Stable Structure of Noncommutative Noetherian Rings

These results are of course well known and well studied in a commutative setting. In this paper, the Stable Range Theorem is proved for several classes of noncommutative rings and the other two theorems are proved for simple and related rings. Section 1 gives some elementary results about semiprime rings that are used throughout the paper. In Section 2, several types of the Stable Range Theorem are proved, all of which follow from the same basic result (Proposition 2.1). The Stable Range Theorem itself is proved for rings known as right ideal invariant. Examples of such rings are given in Section 3 and include Noetherian rings which are either commutative, fully bounded, Asano, or semiprime and hereditary. Strangely, the proof of this result follows closely on the proof of [I 8, Lemma 1.41, which gives a bound on the number of generators of a right ideal in a simple ring. It is not surprising therefore that in the special case of simple rings, or indeed Asano orders, the Stable Range Theorem can be strangthened to give the same stability property for the generators of a right ideal (the Stable Range Theorem for Ideals).