Let $R$ be a ring and $P$ is an $R$-module.

The statement are equivalent:

1-$P$ is projective.

2-For every $R$-module $N$ and for $igeq 1$ , $Ext^i_R(P,N)=0$

3-For every $R$-module $N$ , $Ext^1_R(P,N)=0$

4-For every finitely presented $R$-module $N$, $Ext^1_R(P,N)=0$

5-For every finitely generated ideal $I$ of $R$, $Ext^1_R(P,R/I)=0$

I can prove 1 $Rightarrow$ 2 $Rightarrow$ 3 $Rightarrow$ 4 $Rightarrow$ 5

but I dont know how to prove 5 $Rightarrow$ 1. can anyone help? Thank you.