====== Associative unital algebra ======

An **associative unital algebra** over a commutative unital ring is a monoid internal to the category of modules over that ring with respect to the usual tensor product of such modules.

===== Definition =====

Let $R$ be a commutative unital ring. All $R$-modules considered are supposed to be unital. Let $\otimes$ be a tensor product functor for $R$-modules and let $\alpha$ be the corresponding associator $(\cdot_1\otimes \cdot_2)\otimes \cdot_3\to \cdot_1\otimes(\cdot_2\otimes \cdot_3)$ and $\ell: R\otimes (\cdot)\to (\cdot)$ and $r:(\cdot)\to R\otimes (\cdot)$ the left and right unitors. 

An **associative unital algebra over** $R$ is any triple $(A,m,\eta)$ such that
  * $A$ is an $R$-module,
  * $m$ is an $R$-module morphism $A\otimes A\to A$ (//multiplication// map), 
  * $\eta$ is an $R$-module morphism $R\to A$ (//unit// map) 
and such that the following conditions are met:
  * $m\circ (\mathrm{id}_A\otimes m) \circ \alpha_{A,A,A}=m\circ (m\otimes \mathrm{id}_A)$ (//associativity// axiom),
  * $m\circ (\eta\otimes \mathrm{id}_A)=\ell_A$ (//left unit// axiom),
  * $m\circ (\mathrm{id}_A\otimes \eta)=r_A$ (//right unit// axiom).