# Ideals of Finite Algebras¶

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

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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