Ideals differential under high order derivations (Q1825911)

Let A be a commutative noetherian R-algebra, S a multiplicatively closed subset, and \({\mathcal U}^ a \)subset of the A-module of all R-linear high order derivations \({\mathcal D}_ R^{\infty}(A)\) of A in the sense of Nakai. A \({\mathcal U}\)-differential ideal is an Ideal I of A such that \(\delta\) (I)\(\subset I\) for all \(\delta\in {\mathcal D}_ R^{\infty}(A)\). The author proves that for an \({\mathcal U}\)-differential ideal I also the S- component and consequently all isolated primary components of I are \({\mathcal U}\)-differential ideals. On the other hand he gives an example showing that this is not true for embedded primary components.