Remark 2

Let $k$ be a ring, $A$ a linearly topologized formally smooth $k$-algebra, $M$ an $A$-module whose annihilator is open in $A$. Then every derivation $\delta$ of $k$ into $M$ extends to a derivation of $A$ into $M$. Indeed, put $B = A/Ann(M)$; the mapping $\lambda \mapsto (\lambda 1_B, \delta(\lambda))$ defines a ring homomorphism from $k$ into $B \oplus M$ (No. 1, Example), that is, a $k$-algebra structure on $B \oplus M$. The canonical surjection $\varphi : A \to B$ is continuous, and therefore admits a lifting $\tilde{\varphi} : A \to B \oplus M$; by loc. cit., $pr_2 \circ \tilde{\varphi}$ is a derivation of $A$ into $M$ which extends $\delta$.