On rings with a higher derivation

Abstract:

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