Editing Associative algebra (section)

== Related concepts ==
=== Coalgebras ===
{{Main|Coalgebra}}
An associative algebra over ''K'' is given by a ''K''-vector space ''A'' endowed with a bilinear map {{nowrap|''A'' × ''A'' → ''A''}} having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism {{nowrap|''K'' → ''A''}} identifying the scalar multiples of the multiplicative identity. If the bilinear map {{nowrap|''A'' × ''A'' → ''A''}} is reinterpreted as a linear map (i.e., [[morphism]] in the category of ''K''-vector spaces) {{nowrap|''A'' ⊗ ''A'' → ''A''}} (by the [[Tensor product#Characterization by a universal property|universal property of the tensor product]]), then we can view an associative algebra over ''K'' as a ''K''-vector space ''A'' endowed with two morphisms (one of the form {{nowrap|''A'' ⊗ ''A'' → ''A''}} and one of the form {{nowrap|''K'' → ''A''}}) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using [[categorial duality]] by reversing all arrows in the [[commutative diagram]]s that describe the algebra [[axiom]]s; this defines the structure of a [[coalgebra]].

There is also an abstract notion of [[F-coalgebra|''F''-coalgebra]], where ''F'' is a [[functor]]. This is vaguely related to the notion of coalgebra discussed above.