# Associative algebras¶

class sage.categories.associative_algebras.AssociativeAlgebras(base_category)

The category of associative algebras over a given base ring.

An associative algebra over a ring $$R$$ is a module over $$R$$ which is also a not necessarily unital ring.

Warning

Until trac ticket #15043 is implemented, Algebras is the category of associative unital algebras; thus, unlike the name suggests, AssociativeAlgebras is not a subcategory of Algebras but of MagmaticAlgebras.

EXAMPLES:

sage: from sage.categories.associative_algebras import AssociativeAlgebras
sage: C = AssociativeAlgebras(ZZ); C
Category of associative algebras over Integer Ring


TESTS:

sage: from sage.categories.magmatic_algebras import MagmaticAlgebras
sage: C is MagmaticAlgebras(ZZ).Associative()
True
sage: TestSuite(C).run()

class ElementMethods

An abstract class for elements of an associative algebra.

Note

Magmas.Element.__mul__ is preferable to Modules.Element.__mul__ since the later does not handle products of two elements of self.

TESTS:

sage: A = AlgebrasWithBasis(QQ).example()
sage: a = A.an_element()
sage: a
2*B[word: ] + 2*B[word: a] + 3*B[word: b]
sage: a.__mul__(a)
4*B[word: ] + 8*B[word: a] + 4*B[word: aa] + 6*B[word: ab] + 12*B[word: b] + 6*B[word: ba] + 9*B[word: bb]

AssociativeAlgebras.Unital

alias of Algebras

#### Previous topic

Algebras With Basis

Bialgebras