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# Ideals of Finite Algebras

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

  Bases: `sage.rings.ideal.Ideal_generic`

  An ideal of a `FiniteDimensionalAlgebra`.

  INPUT:

  - `A` – a finite-dimensional algebra
  - `gens` – the generators of this ideal
  - `given_by_matrix` – (default: `False`) whether the basis matrix is given by `gens`

  EXAMPLES:

      sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
      sage: A.ideal(A([0,1]))
      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: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
        sage: I = A.ideal(A([1,1]))
        sage: I.basis_matrix()
        [1 0]
        [0 1]

  `vector_space`()

    Return `self` as a vector space.

    EXAMPLES:

        sage: A = FiniteDimensionalAlgebra(GF(3), [Matrix([[1, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])])
        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]

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