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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On rings with a higher derivation

Author: Shizuka Satô
Journal: Proc. Amer. Math. Soc. 30 (1971), 63-68
MSC: Primary 16.60
MathSciNet review: 0279139
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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$.

References [Enhancements On Off] (What's this?)

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Keywords: Higher derivations, d-differential ideals, quasi-local rings
Article copyright: © Copyright 1971 American Mathematical Society