Let $M$ be a module over a commutative ring (with unity) $R$. Let $\phi : M \to M$ be an $R$-module homomorphism. Then we have a dual map $\phi^* : M^* \to M^*$ given by $\phi^*(f)=f\circ \phi, \forall f \in M^*$. My questions are:
(1) If $\phi $ is surjective, then $\phi^*$ is injective. Under what conditions on $R$ or $M$, can we say that $\phi$ is injective $\implies \phi^*$ is surjective ?
(2) If $M$ is torsion-less, i.e., $\bigcap_{f\in M^*}\ker f=0$, then $\phi^*$ surjective $\implies \phi$ is injective. Under what conditions on $R$ or $M$, can we say that $\phi^*$ is injective $\implies \phi$ is surjective ?
[NOTE: Here, $M^*:=\mathrm{Hom}_R(M,R)$]