PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 118, Number 4, August 1993 COMPLETE PURE INJECTIVITY AND ENDOMORPHISM RINGS J. L. GOMEZ PARDO, NGUYEN V. DUNG, AND R. WISBAUER (Communicated by Maurice Auslander) Abstract. It is shown that if M is a finitely presented completely pure injec- tive object in a locally finitely generated Grothendieck category C such that S = Endc M is von Neumann regular, then 5 is semisimple. This is a gener- alized version of a well-known theorem of Osofsky, which includes also a result of Damiano on PCI-rings. As an application, we obtain a characterization of right hereditary rings with finitely presented injective hull. In [11, 12] Osofsky showed that a ring, all of whose cyclic right modules are injective, is semisimple (Artinian). Faith [6] studied the structure of right PCI-rings, i.e., rings whose proper right cyclic modules are injective, and he left open the question of whether right PCI-rings must be right Noetherian. In [3] Damiano gave an affirmative answer to Faith's question. The key result in [3] was the fact that a proper cyclic finitely presented module Mr over a right PCI-domain 7? has a semisimple endomorphism ring S [3, Proposition]. Damiano's proof uses the von Neumann regularity of S that was observed earlier by Faith [6] and a modification of a constructive technique of Osofsky [12]. In this note we prove a general version for Grothendieck categories of Osofsky's theorem [11, 12], which includes also the above-mentioned result of Damiano. Furthermore, our arguments provide a simple proof of this result. We show that if Af is a finitely presented completely (pure) A7-injective object in a locally finitely generated Grothendieck category C such that S = Endc M is von Neumann regular, then S is semisimple. Consequently, if Af is a projec- tive completely injective object in C, then Af = ©/6/ A,, where Endc Aj are division rings and the subobjects of Ai are linearly ordered. As an application to rings, we obtain a characterization of the right hereditary rings 7? such that the injective hull E(Rr) is finitely presented. This extends a result of Colby and Rutter [2]. Note that even in the module case, Damiano's arguments cannot be applied for proving our main theorem. Let C be a Grothendieck category. An object M of C is finitely presented if it is finitely generated and every epimorphism A —> M, where A is finitely Received by the editors June 20, 1991 and, in revised form, November 29, 1991. 1991 MathematicsSubject Classification. Primary 16D50, 16S50;Secondary 16D80, 16D90, 18E15. The first and second authors gratefully acknowledge the support of the Spanish Ministry of Education and Science(DGICYT PB87-0703). ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 1029 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use