Lemma 91.13.2. Let $P = A[S]$ be a polynomial ring over $A$. Let $M$ be a $(P, P)$-bimodule over $A$. Given $m_ s \in M$ for $s \in S$, there exists a unique $A$-biderivation $\lambda : P \to M$ mapping $s$ to $m_ s$ for $s \in S$.

Proof. We set

$\lambda (s_1 \ldots s_ t) = \sum s_1 \ldots s_{i - 1} m_{s_ i} s_{i + 1} \ldots s_ t$

in $M$. Extending by $A$-linearity we obtain a biderivation. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).