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Proceedings of the American Mathematical Society

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



Two notes on imbedded prime divisors

Author: L. J. Ratliff
Journal: Proc. Amer. Math. Soc. 101 (1987), 395-402
MSC: Primary 13A17; Secondary 13E05
MathSciNet review: 908637
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Abstract: The first note shows that if $ R < T$ are any two Noetherian rings, then there exists a Noetherian ring $ B$ between $ R$ and $ T$ which has a maximal ideal $ N$ such that $ {\text{grade}}(N) \leq 1$ and $ N \cap R$ is a maximal ideal. The second note shows that if $ R$ is a Noetherian ring, then there exists a free quadratic integral extension ring $ B$ of $ R$ such that $ \operatorname{Spec}(B) \cong \operatorname{Spec}(R)$ and such that if $ I$ is any regular ideal in $ R$ and $ {P_1} \cap \cdots \cap {P_g}$ are prime ideals in $ R$ containing $ I$, then there exists an ideal $ J$ in $ B$ integrally dependent on $ IB$ such that the prime ideals corresponding to the $ {P_i}$ are prime divisors of $ {J^n}$ for all $ n \geq 1$.

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Keywords: Cohen-Macaulay ring, flat extension ring, grade of an ideal, integral closure of an ideal, integral extension ring, Notherian ring, prime divisor semilocal ring
Article copyright: © Copyright 1987 American Mathematical Society