On rings with a higher derivation

Abstract

Let R ⊃ O R \supset \mathcal {O} be two rings with the unit 1. Then we set R ( O , R ) = { x ∈ R ; x r ∈ O \mathcal {R}(\mathcal {O},R) = \{ x \in R;{x^r} \in \mathcal {O} for some integer r ≧ 1 } r \geqq 1\} . At first, it is shown that, under some assumptions, d O ⊂ O d\mathcal {O} \subset \mathcal {O} implies d R ( O , R ) ⊂ R ( O , R ) d\mathcal {R}(\mathcal {O},R) \subset \mathcal {R}(\mathcal {O},R) . Next, with the Lying-over Theorem on d-differential ideals, we show: Let (R, M) and ( O , m ) (\mathcal {O},m) be two quasi-local rings and let d be a higher derivation of rank ∞ \infty of the total quotient ring of R such that d O ⊂ O d\mathcal {O} \subset \mathcal {O} . Suppose that R is integral over O \mathcal {O} and O \mathcal {O} is dominated by R. Then d ( m ) ⊂ m d(m) \subset m implies d ( M ) ⊂ M d(M) \subset M .