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Proceedings of the American Mathematical Society

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

 

 

The large condition for rings with Krull dimension


Author: Ann K. Boyle
Journal: Proc. Amer. Math. Soc. 72 (1978), 27-32
MSC: Primary 16A55
DOI: https://doi.org/10.1090/S0002-9939-1978-0503524-X
MathSciNet review: 503524
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Abstract: A module M with Krull dimension is said to satisfy the large condition if for any essential submodule L of M, the Krull dimension of $ M/L$ is strictly less than the Krull dimension of M. For a right noetherian ring R with Krull dimension $ \alpha $ this is equivalent to the condition that every f.g. uniform submodule of $ E({R_R})$ with Krull dimension $ \alpha $ is critical. It is also shown that if R is right noetherian with Krull dimension $ \alpha $ and if $ {I_0}$ is a right ideal maximal with respect to K $ \dim {I_0} < \alpha $, then R satisfies the large condition if and only if $ {I_0}$ is a finite intersection of cocritical right ideals and $ {I_0}$ is closed.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0503524-X
Keywords: The large condition, semicritical module, semiprime ring, nonsingularly k-primitive ring
Article copyright: © Copyright 1978 American Mathematical Society