Complete Pure Injectivity and Endomorphism Rings

Endomorphism rings of completely pure-injective modules

“…The preceding corollary can be regarded as a generalization of [3,Corollary 6]. A more specific extension of this result is the following: Proof.…”

“…It is well known that when every right S-module X is M -invariant, useful information can be passed from M R to S. This is what happens, for example, when M R is a finitely generated projective module, which makes it possible to characterize properties of the endomorphism ring S in terms of M R . This property also holds when M R is finitely presented and S is a (von Neumann) regular ring and this, coupled with Osofsky's theorem [8,9] that asserts that a ring whose cyclic right modules are all injective is semisimple, has been exploited in [3] to obtain an easy proof of the result of Damiano that shows that a right PCI ring (i.e., a ring with each proper cyclic right module injective) is right noetherian. This technique was also (implicitly) applied in [1] to a right hereditary ring R whose injective envelope E(R R ) is projective, showing that R is, in this case, a (two-sided) hereditary artinian QF-3 ring.…”

“…More specifically, we assume that E/JE is a completely pure-injective R-module, i.e., a module such that each pure quotient of itself is pure-injective. We give several applications and we extend [3,Corollary 6] by proving that if R is right hereditary and every finitely generated submodule of E R is finitely presented, then R is right noetherian.…”

“…This technique was also (implicitly) applied in [1] to a right hereditary ring R whose injective envelope E(R R ) is projective, showing that R is, in this case, a (two-sided) hereditary artinian QF-3 ring. An extension in [3,Corollary 6] shows that if E(R R ) is just finitely presented (instead of projective), then R is a right artinian ring with Morita duality. The key point of this proof is to show that R is right finite-dimensional.…”

On the Osofsky–smith Theorem

“…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.…”