrdIdl, RngOrdIdl -> SeqEnum[AlgAssVOrdIdl]
Neighbors(I, p): AlgQuatOrdIdl, RngInt -> SeqEnum[AlgQuatOrdIdl]: Returns the \(p\)-neighbors of the ideal \(I\), namely the ideals \(J \subseteq I\) such that \({\rm Nm}(J) = p {\rm Nm}(I)\). Currently, only implemented for squarefree ideals \(p\) coprime to \({\rm Disc}(O) {\rm Nm}(I)\), where \(O\) is Order(I).

ElementsOfNorm(I, n): AlgAssVOrdIdl, RngElt -> SeqEnum

ModUnits: BoolElt                    Default: false

Returns the elements in \(I\) of norm \(n\). Currently, only implemented when \(I\) is an ideal in a definite quaternion algebra. If ModUnits is set to true, returns only one representative from every orbit under the action of the group of units of norm \(1\) in \(O\), where \(O\) is Order(I).

Bases of Ideals

In this section, \(I\) is an ideal of an order \(O\) in an associative algebra.

Denominator(I): AlgAssVOrdIdl -> RngElt: This returns an element \(d\) of BaseRing(O) such that \(d I \subseteq O\), where \(O\) is Order(I).

PseudoBasis(I): AlgAssVOrdIdl[RngOrd] -> SeqEnum: Given an ideal \(I\) of an order over \(R\), where \(R\) is an order in a number field, this returns data describing \(I\) as an \(R\)-module. It returns a sequence of tuples \(\langle I_i, b_i \rangle\) where \(I_i\) is an ideal of \(R\) and \(b_i\) is an element of Algebra(I), such that \(I\) is the direct sum of the \(R\)-modules \(I_i b_i\). In particular, the \(b_i\) form a basis of Algebra(I) as a vector space over the field of fractions of \(R\). Note that the \(b_i\) may not lie in \(I\).

PseudoMatrix(I): AlgAssVOrdIdl[RngOrd] -> PMat
PseudoMatrix(I): AlgAssVOrdIdl[RngFunOrd] -> PMat: Returns the pseudomatrix corresponding to PseudoBasis(I).

Basis(I): AlgAssVOrdIdl -> SeqEnum

When \(I\) is an ideal of an order over \({\mathbb{Z}}\), this returns a \({\mathbb{Z}}\)-module basis of \(I\).

When the base ring of Order(I) is an extension of \({\mathbb{Z}}\), this returns part of the data returned by PseudoBasis(I): it returns a basis of Algebra(I). Note that this determines nothing about \(I\), while the full data returned by PseudoBasis(I) determines \(I\).

BasisMatrix(I): AlgAssVOrdIdl -> AlgMatElt: This returns Basis(I) as a row matrix.

ZBasis(I): AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]: Returns a \({\mathbb{Z}}\)-module basis for the ideal \(I\).

Generators(I): AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]: Returns a sequence of generators for the ideal \(I\) as a module over its base ring.

LocalBasis(I, p): AlgAssVOrdIdl, RngOrdIdl -> [AlgAssElt]

Type: MonStgElt                    Default: ""

Given an ideal \(I\) of an \(R\)-order in an algebra \(A\), this function returns a basis of a free \(R\)-module \(F\) in \(A\) such that \(I\) and \(F\) agree at the completion at the prime ideal \(p\) of \(R\). If Type is specified, it must either be "Submodule" or "Supermodule". In which case \(F\) is either a sub- or supermodule of \(I\).

PseudoBasis(I, R): AlgAssVOrdIdl[RngOrd], Str -> SeqEnum
PseudoMatrix(I, R): AlgAssVOrdIdl[RngOrd], Str -> PMat
Basis(I, R): AlgAssVOrdIdl, Str -> SeqEnum
BasisMatrix(I, R): AlgAssVOrdIdl, Str -> AlgMatElt: In these variants of the functions described above, the elements of the basis are expressed with respect to the basis of \(R\), where \(R\) is some ring contained in \(A =\) Algebra(I) such that \(R \otimes F = A\) where \(F\) is the base field of \(A\).

Arithmetic for Ideals

I + J: AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl: The sum of the ideals \(I\) and \(J\), which are ideals which share a side in equal orders.

I * J: AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl, AlgAssVOrdIdl
I * J: AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> AlgAssVOrdIdl, AlgAssVOrdIdl: The product of the ideals \(I\) and \(J\), where \(I\) is a right ideal and \(J\) is a left ideal of the same order \(O\). Returns the product given the structure of left and right ideal.

a * I: RngElt, AlgAssVOrdIdl -> AlgAssVOrdIdl
I * a: AlgAssVOrdIdl, RngElt -> AlgAssVOrdIdl
a * I: RngElt, AlgQuatOrdIdl -> AlgQuatOrdIdl
I * a: AlgQuatOrdIdl, RngElt -> AlgQuatOrdIdl: Returns the product of \(a\) and \(I\) as an ideal.

A * I: RngOrdFracIdl, AlgAssVOrdIdl[RngOrd] -> AlgAssVOrdIdl
I * A: RngOrdFracIdl, AlgAssVOrdIdl[RngOrd] -> AlgAssVOrdIdl: Returns the product of a fractional ideal \(A\) of the base ring with \(I\) as an ideal.

Colon(J, I): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> PMat: If \(I,J\) are left ideals, returns the colon \((J:I)=\{x \in A: xI \subset J\}\), similarly defined if \(I,J\) are right ideals.

