Let $P$ be a nonzero projective left $R$-module. I want to show that $Hom(P,R)$ is nonzero.
Let $I$ be an index set and let $R^{I}$ denote $|I|$ copies of $R$. We have an epimorphism $f$ from $R^{I}$ to $P$ and since $P$ is projective there exists $g: P \rightarrow R^{I}$ such that $f \circ g = 1_{P}$. Now consider the projection $\pi: R^{I} \rightarrow R$ then the composition $\pi \circ g$ determines a morphism from $P$ to $R$. Is this nonzero? Why?