Localizations of HNP rings

Kuzmanovich, James

doi:10.1090/S0002-9947-1972-0311699-X

Localizations of HNP rings
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by James Kuzmanovich

Trans. Amer. Math. Soc. 173 (1972), 137-157

DOI: https://doi.org/10.1090/S0002-9947-1972-0311699-X

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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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