It is an interesting fact that a commutative ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective, and if and only if every injective $R$-module is a direct sum of indecomposable injective $R$-modules. In his book Lectures on modules and rings (Theorems 3.46 and 3.48), T.Y.Lam attributes the first one to Bass and Papp and the second one to Matlis and Papp. However, checking out the sources (unfortunately without access to the Papp article) reveals that the history might be more complicated; also the name of Eilenberg pops up somewhere. Hence:
Who was the first to prove these statements, and where were they published for the first time?
EDIT: Carlo Beenakker provided some information. But some points are still not clear to me.
Who was the first to prove that over a noetherian ring direct sums of injectives are injective?
As this appears as Exercise I.7.8 in Cartan-Eilenberg's Homological Algebra it must have been known before 1956.
Who was the first to prove that noetherianness is implied by the property that injectives are direct sums of indecomposable injectives?
This is not mentioned in Carlo's answer, but it might be in Papp's article - can someone confirm this? Lam is also unclear about this point. In Matlis's article only the other implication appears.