Example of invertible modules?

Question

I am trying to find an example of a non-trivial invertible module (let's say over $\mathbb Z$). This seems to be very simple, but after trying and searching around, I do not find any examples. (Many sources of references give proofs without examples.) I would be really helpful if an example can be shared here.

Answer 1

If $R$ is a commutative ring, let $F$ be the total quotient ring of $R$, that is the localisation with respect to the set of regular elements (an element $x\in R$ regular if multiplication by $x$ is injective).

Then for any regular element $x\in R$, $xR$ is invertible (with inverse $\frac{1}{x}R\subset F$)

In particular, if $R$ is an integral domain, $xR$ is invertible for all $x\neq 0$.

A less trivial example is the case of ideals of Dedekind rings: if $R$ is a Dedekind ring, then every nonzero ideal of $R$ is invertible.