Example of finitely generated module over valuation domain

Question

I'm trying to find an example of finitely generated module over valuation domain with its generator. Is there anybody could help me to give the example beside the domain over itself?

Answer 1

Let $D$ be the domain and $M$ be its maximal ideal.

$D/M$ is such a module. Since it is a simple module, every nonzero element is a generator for the entire module.

Or for that matter, any quotient of D by a right ideal that you like.

Or for that matter, any quotient of $D^n$ by whatever submodule you want. Actually, every single finitely generated module arises that way, so you cannot complain it is too trivial.