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.
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.
The order which maps the associative ideal I to itself under left (right) multiplication.
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.
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.
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.
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.
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 )