On a generalisation of self-injective von Neumann regular rings

Abstract

Apart from von Neumann regular rings, rings with infinite identities have not been studied in any detail. We take a first step in that direction by obtaining structure theorems for a class of self-injective rings with infinite identities. These extend the main structure theorems for self-injective von Neumann regular rings. A ring's identity element may be infinitely decomposable in that it cannot be expressed as the sum of (a finite number of) primitive idempotents. For brevity we will call such identity elements infinite and the others finite. This article is the result of an attempt to understand (a little of) the structure of rings with infinite identity elements. Except for von Neumann regular rings such rings have received little attention from algebraists. Von Neumann regular rings with finite identity elements are the semisimple Artinian rings (and are infective). It appears that in the class of rings with infinite identities the self-injective von Neumann regular rings play the same role as the simplest, most basic examples. Therefore our point of departure is to study a class of rings which includes the invective von Neumann regular rings but is sufficiently larger to be interesting. We choose the rings whose finitely generated (left) ideals are quasi-injective and call these rings (left) f Q-rings. They are a generalization of Q-rings which have been of some interest in their own right (see e.g [Ii], [12] or [J] (and their references) the last paper has a survey of results in the area). In ?1 we show that fQ-rings with finite identities are Q-rings and therefore their structure is known completely (except for the rather uninteresting local case). The general case when the ring has infinite identity appears to be very difficult. To simplify the task we introduce two different concepts (dense primitive idempotents and idempotent nonsingular) both of which are intermediate between finite identity and arbitrary infinite identity but in different directions. In ?1 we show that the main structure theorem on invective von Neumann regular rings (the decomposition into Types) can be extended to fQ-rings which are idempotent nonsingular; and in ?2 we determine the structure of indecomposable idempotent nonsingular fQ-rings with dense primitive idempotents and represent them as rings of matrices. Throughout this paper all rings are unital, and all ideals, properties, etc., which are one-sided are left ideals, properties, etc. Right or two-sided ideals, properties, Received by the editors June 7, 1994 and, in revised form, October 18, 1994. 1991 Mathematics Subject Classification. Primary 16D50, 16D70; Secondary 16E60. Honorary Associate at Macquarie University. (?1996 American Mathematical Society