It is known that if $R$ is a DVR with fraction field $K,$ then the $R$-submodules of $K$ are $0,K,x^nR,$ with $n$ any integer and $x$ a generator of the maximal ideal of $R.$ I was wondering if there is a simple structure theorem for the $R$-submodules of $K^n,$ the $n$-dimensional vector space over $K$ (similar to that for finitely generated torsion-free modules over a Dedekind domain). I know that the subroups of $\mathbb{Q}$ are somewhat complicated to describe, but was wondering, in particular, if the situation gets any nicer with, say, $\mathbb{Z}_P$-submodules of $Q^n,$ where $P$ is a nonzero prime ideal of $\mathbb{Z}.$

It is known that if $R$ is a DVR with fraction field $K,$ then the $R$-submodules of $K$ are $0,K,x^nR,$ with $n$ any integer and $x$ a generator of the maximal ideal of $R.$ I was wondering if there is a simple structure theorem for the $R$-submodules of $K^n,$ the $n$-dimensional vector space over $K$ (similar to that for finitely generated torsion-free modules over a Dedekind domain). I know that the subroups of $\mathbb{Q}$ are somewhat complicated to describe, but was wondering, in particular, if the situation gets any nicer with, say, $\mathbb{Z}_P$-submodules of $\mathbb{Q}^n,$ where $P$ is a nonzero prime ideal of $\mathbb{Z}.$