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.
