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

 

Abnormalities in Noetherian rings


Authors: J. T. Arnold and M. B. Boisen
Journal: Proc. Amer. Math. Soc. 73 (1979), 1-6
MSC: Primary 13E05; Secondary 13C15
MathSciNet review: 512046
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Abstract: If $ P \subseteq Q$ are prime ideals in some ring R and if rank $ Q = {\text{rank}}(Q/P) + {\text{rank}}\;P + k$, then $ P \subset Q$ is said to be k-abnormal and k is called the degree of abnormality. The paper consists of two examples. The first example is a Noetherian integral domain in which the set of degrees of abnormality is unbounded. Let P be a prime ideal of R and set $ W = \{ Q/Q$ is a prime ideal and $ P \subset Q$ is abnormal}. The second example is a local domain such that $ \{ k\vert P \subset Q$ is k-abnormal for some $ Q \in W\} \ne \{ k\vert P \subset Q$ is k-abnormal for some Q minimal in W}.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0512046-2
PII: S 0002-9939(1979)0512046-2
Keywords: Prime ideal, Noetherian ring, local ring, rank
Article copyright: © Copyright 1979 American Mathematical Society