The structure of rings with faithful nonsingular modules

It is shown that the existence of a faithful nonsingular uniform module characterizes rings which have a full linear maximal quotient ring. New information about the structure of these rings is obtained and their maximal quotient rings are constructed in an explicit manner. More generally, rings whose maximal quotient rings are finite direct sums of full linear rings are characterized by the existence of a faithful nonsingular finite dimensional module. Introduction. The study of prime rings with nonsingular uniform one-sided ideals was initiated in [5]. With the treatments in [8 and 1], but especially in the latter article, the structure of this class of rings became well understood. In this paper, we will begin by examining rings which possess faithful nonsingular uniform modules but which are not necessarily prime. Our principal discovery is that this is precisely the class of rings whose maximal quotient rings are full linear rings, a class of rings which has been extensively studied from other perspectives (cf. [6, 9]). It is a bit surprising that despite the fact that nonsingular uniform modules play a significant role in module theory (cf. [3]), the consequences for the structure of a ring of the existence of a faithful such module appear not to have been suspected. In [13] the structure of the rings with faithful monoform modules is determined. Since a nonsingular uniform module is monoform, the rings described above form a proper subclass of the rings with faithful monoform modules and hence are (in the terminology of [13]) "dense" rings of linear transformations. As we will soon see, the structure theory of this subclass is actually much richer. In the final section of this paper we study rings whose maximal quotient rings are finite direct sums or arbitrary direct products of full linear rings. One sample result (Corollary 5.3) is that a ring has a maximal quotient ring which is a finite direct sum of full linear rings if and only if it possesses a faithful finite dimensional nonsingular module. An innovation is the introduction in ?2 of "partial" contexts which extend the concept of Morita contexts. We have found this idea to be a rather helpful crutch in the organization of the main results. 1. Preliminaries. It is important to stress that throughout this article rings need not possess identity elements. We will also be careful to write homomorphisms consistently on the side of a module opposite to that of the scalars. Received by the editors June 7, 1982. 1980 Mathematics Subject Classification. Primary 16A42, 16A48, 16A64, 16A53, 16A65, 16A08. ?1983 American Mathematical Society 0002-9947/82/0000-1 939/$04.50