Existence of non-trivial homomorphism from projective module to ring

Question

Let $R$ be a commutative ring and let $P$ be a nonzero projective $R$-module. I want to show that there exists a non-trivial homomorphism from $P$ to $R$.

I don't see how can I start. Can someone give a hint ?

Answer 1

For every $p\in P$, we have $R\to P$, $x\mapsto xp$. From this, you have a surjection $\bigoplus_{p \in P} R \to P$. Lift the identity $P\to P$ to non-trivial $P\to \bigoplus_{p \in P} R $. Project to the individual coordinates, they cannot all be trivial.