Vol. 53, No. 1, 1974

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Localization and splitting in hereditary noetherian prime rings

Kenneth R. Goodearl

Vol. 53 (1974), No. 1, 137–151

The purpose of this paper is to introduce a localization corresponding to any collection X of maximal right ideals in an hereditary noetherian prime ring R. The localized ring RX has only as many simple right modules (up to isomorphism) as R has simple right modules of the form R∕M, where M X. In particular, for a single maximal right ideal M the ring RM has exactly one simple right module (up to isomorphism). These localizations satisfy a globalization property in that a sequence of R-homomorphisms is exact if and only if it is exact when localized at each maximal right ideal of R. These localizations are also the most general possible, for it is shown that every ring between R and its maximal quotient ring has the form RX for suitable X. The relationship between these localizations and other previously introduced localizations for hereditary noetherian prime rings is discussed, and then this localization technique is applied to the question of when an hereditary noetherian prime ring R can be a splitting ring (i.e., a ring such that the singular submodule of every right module is a direct summand). Such a ring is shown to be an iterated idealizer from a ring over which all singular right modules are injective. Finally, hereditary noetherian prime splitting rings are characterized by the properties of possessing a minimal two-sided ideal and having all faithful simple right modules injective.

Mathematical Subject Classification
Primary: 16A08
Received: 5 July 1973
Revised: 10 October 1973
Published: 1 July 1974
Kenneth R. Goodearl
University of California, Santa Barbara
Santa Barbara CA
United States