Localization in Non-Noetherian Rings

Abstract

<p>P. Gabriel constructed rings of quotients by inverting elements of multiplicative sets which satisfy the Ore and the reversibility conditions. We employ this technique in our study of localizations of non-noetherian rings at Goldie semiprime ideals. The three types of clans developed in this thesis enable us to decompose in a unique fashion (weakly) classical sets of prime ideals into (weak) clans which, in essence, are minimal localizable sets of prime ideals, satisfying certain properties. We further show that these (weak) clans are mutually disjoint sets. The different types of rings, brought into consideration, exhibit many interesting properties in the context of our localization theory.</p> <p>We characterize the AR-property for the Jacobson radical of a semilocal ring by considering finitely generated modules. In the study of rings which are module-finite over their centres, we describe expressly the injective hull of the semilocal ring modulo its Jacobson radical. These two facts enable us to establish an interrelationship between the (strongly) classical semiprime ideals of the ring and those of its central subring. Furthermore, we show that under certain conditions the Q-sets are precisely all the minimal localizable sets of prime ideals of the ring. In the case of group rings, the flatness condition can be lifted without jeopardizing the validity of the assertion.</p> <p>Lastly, we apply localization technique to characterize the group theoretic notion of q-nilpotency.</p>