Assess one-sided weakenings of perfectness and heredity

Determine whether replacing perfectness or heredity by its right-sided or left-sided version preserves the strong compactness conclusion for ba-projective modules without forcing the underlying ring to be semisimple.

Background

The paper notes that combining perfectness and heredity can be restrictive, particularly for commutative rings, where the combination implies semisimplicity. It therefore asks whether one-sided variants of these hypotheses can retain the strong compactness result while allowing non-semisimple rings.

A positive answer would identify weaker and more flexible hypotheses under which ba-projectivity implies projectivity for rings of cardinality below bb when ba is bb-strongly compact.