Consider an $R$-module $M$, for a $R$ a commutative ring with identity.
Of course one could define the notion of inner nondegenerate* symmetric product $(-,-)$ in the same fashion of vector spaces.
Is this something that has been found of some interest in mathematics?
Is it true, for example, that the natural map $M\to \text{Hom}(M,R)$ sending $m\mapsto (m,-)$ is an isomorphism when $M$ is projective (or under other conditions)? Note that it is at least injective by nondegeneracy.
Thank you in advance.
*i.e. $\forall m\in M\ \exists n\in M$ s.t. $(m,n)\neq 0$.