We were discussing about the possibility of having an algebra over a field which is associative but has not the unity. Does it exist? 
It has been proposed as a counterexample the set of even numbers. We believe it is wrong because it can't be a vector space.
If an associative but not unital algebra exists, how can we write its associative property?
We usually write it as a commutative diagram or in terms of maps composition. In these terms the identity seams unavoidable.
i.e. 
$$ 
m\circ\left(m\otimes id\right)  =  m\circ\left(id\otimes m\right)\qquad \text{associativity}
$$
with $m : A\otimes A\to A$.