Invertible $R$-module $I$ with a map $I \to R$

Question

Let $R$ be a commutative ring. Let $I$ be an invertible $R$-module which admits a non-zero $R$-linear map $I \to R$. Can we say something about $I$? How strong is this property? The best case would be $I \cong R$, but I assume this is too optimistic? What happens if $R$ is a Dedekind domain, can we say more in that case?

Answer 1

This is true of every invertible $R$-module (unless $R$ is the zero ring). Indeed, since $I$ is invertible, the evaluation map $f:I\otimes I^\vee\to R$ is an isomorphism, and so there must be some $x\in I^\vee$ such that the map $i\mapsto f(i\otimes x)$ is nonzero.