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Transactions of the American Mathematical Society

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



Localizations of HNP rings

Author: James Kuzmanovich
Journal: Trans. Amer. Math. Soc. 173 (1972), 137-157
MSC: Primary 16A08
MathSciNet review: 0311699
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Abstract: In this paper it is shown that every hereditary Noetherian prime ring is the intersection of a hereditary Noetherian prime ring having no invertible ideals with a bounded hereditary Noetherian prime ring in which every nonzero two-sided ideal contains an invertible two-sided ideal. Further, it is shown that this intersection corresponds to a decomposition of torsion modules over such a ring; if $ R$ is an HNP ring with enough invertible ideals, then this decomposition coincides with that of Eisenbud and Robson.

If $ M$ is a maximal invertible ideal of $ R$ where $ R$ is as above, then an overring of $ R$ is constructed which is a localization of $ R$ at $ M$ in a ``classical sense"; that is, it is a ring of quotients with respect to a multiplicatively closed set of regular elements satisfying the Ore conditions. The localizations are shown to have nonzero radical and are also shown to satisfy a globalization theorem. These localizations are generalizations of ones constructed by A. V. Jategaonkar for HNP rings with enough invertible ideals.

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Keywords: Dedekind prime ring, hereditary Noetherian prime ring, ring of quotients, overring, hereditary torsion theory, idempotent kernel functor
Article copyright: © Copyright 1972 American Mathematical Society