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).
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.
The principal left (right) ideal of the associative order O generated by the element e.
In this section, I is an ideal of some order in an associative algebra.
For an ideal I created as an ideal of an order O, this returns O.
This returns Algebra(Order(I)).
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 ⊆I }.
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.
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".
In this section, I is an ideal of an order O in an associative algebra.
This returns an element d of BaseRing(O) such that d I ⊆O, where O is Order(I).
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 < Ii, bi > where Ii is an ideal of R and bi is an element of Algebra(I), such that I is the direct sum of the R-modules Ii bi. In particular, the bi form a basis of Algebra(I) as a vector space over the field of fractions of R. Note that the bi may not lie in I.
Returns the pseudomatrix corresponding to PseudoBasis(I).
When I is an ideal of an order over Z, this returns a Z-module basis of I.When the base ring of Order(I) is an extension of 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.
This returns Basis(I) as a row matrix.
Returns a Z-module basis for the ideal I.
Returns a sequence of generators for the ideal I as a module over its base ring.
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.
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 tensor F eq A where F is the base field of A.
The sum of the ideals I and J, which are ideals which share a side in equal orders.
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.
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.
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.
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 )