On the Osofsky–smith Theorem

Abstract

Abstract. We recall a version of the Osofsky-Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which characterizes semisimple rings as those rings whose every cyclic module is injective.2002 Mathematics Subject Classification. 16D50, 16S90. Introduction.In the late 1960s, Osofsky showed her classical result which asserts that a ring is semi-simple if and only if every cyclic module is injective [8, Theorem], [9, Corollary]. Among the categorical generalizations of the Osofsky theorem, we mention the version established by Gómez Pardo et al. [5]. They showed that if C is a locally finitely generated Grothendieck category and M is a finitely presented object of C which is completely (pure-)injective and has a von Neumann regular endomorphism ring S, then S is a semi-simple ring [5, Theorem 1]. In the early 1990s, Osofsky and Smith established a module counterpart of the original Osofsky theorem. They proved that if M is a cyclic module with the property that every cyclic submodule of M is completely extending, then M is a finite direct sum of uniform modules [10]. As a consequence, if M is a module with every quotient of a cyclic submodule injective, then M is semi-simple. In the same paper, Osofsky and Smith noted that their result still holds in a more general categorical setting.The purpose of this paper is to discuss some categorical version of the OsofskySmith theorem and give several applications. We first consider the setting of a locally finitely generated Grothendieck category C and deduce that if C has a family of completely injective finitely generated generators, then C is semi-simple. As an application, we give a positive partial answer to the following question raised by

The Osofsky-Smith Theorem for Modular Lattices, and Applications (II)

“…In order to show that X is semi-simple, it is sufficient to prove that so is any finitely generated subobject of X. Consequently, without loss of generality, we may assume that X is finitely generated. Then X is an epimorphic image of a direct sum of finitely many completely injective generators of , so it is itself completely injective by Crivei et al [15,Proposition 2.2]. Therefore, any finitely generated subfactor of X is injective, and a fortiori X-injective.…”

“…We end this section by referring to some statements related to the Categorical Osofsky-Smith Theorem from Osofsky and Smith [28] and Crivei et al [15], that seem not to be in order.…”

“…Next, observe that the category 0 from Remark 4.15 is a counterexample to [15,Corollary 2.9], which is a consequence of [15,Theorem 2.7], because the hypothesis of that corollary holds vacuously, but its conclusion does not hold since the category 0 has neither simple nor nonzero semi-simple objects. This fact makes it doubtful the claim that [15,Theorem 2.7] can be proven as in the known proof for modules, and it also casts suspicion on its [15, Corollary 2.8].…”

“…This fact makes it doubtful the claim that [15,Theorem 2.7] can be proven as in the known proof for modules, and it also casts suspicion on its [15, Corollary 2.8].…”

On Krull–schmidt Finitely Accessible Categories

“…Some of the most important ones have been its module counterpart called the Osofsky-Smith theorem [16], a ring version by Gómez Pardo and Guil Asensio [12] where injectivity is replaced by pure-injectivity, and some categorical version for locally finitely generated Grothendieck categories given by Gómez Pardo et al [11]. Also, in a recent paper, Crivei et al [7] discussed the Osofsky-Smith theorem in locally finitely generated Grothendieck categories.…”