Correspondences of closed submodules

If N is an M-faithful R-module, then there is an order-preserving correspondence between the closed R-submodules of N and the closed Ssubmodules of HomR(M, N), where S = EndRM. There is a considerable body of research dealing with correspondences between the lattice of submodules of an R-module M and the lattice of left ideals of its endomorphism ring S. This literature includes the well-known Morita theory and its generalizations. When a complete correspondence between all submodules and all left ideals fails to hold, one may still ask whether there is a correspondence between designated sublattices of submodules and left ideals. One particular situation that has attracted much attention has been cases when there is a correspondence between the closed (i.e., essentially closed) submodules of M and the closed left ideals of S. It is known that such a correspondence exists when M is a semisimple module (an elementary observation), when M is a free module [2], when M is a nonsingular retractable module satisfying an additional condition [7], and when M is a nondegenerate module [8]. This article began as a search for a conceptual link among these special cases. Eventually, it was realized that a common denominator is the notion of a self-faithful module, a concept first introduced for generators in [6], and recently exploited to good effect in [3], [4] and [5]. The principal contribution of this article is to demonstrate that a natural correspondence of closed submodules with closed left ideals occurs whenever M is a self-faithful module. In fact, taking a cue from the approach in [1], we show more generally in Theorem 1.2 that when N is an M-faithful R-module, then there exists an order-preserving correspondence between the closed R-submodules of M and the closed S-submodules of HomR(M, N), where S = EndRM. Taking N = M then specializes to the self-faithful case. Additional examples of M being self-faithful, and of the desired correspondence holding, occur when M is a quasi-projective retractable module (Proposition 1.2 in [3]) and when M is a polyform retractable module (Corollary 2.3). Of purely technical interest is the fact that the results contained in this paper remain true even over rings which fail to have an identity element. Received by the editors March 7, 1995. 1991 Mathematics Subject Classification. Primary 16S50; Secondary 16D70. ?)1996 American Mathematical Society