A Shorter Proof of Goldie's Theorem

Abstract

In this note we present an extremely short proof of Goldie's theorem on the structure of semiprime Noetherian rings [1]. The outline of the proof was given by Procesi and Small in [4]. By utilizing the concept of the singular ideal of a ring we have been able to weaken the hypotheses of many of the steps in [4]. Most significantly, we are able to avoid a reduction to the case of prime rings, and in Lemma 5 we give an informative list of the relationship between regular elements and essential ideals of semiprime rings.