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Subsections
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, 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.
e * O : RngElt, AlgAssVOrd -> AlgAssVOrdIdl
The principal left (right) ideal of the associative order O generated by the
element e.
Returns a "random" right ideal of the order O, generated by
elements with small coefficients.
The container algebra of the associative ideal I.
The associative order the associative ideal I was created as an ideal of.
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, 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, 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, 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, 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.
Returns a Z-basis for the ideal I.
Returns a sequence of generators for the ideal I as a module over its base ring.
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.
The sum of the ideals I and J, which are ideals which share a side
in equal orders.
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.
I * a: AlgAssVOrdIdl, RngElt -> AlgAssVOrdIdl
Returns the product of a and I as an ideal.
If I, J are left ideals, returns the colon
(J:I)={x ∈A: xI ⊂J}, similarly defined if I, J
are right ideals.
Returns the colon (I:I) of the ideal I, the set of all elements
which multiply I into I.
IsRightIdeal(I) : AlgAssVOrdIdl -> BoolElt
IsTwoSidedIdeal(I) : AlgAssVOrdIdl -> BoolElt
Return true if the associative ideal I is a left, right or two sided ideal
(respectively).
Return true if the associative ideals I and J are equal.
Returns true if and only if the ideal I is contained in the
ideal J.
a notin I : AlgAssVElt, AlgAssVOrdIdl -> BoolElt
Return true (false) if the element a of an associative algebra is
contained in the associative ideal I.
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.
> 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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