Inner derivation through the universal bimodule derivation

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 ${\Omega }_R^{\mathrm{nc}}\subseteq R{\otimes }_{k}R$.