Proposition 3

Let $d : A \to E$ be an $\varepsilon$-derivation of degree $\delta$. Then $\mathrm{Ker}(d)$ is a graded subalgebra of A; if A admits a unit element, it belongs to $\mathrm{Ker}(d)$.

Clearly $\mathrm{Ker}(d)$ is a graded sub-K-module of A; further, relation (8) of no. 2 shows that, if $x$ and $y$ are two homogeneous elements belonging to $\mathrm{Ker}(d)$, then $d(xy) = 0$ and hence $xy \in \mathrm{Ker}(d)$. Finally, if A admits a unit element 1 (of degree 0, cf. § 3, no. 1), relation (8) of no. 2, where $x$ and $y$ are replaced by 1, gives $d(1) = d(1) + d(1)$ and hence $d(1) = 0$.