# Filtered Algebras¶

class sage.categories.filtered_algebras.FilteredAlgebras(base_category)

The category of filtered algebras.

An algebra $$A$$ over a commutative ring $$R$$ is filtered if $$A$$ is endowed with a structure of a filtered $$R$$-module (whose underlying $$R$$-module structure is identical with that of the $$R$$-algebra $$A$$) such that the indexing set $$I$$ (typically $$I = \NN$$) is also an additive abelian monoid, the unity $$1$$ of $$A$$ belongs to $$F_0$$, and we have $$F_i \cdot F_j \subseteq F_{i+j}$$ for all $$i, j \in I$$.

EXAMPLES:

sage: Algebras(ZZ).Filtered()
Category of filtered algebras over Integer Ring
sage: Algebras(ZZ).Filtered().super_categories()
[Category of algebras over Integer Ring,
Category of filtered modules over Integer Ring]


REFERENCES:

class ParentMethods
graded_algebra()

Return the associated graded algebra to self.

Todo

Implement a version of the associated graded algebra which does not require self to have a distinguished basis.

EXAMPLES:

sage: A = AlgebrasWithBasis(ZZ).Filtered().example()