When R is a testing module for projectivity?

A right R-module is called R-projective if it is projective relative to the right R-module RR. A ring R is called right testing for projectivity if every right R-projective module is projective. Every right perfect ring is right testing and there are examples of local rings that are not right testing. In this paper an attempt is made to understand the class of right testing rings and several examples are provided to highlight the set theoretic obstacles in characterizing such rings.