Editing Associative algebra (section)

== Dual of an associative algebra ==
Let ''A'' be an associative algebra over a commutative ring ''R''. Since ''A'' is in particular a module, we can take the dual module ''A''<sup>*</sup> of ''A''. A priori, the dual ''A''<sup>*</sup> need not have a structure of an associative algebra. However, ''A'' may come with an extra structure (namely, that of a Hopf algebra) so that the dual is also an associative algebra.

For example, take ''A'' to be the ring of continuous functions on a compact group ''G''. Then, not only ''A'' is an associative algebra, but it also comes with the co-multiplication {{nowrap|1=Δ({{itco|''f''}})(''g'', ''h'') = {{itco|''f''}}(''gh'')}} and co-unit {{nowrap|1=''ε''({{itco|''f''}}) = {{itco|''f''}}(1)}}.{{sfn|Tjin|1992|loc=Example 1|ps=none}} The "co-" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom. Hence, the dual ''A''<sup>*</sup> is an associative algebra. The co-multiplication and co-unit are also important in order to form a tensor product of representations of associative algebras (see ''{{slink|#Representations}}'' below).