Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?
Analogously, under what assumptions on R does a countable product of projective left R-modules necessarily have a finite projective dimension?
These questions arise in the study of the coderived and contraderived categories of (CDG-)modules, or, if one wishes, the homotopy categories of unbounded complexes of injective or projective modules.
There are some obvious sufficient conditions and some less-so-obvious ones. For both #1 and #2, it clearly suffices that R have a finite left homological dimension.
More interestingly, in both cases it suffices that R be left Gorenstein, i.e., such that the classes of left R-modules of finite projective dimension and left R-modules of finite injective dimension coincide.
For #1, it also suffices that R be left Noetherian. For #2, it suffices that R be right coherent and such that any flat left module has a finite projective dimension.
Any other sufficient conditions?