JOURNAL OF ALGEBRA 135, 44&455 (1990) Local-Global Results for Modules over Algebras and Azumaya Rings ROBERT WISBAUER Mathematisches Institut, Universitht Diisseldorf; Universitdtsstr. 1, 4000 Diisseldorf, Federal Republic of Germany Communicated by Kent R. Fuller Received September 5, 1988 For an R-module A4 the category of all submodules of M-generated modules is denoted by o[M]. The study of a[M] leads to a remarkable refinement of classical module theory. Of special interest are modules M for which the functor Hom(M, - ): o[M] + End(M)-Mod is an equivalence of categories. They are called self-progenerators or quasi-progenerators. In this article we give local-global characterizations for self-progener- ators over Z-algebras R, 2 a commutative ring, considering localization with respect to the maximal ideals of 2 and with respect to the maximal ideals in the ring of idempotents of 2 (Pierce stalks, Theorem 2.1). We call an R-module M an ideal module if there is a bijection between the submodules of M and the left ideals of End(M). It is shown that a finitely generated ideal module with right perfect endomorphism ring is a self-progenerator. A not necessarily associative Z-algebra A can be considered as a module over the multiplication algebra M(A); o[A] will denote the category of submodules of A-generated M(A)-modules. In caseA is associative and commutative the category o[A] is just the category of all A-modules. Hence the study of o[A] may be taken as a generalization of module theory over commutative rings which parallels the investigation of left modules and refines the study of bimodzdes over arbitrary rings A. For example, in this setting the central closure of a semiprime ring A (in the senseof Martindale) can be obtained as the injective envelope of A in the category @[A] (e.g., [21]). Here we apply the above mentioned results on a[M] to examine the relationship between the ring A and a[A]. A is called an Azumaya ring if it is a self-progenerator as M(A)-module, and an Azumaya algebra (over its center) if it is a progenerator for all M(A)-modules. We obtain local-global characterizations for (non-associative) Azumaya rings (Corollary 3.3), 440 0021-8693/90 $3.00 Copyright 0 1990 by Academic Press, Inc. AI1 rights of reproduction in any form reserved.