Editing Associative algebra (section)

=== As a monoid object in the category of modules ===
The definition is equivalent to saying that a unital associative ''R''-algebra is a [[monoid (category theory)|monoid object]] in [[category of modules|'''''R''-Mod''']] (the [[monoidal category]] of ''R''-modules). By definition, a ring is a monoid object in the [[category of abelian groups]]; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the [[category of modules]].

Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra ''A''. For example, the associativity can be expressed as follows. By the universal property of a [[tensor product of modules]], the multiplication (the ''R''-bilinear map) corresponds to a unique ''R''-linear map
: <math>m : A \otimes_R A \to A</math>.
The associativity then refers to the identity:
: <math>m \circ ({\operatorname{id}} \otimes m) = m \circ (m \otimes \operatorname{id}).</math>