Regular elements in semiprime rings

Abstract

In the proof of Goldie's theorem [1, Theorem 4.1], one of the crucial steps is to establish that every large right ideal contains a regular element [1, Theorem 3.9]. Recently, S. A. Amitsur told one of the authors he had proved, using the weaker conditions of the ACC on left and right annihilators, that every prime ring contains a left regular element a (i.e., the left annihilator a' of a is zero) and a right regular element b (i.e. the right annihilator br of b is zero). In this note, we generalize Amitsur's result as follows: