Mumford in his Red Book gives on page 144 an example of computation of a module of Kähler differential forms.
Namely, he lets $k$ be a field and considers the quotient algebra $B=k[X,Y]/(XY)$.
He defines $\omega=XdY=-YdX\in \Omega_{B/k} $ and claims there is a short exact sequence $$0\to k\cdot\omega \to \Omega_{B/k}\to k[X]dX\oplus k[Y]dY\to 0 $$ I more or less understand the idea behind this exact sequence (using the standard computation of the Kähler differential module of a quotient of a polynomial ring) , but I would like to see this exact sequence in a broader, more general context rather than the ad hoc explanations given by Mumford.
Also, he claims that his exact sequence (of $B$-modules I guess) does not split. Why is that?