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Ideals of Orders

Subsections

Creation 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.
RandomRightIdeal(O) : AlgAssVOrd -> AlgAssVOrdIdl
Returns a "random" right ideal of the order O, generated by elements with small coefficients.

Attributes of Ideals

Algebra(I) : AlgAssVOrdIdl -> AlgAssV
The container algebra of the associative ideal I.
Order(I) : AlgAssVOrdIdl -> AlgAssVOrd
The associative order the associative ideal I was created as an ideal of.
LeftOrder(I) : AlgAssVOrdIdl[RngOrd] -> AlgAssVOrd
LeftOrder(I) : AlgAssVOrdIdl -> AlgAssVOrd
RightOrder(I) : AlgAssVOrdIdl[RngOrd] -> AlgAssVOrd
RightOrder(I) : AlgAssVOrdIdl -> AlgAssVOrd
The order which maps the associative ideal I to itself under left (right) multiplication.
Basis(I) : AlgAssVOrdIdl -> SeqEnum
Basis(I, R) : AlgAssVOrdIdl, Str -> SeqEnum
The basis of the associative ideal I. This will be returned as elements of the order or algebra R if this second argument is given, otherwise as elements of the algebra of I.
BasisMatrix(I) : AlgAssVOrdIdl -> AlgMatElt
BasisMatrix(I, R) : AlgAssVOrdIdl, Str -> AlgMatElt
The basis matrix of the associative ideal I. This will be with respect to the basis of the order or algebra R if this second argument is given, otherwise with respect to the basis of the order I was created as an ideal of.
PseudoBasis(I) : AlgAssVOrdIdl[RngOrd] -> SeqEnum
PseudoBasis(I, R) : AlgAssVOrdIdl[RngOrd], Str -> SeqEnum
Return a sequence of tuples of the coefficient ideals and the basis elements of the associative ideal I. If a second argument is given, an order or algebra R, then the basis elements will be in R, otherwise the algebra of I.
PseudoMatrix(I) : AlgAssVOrdIdl[RngOrd] -> PMat
PseudoMatrix(I, R) : AlgAssVOrdIdl[RngOrd], Str -> PMat
Return a pseudo matrix describing the basis of the associative ideal I. If a second argument is given, an order or algebra R, then the basis matrix will be with respect to the basis of R, otherwise the order I was created as an ideal of.
ZBasis(I) : AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]
Returns a Z-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.
Denominator(I) : AlgAssVOrdIdl -> RngElt
Return the denominator of the ideal I. This is the minimal element d of the coefficient ring of O such that d * I ⊆O where O is the order I was created as an ideal of.

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
Returns the product of a and I as an ideal.
Colon(J, I): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> PMat
If I, J are left ideals, returns the colon (J:I)={x ∈A: xI ⊂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 AlgAss_sumandadjoin (H81E9)


> 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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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013