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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Contracted ideals and purity for ring extensions


Authors: J. W. Brewer and D. L. Costa
Journal: Proc. Amer. Math. Soc. 53 (1975), 271-276
MSC: Primary 13B99
MathSciNet review: 0384774
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Abstract: In this paper an example is given of a pair of commutative noetherian rings $ R \subseteq S$ with $ S$ a finite $ R$-module and $ IS \cap R = I$ for each ideal $ I$ of $ R$, but having the property that $ 0 \to R \to S$ is not a pure sequence of $ R$-modules. Purity of the sequence $ 0 \to R \to S$ is equivalent to $ R[X]$ being ``ideally closed'' in $ S[X],\;X$ an indeterminate. Therefore, the example renders appealing the proposition that for $ R$ noetherian and $ S$ a noetherian torsion-free $ R$-algebra containing $ R$, if $ \alpha S \cap R = \alpha R$ for each non-zero-divisor $ \alpha \epsilon R$, then the extension $ R[X] \subseteq S[X]$ has the same properties. Finally, it is also shown that for $ R$ noetherian and $ 0 \to R \to S$ pure, with $ S$ an $ R$-algebra, then $ R[[{X_1}, \ldots ,{X_n}]]$ is pure in $ S[[{X_1}, \ldots ,{X_n}]]$ for each positive integer $ n$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0384774-X
PII: S 0002-9939(1975)0384774-X
Article copyright: © Copyright 1975 American Mathematical Society