Weak normalization and the power series rings (Q5961640)

We follow the notations of \textit{D. E. Dobbs} and \textit{M. Roitman} in Can. Math. Bull. 38, No. 4, 429-433 (1995; Zbl 0848.13019) with the difference that A and B are commutative rings and not necessarily integral domains. The main theorem says that if \(A \subset B\), then the weak normalization of \(A[[X]]\) in \(B[[X]]\) is contained in the formal power series ring of the weak normalization \(_B^*A\) of \(A\) in \(B\) and if \(_B^*A\) is finite over \(A\), then equality holds.NEWLINENEWLINENEWLINEThe proof given here is quite different from that of Dobbs and Roitman referred above. As an application it is shown that if \(^*A\) is the weak normalization of \(A\) in its total quotient ring \(K\) and \(D\) is a higher derivation of \(A\) then \(D\) extends to \(^*A\).