# Ideals of Finite 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` – (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]
```