Countable injective modules are sigma injective

Abstract:

In this note we show that a countable injective module is $\sum$-injective and consequently a ring $R$ is left noetherian if the category of left $R$-modules has a countable injective cogenerator. Our proof can be modified to establish the corresponding result for quasi-injective modules. We also give an example of a nonnoetherian commutative ring $R$ such that the category of $R$-modules has a countable cogenerator.