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.
A module is called $\Sigma$-injective if a direct sum of arbitrarily (equivalently, countably infinitely) many copies of that module is injective. So it suffices to find an example of a $\Sigma$-injective module over a non-noetherian ring. Apart from silly examples such as a direct product of two rings one of which is one-sided noetherian and the other of which is not, the main theorem of C. Megibben, “Countable injective modules are $\Sigma$-injective,” Proc. Amer. Math. Soc. 84 (1982), no. 1, 8–10, says what the title indicates. This gives all sorts of examples of $\Sigma$-injective modules over non-noetherian rings.