Is there any condition that guarantees that a commutative ring $R$ satisfies the condition that a contravariant hom functor ${\rm Hom}_R(-,R):{\rm Mod}_R^{op} \to {\rm Mod}_R$ is a conservative functor, that is if ${\rm Hom}_R(f,R)$ is an isomorphism, so is $f$?
If $R$ is an integral domain that is not a field, then since ${\rm Hom}_R(R/I,R)=0$ for a nontrivial ideal $I$, we can see that the contravariant hom functor of interest is not conservative by applying it to a morphism $R/I \to 0$, which is not an isomorphism here.
I have no clue other than this fact and I am even not sure whether the dual functor ${\rm Hom}_k(-,k) $ is conservative for a general field $k$.
I would really appreciate it if you would provide necessary and/or sufficient conditions for a ring to satisfy the condition that the contravariant hom functor of interest is conservative.
Thank you in advance.