On the kernels of some higher derivations in polynomial rings (Q635466)

The author generalizes some results known on locally nilpotent derivations to locally iterative higher derivations, and from fields to commutative rings. In particular, he proves the following: Theorem 1.1. Let \(D\) be a rational higher \(R\)-derivation on the polynomial ring \(A:=R[x_1,\dots,x_n]\) where \(R\) is an integral domain, and let \(\overline D\) be the extension of \(D\) to \(Q(A)\). Assume that \(\text{trdeg}_RAD\geq n-1\). Then \(Q(A)^{\overline D}=Q(A^D)\). Theorem 1.2. Let \(A=R[x,y]\) be the polynomial ring in two variables over an integral domain \(R\) with unit and let \(B\) be an \(R\)-subalgebra of \(A\). Then the following conditions (1) and (2) are equivalent: (1) There exists a rational higher \(R\)-derivation \(D\) on \(A\) such that \(B=A^D\). (2) \(B\) is integrally closed in \(A\), \(Q(B)\cap A=B\), and \(Q(A)\) is a separable extension of \(Q(B)\). -- Moreover, if \(R\) is a field, and \(\text{trdeg}_R(B)\leq 1\), then (1) is equivalent to (3) \(B\) is integrally closed in \(A\) and \(Q(A)\) is a separable extension of \(Q(B)\).