Corollary

*Suppose that $\mathbf{A}$ is a unital commutative associative algebra and $\mathbf{E}$ an $\mathbf{A}$-module. If $d : \mathbf{A} \to \mathbf{E}$ is a derivation, then, for every integer $n \geq 0$,

It suffices to give $\mathbf{A}$ the trivial graduation and apply (21) with all the $x_i$ equal to $x$.

We return to the general case of an $\varepsilon$-derivation $d : \mathbf{A} \to \mathbf{E}$ of degree $\delta$. Let $Z_\delta$ be the set of $a \in \mathbf{A}$ such that for every homogeneous component $a_\alpha$ of $a$ of degree $\alpha$, for every homogeneous $x$ in $\mathbf{E}$,

If $A$ is a unital associative graded algebra and $E$ a graded $(A, A)$-bimodule it follows immediately from this definition that $Z_\varepsilon$ is a graded subalgebra of $A$ containing the unit element.