Noetherian Prime Rings of Krull Dimension One

Abstract

<p>A right noetherian prime ring R is, by Goldie's Theorem, a right order in a simple artinian ring Q: Q is obtained from R by inverting all non-zero-divisors. O can be described as the quotient ring of R at a torsion theory, the Goldie torsion theory. If R has right Krull dimension one, the Goldie torsion theory is generated by the class of all simple right R-modules.</p> <p>In this thesis we develop a theory of localization for (right) noetherian prime rings of (right) Krull dimension one, based on the direct decompositions of the Goldie torsion theory. We characterize these decompositions, using a natural partition of the class of all simple modules, and show that the quotient rings at the components remain right noetherian, prime and of right Krull dimension one. Other desirable properties of these localizations are determined: they are perfect, they preserve the two-sidedness of ideals, and they are well behaved on the simple modules. We further show that they generalize the localizations at classical semiprime ideals.</p> <p>A criterion is given for a right quotient ring at a component of a decomposition of the right Goldie torsion theory to be also a left quotient ring at a component of the left Goldie torsion theory. We show that this criterion is satisfied if the ring has global dimension no longer than two.</p> <p>Finally, we study hereditary noetherian prime rings in the context of our localization theory.</p>