# homological algebra – Ext functor and Projective Module

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.