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.
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.
sage: from sage.categories.associative_algebras import AssociativeAlgebras sage: C = AssociativeAlgebras(ZZ); C Category of associative algebras over Integer Ring
sage: from sage.categories.magmatic_algebras import MagmaticAlgebras sage: C is MagmaticAlgebras(ZZ).Associative() True sage: TestSuite(C).run()
An abstract class for elements of an associative algebra.
Magmas.Element.__mul__ is preferable to Modules.Element.__mul__ since the later does not handle products of two elements of self.
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]