An additivity principle for Goldie rank

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 , where rk denotes Goldie rank. This applies to the study of primitive ideals in the enveloping algebra of a complex semisimple Lie algebra. This paper was written while the authors were guests of the Institute for Advanced Studies, The Hebrew University of Jerusalem. The first author was on leave of absence from the Centre Nationale de la Recherche Scientifique, France.