{title2=$D:R\to\Omega_R^{\mathrm{nc}}$}

Let $\mu:R\otimes_kR\to R$ be multiplication and $\Omega_R^{\mathrm{nc}}=\ker\mu$, with the outer $R$-bimodule structure. Then
$$
D(r)=r\otimes1-1\otimes r
$$
is a derivation and composition with $D$ gives a natural isomorphism
$$
\operatorname{Hom}_{R-R}(\Omega_R^{\mathrm{nc}},M)
\cong\operatorname{Der}_k(R,M).
$$
The inverse sends $d$ to
$$
\theta_d\left(\sum_ir_i\otimes s_i\right)=\sum_id(r_i)s_i.
$$
The condition $\sum_ir_is_i=0$ makes this map left as well as right $R$-linear. Every element of $\Omega_R^{\mathrm{nc}}$ is a sum of terms $D(r)s$, which proves uniqueness.


 Universal bimodule derivation

ID: universal-bimodule-derivation