Characterize ring classes ensuring hereditary and ultraproduct-closure properties

Determine what classes of rings, apart from perfect and hereditary rings, ensure both that submodules of projective modules are projective and that the class of projective modules is closed under ultraproducts, thereby yielding the strong compactness result for ba-projective modules.

Background

The strong compactness argument establishes that if ba is bb-strongly compact, the ring has cardinality less than bb, and the ring is hereditary with projective modules closed under ultraproducts, then every ba-projective module is projective. The paper obtains these two structural properties from the stronger assumptions that the ring is perfect and hereditary, but asks which other classes of rings impose them.

Identifying broader ring classes would extend the strong compactness theorem beyond the perfect-hereditary setting and clarify which hypotheses are genuinely necessary.