Let be a ring with 1 and let and be (left, unital) -modules. Prove that is projective if and only if and are projective.
Recall that, as it happens, a module is projective if and only if it is a direct summand of a free module.
Suppose is projective. Then for some module , we have free. In particular, both and are direct summands of a free module, and are thus projective.
Conversely, suppose and are projective; then there exist modules and such that and are free. As we proved in this previous exercise, is free. Thus is projective.