An additivity principle for Goldie rank

Abstract

LetA be a noetherian ring. In generalA will not admit a classical Artinian ring of quotients. Yet a problem in enveloping algebras leads one to consider the possible embedding ofA in a prime ringB which is finitely generated as a left and a rightA module. Under certain additional technical assumptions, it is shown that the setS of regular elements ofA is regular inB and is an Ore set in bothA andB withS −1 A andS −1 B Artinian. This enables one to establish the following additivity principle for Goldie rank. Let {P 1,P 2, …P 1} be the set of minimal primes ofA. Then under the above conditions it is shown that there exist positive integersz 1,z 2, …,z, such that $$\sum\limits_{i = 1}^r {z_i rk} (A/Pi) = rk B,$$ , where rk denotes Goldie rank. This applies to the study of primitive ideals in the enveloping algebra of a complex semisimple Lie algebra.