Elsevier

Journal of Algebra

Volume 460, 15 August 2016, Pages 128-142
Journal of Algebra

Rings whose cyclic modules are pure-injective or pure-projective

https://doi.org/10.1016/j.jalgebra.2016.03.044Get rights and content
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Abstract

A famous theorem of algebra due to Osofsky states that “if every cyclic left R-module is injective, then R is semisimple”. Therefore, a natural question of this sort is: “What is the class of rings R for which every cyclic left R-module is pure-injective or pure-projective?” The goal of this paper is to answer this question. For instance, we show that if every cyclic left R-module is pure-injective, then R is a left perfect ring. As a consequence, a commutative coherent ring R is Artinian if and only if every cyclic R-module is pure-injective. Also, a commutative ring R is pure-semisimple (i.e., every R-module is pure-injective) if and only if all cyclic R-modules and all indecomposable R-modules are pure-injective. We obtain some generalizations of Osofsky's theorem in the cases R is semiprimitive or commutative coherent or a commutative semiprime Goldie ring. Finally, we show that a ring R is left Noetherian if and only if every cyclic left R-module is pure-projective. As a corollary of this result we obtain: if every cyclic left R-module is pure-injective and pure-projective, then R is a left Artinian ring. The converse is also true when R is commutative.

MSC

16D80
16D40
13C11

Keywords

Cyclic module
Pure-injective module
Pure-projective module
Left perfect ring
Left pure-semisimple ring
Left Köthe ring

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This research was in part supported by a grant from IPM (No. 94160056).