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Transactions of the American Mathematical Society

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



Noetherian ring extensions with trace conditions

Author: Robert B. Warfield
Journal: Trans. Amer. Math. Soc. 331 (1992), 449-463
MSC: Primary 16P40; Secondary 16D20, 16D30
MathSciNet review: 1080737
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Abstract: Finite ring extensions of Noetherian rings with certain restrictions on the corresponding trace ideals are studied. This setting includes finite free extensions and extensions arising from actions of finite groups when the order of the group is invertible. In this setting we establish the following results which were previously obtained (for finite extensions without trace conditions) only under strong restrictions on the rings involved. Let $ R \subset S$ be an extension of Noetherian rings such that $ S$ is finitely generated as a left $ R$-module and such that the left trace ideal of $ S$ in $ R$ is equal to $ R$. If $ S$ is right fully bounded, or is a Jacobson ring, then $ R$ has the same property; furthermore, $ R$ and $ S$ have the same classical Krull dimension. If $ S$ is finitely generated as both a right and a left $ R$-module, if both trace ideals of $ S$ in $ R$ are equal to $ R$, and if $ S$ satisfies the strong second layer condition, then this condition also holds in $ R$. Finally, we compare the link graphs of $ R$ and $ S$

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Keywords: Noetherian ring, ring extension, Noetherian bimodule, trace ideal, Jacobson ring, Krull dimension, fully bounded ring, second layer condition, link, prime ideal
Article copyright: © Copyright 1992 American Mathematical Society