On some invariant ideals and on extension of differentiations to seminormalization
The authors consider differentiations \textbf{D}\(=(1,D_ 1,D_ 2,...)\) in the sense of Hasse and F. K. Schmidt of a noetherian integral domain A, the integral closure of which \(\bar A\) in its quotient field is a finitely generated A-module. While the conductor \(\alpha\) of \(\bar A\) into A is \textbf{D}-invariant and \textbf{D} always extends to \(\bar A,\) this is not true in general for rings B with \(A\subset B\subset \bar A\) instead of \(\bar A.\) In the first part of the paper the authors show that the A-radical \({}^ A\sqrt{\alpha}\) of the conductor \(\alpha\) of B into A is \textbf{D}- invariant for any differentiation \textbf{D} of A. - In the second part they show that any differentiation of A extends to the seminormalization of A. - Finally in a third part the authors prove relations between the radicals of the various conductors of B and \(\bar A\) into A and B and give conditions when they are invariant with respect to differentiations of A or B. By some counterexamples they show that their conditions cannot be removed.
differentiation of a noetherian integral domain
scientific article; zbMATH DE number 4191842
scientific article