Example 5

Returning to the general situation of Definition 1 of no. 2, let $\overline{K}$ be another commutative ring and $\rho : K \to \overline{K}$ a ring homomorphism; let $\overline{A}, \overline{A}', \overline{A}'', \overline{B}, \overline{B}', \overline{B}''$ denote the graded $\overline{K}$-modules obtained respectively from $A, A', A'', B, B', B''$ by extending the ring of scalars to $\overline{K}$ (II, § 11, no. 5); we derive from $\mu, \lambda_1$ and $\lambda_2$ $\overline{K}$-bilinear mappings

by considering the tensor products by $l_{\overline{K}}$ of the corresponding $K$-linear mappings to $\mu, \lambda_1$ and $\lambda_2$ (II, § 5, no. 1). Then, if $d$ is an $\varepsilon$-derivation of degree $\delta$ of $(A, A', A'')$ into $(B, B', B'')$, the mapping $\bar{d} = d \otimes l_{\overline{K}}$ of $\overline{A} \oplus \overline{A}' \oplus \overline{A}''$ into $\overline{B} \oplus \overline{B}' \oplus \overline{B}''$ is an $\varepsilon$-derivation of degree $\delta$ of $(\overline{A}, \overline{A}', \overline{A}'')$ into $(\overline{B}, \overline{B}', \overline{B}'')$.