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 $\delta = \{ {\delta _i}\}$ 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 $\delta$. Then there exists a prime ideal p’ in R’ such that p’ is $\delta$-differential and $p’ \cap R = p$.