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


Characterizing when $ R[X]$ is integrally closed

Author: Thomas G. Lucas
Journal: Proc. Amer. Math. Soc. 105 (1989), 861-867
MSC: Primary 13B20; Secondary 13B30
MathSciNet review: 942636
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Abstract: Unlike the situation when dealing with integral domains, it is not always the case that the polynomial ring $ R\left[ X \right]$ is integrally closed when $ R$ is an integrally closed commutative ring with nonzero zero divisors. In the main theorem it is shown that for an integrally closed reduced ring $ R$, $ R\left[ X \right]$ is not integrally closed if and only if there exists a finitely generated dense ideal $ J$ and an $ R$-module homomorphism $ s \in {\text{Ho}}{{\text{m}}_R}\left( {J,R} \right)$ such that $ s$ is integral over $ R$ and $ s$ is not defined by multiplication by a fixed element of $ R$. As a corollary it is shown that $ R\left[ X \right]$ is integrally closed if and only if $ R$ is integrally closed in $ T\left( {R\left[ X \right]} \right)$, the total quotient ring of $ R\left[ X \right]$.

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Additional Information

PII: S 0002-9939(1989)0942636-0
Keywords: Reduced ring, integrally closed, complete ring of quotients, dense ideal
Article copyright: © Copyright 1989 American Mathematical Society

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