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Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings

Published online by Cambridge University Press:  20 November 2018

A. G. Heinicke
A. G. Heinicke*
Affiliation:
University of Western Ontario, London, Ontario
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If P is a right localizable prime ideal in a right Noetherian ring R, it is known that the ring RP is right Noetherian, that its Jacobson radical is the only maximal ideal, and that RP/J(RP) is simple Artinian: in short it has several properties of the commutative local rings.

In the present work we examine the properties of RP under the additional assumption that P is generated by, or is a minimal prime above, a normalizing sequence. It is shown that in such cases J(RP) satisfies the AR-property (i.e., P is classical) and that the rank of P coincides with the Krull dimension of RP. The length of the normalizing sequence is shown to be an upper bound for the rank of P, and if P is generated by a normalizing sequence x1, x2, …, xn then the rank of P equals n if and only if the P-closures of the ideals Ij generated by x1, x2, …, xj (j = 0, 1, …, n), are all distinct primes.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 325 - 346
Copyright
Copyright © Canadian Mathematical Society 1981

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