# I have to prove that if P is a R-module

I have to prove that if P is a R-module , P is projective right there is a family $\left\{{x}_{i}\right\}$ in P and morphisms ${f}_{i}:P⇒R$ such that for all $x\in P$
$x=\sum _{i\in I}{f}_{i}\left(x\right){x}_{i}$
where for each  for almost all $i\in I$.

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Step 1
Put the ${f}_{i}$ together to form one giant f from P to ${R}^{\left(I\right)}$, the direct sum of I copies of the ring R. The condition that
$x\sum {f}_{i}\left(x\right){x}_{i}$
just means that there is some $g:{R}^{\left(I\right)}⇒P$ such that $g\left(f\left(x\right)\right)=x$, namely . In other words, P is a direct summand of the free module ${R}^{\left(I\right)}$
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