Proposition 14

*Let $A$ be a commutative ring, $M$ an $A$-module, $B$ the $A$-algebra $T(M)$ (resp. $S(M)$, resp. $\Lambda(M)$) and $E$ a $(B, B)$-bimodule. Let $d_0 : A \to E$ be a derivation of the ring $\mathbf{A}$ into the $\mathbf{A}$-module $\mathbf{E}$ and $d_1 : M \to \mathbf{E}$ an additive group homomorphism such that, for all $a \in \mathbf{A}$ and all $x \in M$,

and further, when $B = S(M)$,

for all $x, y$ in $M$, and, when $B = \bigwedge(M)$,

for all $x \in M$. Then there exists one and only one derivation $d$ of $B$ (considered as a $\mathbf{Z}$-algebra) into the $(B, B)$-bimodule $\mathbf{E}$ such that $d|_{\mathbf{A}} = d_0$ and $d|_{M} = d_1$.

We take on the $\mathbf{Z}$-module $B \oplus \mathbf{E}$ the associative $\mathbf{Z}$-algebra structure defined by

which has $(1, 0)$ as unit element (no. 8, Proposition 13). Under the canonical injection $t \mapsto (0, t)$, $\mathbf{E}$ is identified with a two-sided ideal of $B \oplus \mathbf{E}$ such that $\mathbf{E}^2 = \{0\}$. On the other hand, the mapping $h_0 : B \oplus \mathbf{E}$ defined by $h_0(a) = (a, d_0(a))$ is a unital ring homomorphism (no. 8, Proposition 12); under this mapping, $B \oplus \mathbf{E}$ then becomes an $\mathbf{A}$-algebra. Moreover, if, for all $x \in M$, we write $h_1(x) = (x, d_1(x))$, it follows from the definition of $h_0$ and (29) that $h_1(ax) = h_0(a)h_1(x)$; in other words $h_1$ is an $\mathbf{A}$-linear mapping of $M$ into $B \oplus \mathbf{E}$. Then there exists one and only one $\mathbf{A}$-algebra homomorphism, $h : B \to B \oplus \mathbf{E}$ such that $h|_M = h_1$ (and necessarily $h|_{\mathbf{A}} = h_0$): for, if $B = T(M)$, this follows from $§ 5$, no. 1, Proposition 1; if $B = S(M)$, condition (30) shows that $h(x)h(y) = h(y)h(x)$ for all $x, y$ in $M$ and the conclusion follows from $§ 6$, no. 1, Proposition 2; finally if $B = \bigwedge(M)$, condition (31) shows that $(h(x))^2 = 0$ for all $x \in M$, since $x \wedge x = 0$ and the conclusion follows from $§ 7$, no. 1, Proposition 1. The homomorphism $h$ is such that the composition $p \circ h : B \to B$ with the augmentation $\rho : B \oplus \mathbf{E} \to B$ is the identity $l_B$, for $p \circ h$ and $l_B$ coincide by definition for the elements of $\mathbf{A}$ and those of $M$ and the set of these elements is a generating system of $B$. We can therefore write $h(b) = (b, d(b))$ for all $b \in B$ and the mapping $b \mapsto d(b)$ of $B$ into $\mathbf{E}$ is a derivation with the required properties, by virtue of Proposition 12 of no. 8.