Doc, you ain't write no evaluation map. If $R$ is commutative, you write the trace map. If $R$ is noncom, god knows what you write. The evaluation map, that is iso for a f.g. projective and a hom of $R$-$R$-bimodules, in general, is $$ p \otimes \phi \mapsto \phi (p), \ P \otimes_{End_RP} P^{\vee} \rightarrow R \, . $$ Use it with care.
In terms of Leonid's answer, you have a canonical $S:=End_RP$, lying around.