Take P a free module over an arbitrary ring. Show P is projective. Definition of Projective: If f: M-->N and g: P-->N are module homomorphisms with f surjective, then if P is projective there exists homomorphism h such that h: P-->M with f(h(x))=g(x) for all x. Obviously f has a right inverse, and so it is easy to find an h that works, but not a homomorphism h that works. I really have no idea about this and help would be appreciated. Second bit: Conversely, show that if P is a projective module over a Principal Ideal Domain, then P is free.