I have been looking for a couple of days for the answer to this question to no avail. Let me define what $\Sigma$-injective is.
Let $R$ be a unital, not necessarily commutative ring. A left $R$-module $I$ is called $\Sigma$-injective if an arbitrary direct sum of copies of itself is again injective.
The Bass-Papp theorem states that $R$ is left noetherian if and only if the direct sum of an arbitrary family of injective left $R$-modules is again injective, so noetherian rings have the property I am looking for, but I was wondering if the fact that every injective is $\Sigma$-injective is equivalent to some classical ring-theoretical property.