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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ \Delta$-closures of ideals and rings

Author: Louis J. Ratliff
Journal: Trans. Amer. Math. Soc. 313 (1989), 221-247
MSC: Primary 13A15; Secondary 13B20, 13C99
MathSciNet review: 961595
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Abstract: It is shown that if $ R$ is a commutative ring with identity and $ \Delta $ is a multiplicatively closed set of finitely generated nonzero ideals of $ R$, then the operation $ I \to {I_\Delta } = { \cup _{K \in \Delta }}(IK:K)$ is a closure operation on the set of ideals $ I$ of $ R$ that satisfies a partial cancellation law, and it is a prime operation if and only if $ R$ is $ \Delta $-closed. Also, if none of the ideals in $ \Delta $ is contained in a minimal prime ideal, then $ {I_\Delta } \subseteq {I_a}$, the integral closure of $ I$ in $ R$, and if $ \Delta $ is the set of all such finitely generated ideals and $ I$ contains an ideal in $ \Delta $, then $ {I_\Delta } = {I_a}$. Further, $ R$ has a natural $ \Delta $-closure $ {R^\Delta },A \to {A^\Delta }$ is a closure operation on a large set of rings $ A$ that contain $ R$ as a subring, $ A \to {A^\Delta }$ behaves nicely under certain types of ring extension, and every integral extension overring of $ R$ is $ {R^\Delta }$ for an appropriate set $ \Delta $. Finally, if $ R$ is Noetherian, then the associated primes of $ {I_\Delta }$ are also associated primes of $ {I_\Delta }K$ and $ {(IK)_\Delta }$ for all $ K \in \Delta $.

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Keywords: ACC on colon ideals, cancellation law, closure operation, directed set, flat extension ring, integral closure of an ideal, integral extension ring, Noetherian ring, polynomial extension ring, prime divisor, prime operation, reduction of an ideal, Rees ring, semiprime operation
Article copyright: © Copyright 1989 American Mathematical Society

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