quantum groups - Classical Calculi as Universal Quotients

In the classical case, if $Omega(A)$ is the kernel of the multiplication map $m:Aotimes Ato A$, then—since $A$ is commutative, so that $m$ is not only a map of $A$-bimodules but also a morphism of $k$-algebras,—it turns out that $Omega(A)$ is an ideal of $Aotimes A$, not only a sub-$A$-bimodule. In particular, you can compute its square $(Omega(A))^2$. Then the classical module of Kähler differentials $Omega^1_{A/k}$ is the quotient $Omega(A)/Omega(A)^2$.

(This is the construction used by Grothendieck in EGA IV, for example)