On rings with a higher derivation
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- by Shizuka Satô PDF
- Proc. Amer. Math. Soc. 30 (1971), 63-68 Request permission
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
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P. Ribenboim, Higher derivations of rings, Queen’s Math. Preprints #1969-20, Queen’s University, Kingston, Ontario, Canada.
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Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 63-68
- MSC: Primary 16.60
- DOI: https://doi.org/10.1090/S0002-9939-1971-0279139-1
- MathSciNet review: 0279139