NONCOMMUTATIVE RINGS - LOCALIZATION AND NORMALIZING EXTENSIONS (SEMIPRIME IDEALS, NOETHERIAN)

This dissertation deals with two main topics: the localization of a semiprime ideal N in a right Noetherian ring R and normalizing extensions of rings.;With respect to a normalizing extension S of a subring R, we obtain an injective R-submodule, L, of an injective R-module M. In the case where S is a free normalizing extension of R and M is an indecomposable injective S-module, a decomposition of M(,R) into a finite direct sum of semilinearly isomorphic indecomposable injective R-modules is presented. Furthermore, when S is a flat centralizing extension of R we decompose M(,R) into a finite direct sum of isomorphic indecomposable injective R-modules. When R is a right Noetherian ring we relate the associated prime ideal to M(,S) with the associated prime ideals to a family of R-modules.;Concerning the first topic, we relate the nilpotency of N with that of J((LAMDA)), the Jacobson radical of the endomorphism ring, (LAMDA), of the injective hull, E(R/N) of the right R-module R/N. We also prove that if J((LAMDA)) is nilpotent and N is right localizable in R, then E(R/N) is N-primary. We give a characterization for the perfectness of the topology, D(,E(R/N)), cogenerated by E(R/N) and present a condition for the right closure of N, with respect to D(,E(R/N)), to be an ideal of R(,N), the ring of right quotients of R with respect to D(,E(R/N)). Two conditions are given for the localization in R of a semiprime ideal of square zero. We show a new characterization for weak ideal invariance and present a special criterion for N to be weakly ideal invariant. A condition for the localization of a prime ideal having the right AR-property is given. We introduce a more general setting of localization in terms of the elements of R which are regular on an injective R-module V. We characterize the set of regular elements on V and also the topology cogenerated by V.