Proposition 13

For the algebra $A \oplus E(\delta)$ to be associative and unital, it is necessary and sufficient that $A$ be associative and unital, and that the mappings $(a, x) \mapsto a.x$ and $(a, x) \mapsto x.a$ define on $E$ an $(A, A)$-bimodule structure; the unit element of $A \oplus E(\delta)$ is then $(1, 0)$.

If an element $(u, m) \in A \oplus E(\delta)$ is written as unit element of this algebra, it is immediately found that $u$ must be the unit element of $A$; writing $(u, m).(0, x) = (0, x).(u, m) = (0, x)$, we obtain $u.x = x.u = x$ for $x \in E$ and, writing $(u, m).(u, 0) = (u, 0).(u, m) = (u, 0)$, we obtain $m = 0$. The fact that $A$ is associative when $A \oplus E(\delta)$ is follows from the fact that the augmentation is a surjective homomorphism. The condition $(x.a').a'' = x.(a'.a'')$ is then equivalent to $((0, x)(a', 0))(a'', 0) = (0, x)((a', 0)(a'', 0))$ and similarly the condition $a.(a'.x) = (a.a').x$ is equivalent to

finally the condition $a.(x.a') = (a.x).a'$ is equivalent to