How to compute module of Kahler differentials

Question

Definition of modulo of Kahler differentials $\Omega_{S/R}$ when a ring homomorphism $\phi:R\rightarrow S$ exists is the $S$-free module generated by elements of $R$ modulo the relations $D(s-s')-Ds-Ds',D(rs)-rDs,D(ss')-sDs'-s'Ds$. But how can you compute a modulo of Kahler differentials sxplicitly. Say one picks up the identity homomorphism $R\rightarrow R$, then what will happen?

Answer 1

If $S$ is a finitely presented $R$-algebra: $\;S\simeq R[X_1,\dots,X_n]/(f_1,\dots,f_r)$, then $$\Omega_{S/R}\simeq \frac{S\, \mathrm dX_1\oplus\dots\oplus S\, \mathrm dX_n}{S\,\mathrm df_1+\dots+S\,\mathrm df_r}.$$