Inversive localization at semiprlme goldie ideals

Abstract

P. M. Cohn [6] introduced a method of localizing at a semiprime ideal of a noncommutative Noetherian ring by inverting certain matrices. This paper continues the study of the technique of inversive localization, in a more general setting. The inversive localization is characterized by its structure modulo its Jacobson radical. This is in marked contrast to the torsion theoretic localization, and the two constructions coincide only when the localization can actually be obtained by inverting elements rather than matrices. The inversive localization is computed for the class of left Artinian rings, and it is then shown that at a minimal prime ideal of an order in a left Artinian ring the inversive localization must be left Artinian. On the other hand, the inversive localization at a semiprime ideal of a left Noetherian ring need not be left Noetherian.