Proposition 18

Let $A$ be a commutative $K$-algebra and $\mathfrak{J}$ the ideal the kernel of the surjective canonical homomorphism $m : A \otimes_K A \to A$, so that $A$ is isomorphic to $(A \otimes_K A)/\mathfrak{J}$ and $\mathfrak{J}/\mathfrak{J}^2$ has canonically an $A$-module structure. Let $d_{A/K} : A \to I/I^2$ be the $K$-linear mapping which associates with every $x \in A$ the class of $x \otimes 1 - 1 \otimes x$ modulo $\mathfrak{J}^2$. The mapping $d_{A/K}$ is a $K$-derivation and, for every $A$-module $E$ and every $K$-derivation $D : A \to E$, there exists one and only one $A$-linear mapping $g : \mathfrak{J}/\mathfrak{J}^2 \to E$ such that $D = g \circ d_{A/K}$.