Invertible sheaves on affine varieties

Question

Let $X:=\rm{Spec}(A)$ be an integral, noetherian, affine variety, and let $L$ be an invertible sheaf on $X$, I try to find an example where $L$ is not isomorphic to the structure sheaf of $X$.

In the language of commutative algebra, this simply means to find an $A$-module $M$ which is locally free, but $M$ is not isomorphic to $A$ as a module.

Because when $A$ is UFD, the Weil class group is trivial, such example should not exists in this case. Also, I know for affine toric variety, the Picard group is also trivial.

Answer 1

Take any non-principal ideal in your favorite Dedekind domain with nontrivial class group.

Answer 2

Any commutative group yields a counterexample!
Indeed , a mind-boggling theorem proved by Claborn in 1966 states that:

Any abelian group is isomorphic to the class group of some Dedekind ring.