I have to prove that if P is a R-module , P is projective ⇔ there is a family {xi} in P and morphisms fi:P→R such that for all x∈Px=∑i∈Ifi(x)xiwhere for each x∈P, fi(x)=0 for almost all i∈I.

Step 1Put the fi together to form one giant f from P to R(I), the direct sum of I copies of the ring R. The condition thatx∑fi(x)xijust means that there is some g:R(I)→P such that g(f(x))=x, namely g(r1 r2⋯)=r1x1+r2x2+⋯. In other words, P is a direct summand of the free module R(I)