Ideals of Finite Dimensional Algebras - Algebras

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# Ideals of Finite Dimensional Algebras

It is necessary to use algebras in the category of associative algebras.

class sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_ideal.FiniteDimensionalAlgebraIdeal(A, gens=None, given_by_matrix=False)

  Bases: `Ideal_generic`

  An ideal of a `FiniteDimensionalAlgebra`.

  INPUT:

  - `A` – a finite-dimensional algebra

  - `gens` – the generators of this ideal

  - `given_by_matrix` – boolean (default: `False`); whether the basis matrix is given by `gens`

  EXAMPLES:

  Sage

      sage: cat = CommutativeAlgebras(GF(3)).FiniteDimensional().WithBasis()
      sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]),
      ....:                                      Matrix([[0, 1], [0, 0]])],
      ....:                              category=cat)
      sage: A.ideal(A([0,1]))
      Ideal (e1) of Finite-dimensional algebra of degree 2 over Finite Field of size 3

  Python

      >>> from sage.all import *
      >>> cat = CommutativeAlgebras(GF(Integer(3))).FiniteDimensional().WithBasis()
      >>> A = FiniteDimensionalAlgebra(GF(Integer(3)), [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
      ...                                      Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])],
      ...                              category=cat)
      >>> A.ideal(A([Integer(0),Integer(1)]))
      Ideal (e1) of Finite-dimensional algebra of degree 2 over Finite Field of size 3

  basis_matrix()

    Return the echelonized matrix whose rows form a basis of `self`.

    EXAMPLES:

    Sage

        sage: cat = CommutativeAlgebras(GF(3)).FiniteDimensional().WithBasis()
        sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]),
        ....:                                      Matrix([[0, 1], [0, 0]])],
        ....:                              category=cat)
        sage: I = A.ideal(A([1,1]))
        sage: I.basis_matrix()
        [1 0]
        [0 1]

    Python

        >>> from sage.all import *
        >>> cat = CommutativeAlgebras(GF(Integer(3))).FiniteDimensional().WithBasis()
        >>> A = FiniteDimensionalAlgebra(GF(Integer(3)), [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
        ...                                      Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])],
        ...                              category=cat)
        >>> I = A.ideal(A([Integer(1),Integer(1)]))
        >>> I.basis_matrix()
        [1 0]
        [0 1]

  vector_space()

    Return `self` as a vector space.

    EXAMPLES:

    Sage

        sage: cat = CommutativeAlgebras(GF(3)).FiniteDimensional().WithBasis()
        sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]),
        ....:                                      Matrix([[0, 1], [0, 0]])],
        ....:                              category=cat)
        sage: I = A.ideal(A([1,1]))
        sage: I.vector_space()
        Vector space of degree 2 and dimension 2 over Finite Field of size 3
        Basis matrix:
        [1 0]
        [0 1]

    Python

        >>> from sage.all import *
        >>> cat = CommutativeAlgebras(GF(Integer(3))).FiniteDimensional().WithBasis()
        >>> A = FiniteDimensionalAlgebra(GF(Integer(3)), [Matrix([[Integer(1), Integer(0)], [Integer(0), Integer(1)]]),
        ...                                      Matrix([[Integer(0), Integer(1)], [Integer(0), Integer(0)]])],
        ...                              category=cat)
        >>> I = A.ideal(A([Integer(1),Integer(1)]))
        >>> I.vector_space()
        Vector space of degree 2 and dimension 2 over Finite Field of size 3
        Basis matrix:
        [1 0]
        [0 1]