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• W2417285923 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." @default.
• W2417285923 created "2016-06-24" @default.
• W2417285923 creator A5004616343 @default.
• W2417285923 date "2016-08-01" @default.
• W2417285923 modified "2023-09-27" @default.
• W2417285923 title "Rings whose cyclic modules are pure-injective or pure-projective" @default.
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• W2417285923 doi "https://doi.org/10.1016/j.jalgebra.2016.03.044" @default.
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