When is every injective module $\Sigma$-injective?

Question

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.

Answer 1

A result of Faith and Walker (page 205 in C. Faith, E. A. Walker, Direct-sum representations of injective modules, J. Algebra 5 (1967), 203-221) answers your question:

If each injective left $R$-module is $\Sigma$-injective, then $R$ is left noetherian.

(In fact, a sufficient condition for $R$ to be left noetherian is that each injective cogenerator of the category of left $R$-modules is $\Sigma$-injective - see Proposition 2.2 in this article)