Universal bimodule derivation ($D:R\to {\Omega }_R^{\mathrm{nc}}$)

Let $\mu :R{\otimes }_{k}R\to R$ be multiplication and ${\Omega }_R^{\mathrm{nc}}=\ker \mu$, with the outer $R$-bimodule structure. Thenis a derivation and composition with $D$ gives a natural isomorphismThe inverse sends $d$ toThe condition ${\sum }_i{r}_i{s}_i=0$ 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.

Under the universal isomorphism, the inner derivation $d_m(r)=rm-mr$ corresponds toThus inner derivations correspond exactly to bimodule maps on ${\Omega }_R^{\mathrm{nc}}$ that extend across the inclusion ${\Omega }_R^{\mathrm{nc}}\subseteq R{\otimes }_{k}R$.