Hi. I know, by the Bass-Papp theorem, that if every direct sum of injective $R$-modules is injective then $R$ is Noetherian. I would like to know if there exists a direct sum of injective $R$-modules injective with $R$ non-Noetherian. Of course if the sum is a finite sum of injective modules, then it is injective; so I assume that the sum is infinite (let's say that all the modules are nonzero).

By the Bass-Papp theorem, if every direct sum of injective $R$-modules is injective then $R$ is Noetherian. I would like to know if there exists an injective module over $R$ non-Noetherian, that splits as infinite direct sum of nonzero (injective) $R$-modules.