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Proceedings of the American Mathematical Society

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

 

 

A note on integral closure


Author: Judith Sally
Journal: Proc. Amer. Math. Soc. 36 (1972), 93-96
MSC: Primary 13B20
DOI: https://doi.org/10.1090/S0002-9939-1972-0311639-9
MathSciNet review: 0311639
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Abstract: Let R be an integrally closed domain and $ {x_i},{y_j}(1 \leqq i \leqq n,1 \leqq j \leqq m)$ R-sequences. Let

$\displaystyle T = R[x_1^{{\alpha _1}} \cdots x_n^{{\alpha _n}}/y_1^{{\beta _1}} \cdots y_m^{{\beta _m}}],$

where the $ {\alpha _i}$ and $ {\beta _j}$ are positive integers. If T is integrally closed then

$\displaystyle {\alpha _1} = \cdots = {\alpha _n} = 1\quad {\text{or}}\quad {\beta _1} = \cdots = {\beta _m} = 1.$ ($ *$)

$ ( ^\ast )$ is sufficient for T to be integrally closed in the following cases:

(1) R is Noetherian and the $ ({x_i},{y_j})R$ are distinct prime ideals,

(2) R is a polynomial ring over an integrally closed domain and the $ {x_i}$ and $ {y_j}$ are indeterminates.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0311639-9
Keywords: Integrally closed domain, Noetherian domain, R-sequence, monoidal transform, complete ideal
Article copyright: © Copyright 1972 American Mathematical Society