Let $k$ be any field and consider $R=\prod_{\mathbb{N}}k$ the product of copies of $k$ indexed over the natural numbers. The direct sum $\sum_{\mathbb{N}}k=I$ is an ideal of $R$, hence we can form the quotient $R/I$. My question is the following: is the ring $R/I$ injective? Since $R\rightarrow R/I$ is a flat ring homomorphism, $R/I$ being injective as $R$-module is equivalent to being self-injective. I believe that these should be a result in the literature pointing toward this, yet I did not manage to find anything useful.
I am also interested in this generalization: let $S$ be a commutative, von Neumann regular ring which is not semi-artinian (i.e. $\text{Spec}(S)$ is not scattered) and consider $J$ to be the intersection of all maximal ideals of $S$ which are belong to the perfect subset of $\text{Spec}(S)$. Is the ring $S/J$ injective?
Observe that in the case of $R$ the Zariski spectrum is homeomorphic to $\beta \mathbb{N}$, the set of ultrafilters on $\mathbb{N}$ with the Stone topology. This space consists in its isolated points, corresponding to the principal ultrafilters, and its perfect subset whose points correspond to the non-principal ultrafilters. The fact that $I$ coincides with the intersection of all ideals associated to non-principal ultrafilters follows from the fact that the cofinite filter coincides with the intersection of all non-principal ultrafilters. See this old question: cofinite filter is intersection of all non-principal ultrafilter.
The main theorem of
Osofsky, B. L., Noninjective cyclic modules, Proc. Am. Math. Soc. 19, 1383-1384 (1968)
implies that $\prod_{\mathbb{N}}k/\sum_{\mathbb{N}}k$ is not an injective $\prod_{\mathbb{N}}k$-module.