Hereditary Noetherian prime rings

Abstract

In the study of hereditary Noetherian rings, it is clear that hereditary Noetherian prime rings will play a central role (see, for example, [12]). Here we study the (two-sided) ideals of an hereditary Xoetherian prime ring and, as a consequence, ascertain the structure of factor rings and torsion modules. The torsion theory represents a generalization of similar results about Dedekind prime rings ([3], Section 3). The basic results are concerned with ideals and come in Sections 1, 2, and 4. Each ideal is a product of an invertible ideal and an ideal some power of which is idempotent; the invertible ideals generate an Abelian group; and a maximal invertible ideal is either a maximal ideal or else a finite intersection of idempotent maximal ideals of a specified form. We will say that a ring has enough invertible ideals if every nonzero ideal contains an invertible ideal. All the examples of hereditary Noethcrian prime rings of which we know have enough invertible ideals. They are described in Section 5. In Section 3 we show that every finitely generated torsion module of an hereditary Noetherian prime ring with enough invertible ideals is a direct sum of cyclic modules. The proof involves showing that each factor ring is generalized uniserial. In Section 6 it is shown that a factor ring of an arbitrary hereditary Noetherian prime ring is the direct product of two rings, one generalized uniserial, the other generalised triangular.