TITLE: Direct sum of injective modules over non-Noetherian rings
QUESTION [8 upvotes]: 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.

REPLY [12 votes]: 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.