A Right PCI Ring is Right Noetherian

Damiano, Robert

doi:10.2307/2042705

Cited by 6 publications

(10 citation statements)

References 3 publications

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“…Faith called a ring R a right PCI ring if every cyclic right R-module is injective or isomorphic to R R . By his result in [11] and a result of Damiano in [6] such a ring is either semisimple or it is a non-artinian, right hereditary, right noetherian, simple domain. The latter ring is usually called a right PCI domain.…”

Section: Some Remarksmentioning

confidence: 99%

A characterization of noetherian rings by cyclic modules

Huynh

1996

Proceedings of the Edinburgh Mathematical Society

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Section: Some Remarksmentioning

confidence: 99%

A characterization of noetherian rings by cyclic modules

Huynh

1996

Proceedings of the Edinburgh Mathematical Society

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“…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?…”

mentioning

confidence: 99%

“…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].…”

mentioning

confidence: 99%

Complete pure injectivity and endomorphism rings

Pardo¹,

Dũng²,

Wisbauer³

1993

Proc. Amer. Math. Soc.

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Abstract. It is shown that if M is a finitely presented completely pure injective 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 generalized 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 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 projective 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.

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“…More recently, in Huynh-Dung [4], an attempt was made to generalize Osofsky's theorem to cyclic injective modules. Using a result of Damiano [3], it was shown in [4] that a cyclic finitely presented module is semisimple if all quotients of cyclic submodules of M are injective. In the recent work [8], Osofsky and Smith have shown that the hypothesis "finitely presented" can be removed.…”

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confidence: 99%

Modules whose closed submodules are finitely generated

Dũng

1991

Proceedings of the Edinburgh Mathematical Society

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A module M is called a CC-module if every closed submodule of M is cyclic. It is shown that a cyclic module M is a direct sum of indecomposable submodules if all quotients of cyclic submodules of M are CC-modules. This theorem generalizes a recent result of B. L. Osofsky and P. F. Smith on cyclic completely CS-modules. Some further applications are given for cyclic modules which are decomposed into projectives and injectives.1980 Mathematics subject classification: Primary 16A52, 16A53.In [6,7] Osofsky proved that a ring R is semisimple Artinian if every cyclic right J?-module is injective. Since that time, this important theorem has been extensively investigated and extended by several authors (see e.g. The purpose of this note is to present some extensions of Osofsky-Smith's theorem in [8]. First we show that a cyclic module M is a direct sum of indecomposable submodules if all quotients of cyclic submodules of M have closed submodules cyclic. Our arguments use the idea of proof in [8]. A similar result holds also for finitely generated modules with closed submodules finitely generated. Further we prove that a finitely generated module M has finite uniform dimension if every quotient of a cyclic submodule of M is a direct sum of a projective module and a CS-module. As a consequence, we obtain a module-theoretic version of a result in [8] that right CDPIrings are right Noetherian. Finally, an application is given for right linearly topologized rings.

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A Right PCI Ring is Right Noetherian

Cited by 6 publications

References 3 publications

A characterization of noetherian rings by cyclic modules

A characterization of noetherian rings by cyclic modules

Complete pure injectivity and endomorphism rings

Modules whose closed submodules are finitely generated

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