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

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.