A note on higher derivations and integral dependence

Abstract

In this note we prove the following: Theorem. Let R’ be an associative commutative ring with identity. Suppose R’ is an integral extension of R, and δ = { δ i } \delta = \{ {\delta _i}\} is a higher derivation on R’ which restricts to a higher derivation on R. Suppose p is a prime ideal in R which is differential under δ \delta . Then there exists a prime ideal p’ in R’ such that p’ is δ \delta -differential and p ′ ∩ R = p p’ \cap R = p .