MultiplicatorRing(I): AlgAssVOrdIdl -> AlgAssVOrd: Returns the colon \((I:I)\) of the ideal \(I\), the set of all elements which multiply \(I\) into \(I\).

Predicates on Ideals

IsLeftIdeal(I): AlgAssVOrdIdl -> BoolElt
IsRightIdeal(I): AlgAssVOrdIdl -> BoolElt
IsTwoSidedIdeal(I): AlgAssVOrdIdl -> BoolElt: Return true if the associative ideal \(I\) is a left, right, or two sided ideal (respectively).

I eq J: AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt: Return true if the associative ideals \(I\) and \(J\) are equal.

I subset J: AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt: Returns true if and only if the ideal \(I\) is contained in the ideal \(J\).

a in I: AlgAssVElt, AlgAssVOrdIdl -> BoolElt
a notin I: AlgAssVElt, AlgAssVOrdIdl -> BoolElt: Return true (false) if the element \(a\) of an associative algebra is contained in the associative ideal \(I\).

Other Operations on Ideals

Norm(I): AlgAssVOrdIdl[RngOrd] -> RngOrdIdl: Returns the norm of the ideal \(I\), the ideal of the base number ring of \(I\) generated by the norms of the elements in \(I\).

Example: One-Sided Ideals of an Order (ex-d618e7)

Run in calculator (opens in a new tab)

> F<w> := CyclotomicField(3);
> R := MaximalOrder(F);
> A := Algebra(FPAlgebra<F, x, y | x^3-3, y^3+5, y*x-w*x*y>);
> O := Order([A.i : i in [1..9]]);
> MinimalPolynomial(O.2);
$.1^3 + 5/1*R.1
> I := rideal<O | O.2>;
> IsLeftIdeal(I), IsRightIdeal(I), IsTwoSidedIdeal(I);
false true false
> MultiplicatorRing(I) eq O;
true
> PseudoBasis(I);
[
    <Principal Ideal of R
    Generator:
        R.1, (0 R.1 0 0 0 0 0 0 0)>,
    <Principal Ideal of R
    Generator:
        R.1, (0 0 0 R.1 0 0 0 0 0)>,
    <Principal Ideal of R
    Generator:
        R.1, (0 0 0 0 -R.1 - R.2 0 0 0 0)>,
    <Principal Ideal of R
    Generator:
        R.1, (-5/1*R.1 0 0 0 0 0 0 0 0)>,
    <Principal Ideal of R
    Generator:
        R.1, (0 0 0 0 0 0 -R.1 - R.2 0 0)>,
    <Principal Ideal of R
    Generator:
        R.1, (0 0 0 0 0 0 0 R.2 0)>,
    <Principal Ideal of R
    Generator:
        R.1, (0 0 5/1*R.1 + 5/1*R.2 0 0 0 0 0 0)>,
    <Principal Ideal of R
    Generator:
        R.1, (0 0 0 0 0 0 0 0 R.2)>,
    <Principal Ideal of R
    Generator:
        R.1, (0 0 0 0 0 -5/1*R.2 0 0 0)>
]
> ZBasis(I);
[ [0 R.1 0 0 0 0 0 0 0], [0 R.2 0 0 0 0 0 0 0], [0 0 0 R.1 0 0 0 0 0], [0 0 0
    R.2 0 0 0 0 0], [0 0 0 0 -R.1 - R.2 0 0 0 0], [0 0 0 0 R.1 0 0 0 0],
    [-5/1*R.1 0 0 0 0 0 0 0 0], [-5/1*R.2 0 0 0 0 0 0 0 0] ]
> Norm(I);
Principal Ideal of R
Generator:
    15625/1*R.1
> J := rideal<O | O.3>;
> Norm(J);
Principal Ideal of R
Generator:
    729/1*R.1
> A!1 in I+J;
false
> Denominator(1/6*I);
[1, 0]
> Colon(J,I);
Pseudo-matrix over Maximal Equation Order with defining polynomial x^2 + x + 1
over its ground order
Principal Ideal of R
Generator:
    3/1*R.1 * ( R.1 0 0 0 0 0 0 0 0 )
Principal Ideal of R
Generator:
    3/1*R.1 * ( 0 R.1 0 0 0 0 0 0 0 )
Principal Ideal of R
Generator:
    R.1 * ( 0 0 R.1 0 0 0 0 0 0 )
Fractional Principal Ideal of R
Generator:
    3/5*R.1 * ( 0 0 0 R.1 0 0 0 0 0 )
Principal Ideal of R
Generator:
    R.1 * ( 0 0 0 0 R.1 0 0 0 0 )
Principal Ideal of R
Generator:
    R.1 * ( 0 0 0 0 0 R.1 0 0 0 )
Fractional Principal Ideal of R
Generator:
    -1/5*R.1 * ( 0 0 0 0 0 0 R.1 0 0 )
Principal Ideal of R
Generator:
    R.1 * ( 0 0 0 0 0 0 0 R.1 0 )
Fractional Principal Ideal of R
Generator:
    1/5*R.1 * ( 0 0 0 0 0 0 0 0 R.1 )

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Ideals of Orders#

Construction of Ideals#

Attributes of Ideals#

Bases of Ideals#

Arithmetic for Ideals#

Predicates on Ideals#

Other Operations on Ideals#