For purposes of this post, I'll confine myself to talking about torsion Abelian groups and torsion modules over a Dedekind domain.
If $A$ is an Abelian $p$-group, then heights of elements play an important role in studying the group. So the $p$-height of an element is finite, or $h_p (a) = n$, if $p^n x = a$ has a solution $x \in A$. Equivalently $a \in p^n A$ but not $p^{n+1}A$.
The first Ulm subgroup of $A$ contains the elements of infinite height $A^1 = \{ a \in A \mid h_p (a) = \infty \} = \{ a \in A \mid a \in \bigcap_{n \geq 0 } p^n A \} $.
Now every Abelian group is a $\mathbb{Z}$-module, $\mathbb{Z}$ is a PID, and every PID is a Dedekind domain. Unless I'm mistaken, we can define $p$-height and the first Ulm submodule in an identical manner for a PID. Our prime $p$ comes from a prime ideal $\mathfrak{p}$ where $\mathfrak{p} = \langle p \rangle$.
--------------------At last, here is my set-up and question--------------------
If $A_R$ is a torsion module over a Dedekind domain $R$, and $\mathfrak{p}$ is a prime ideal of $R$, then can we say that $a \in A$ has finite $\mathfrak{p}$-height $n$ if and only if $a \in \mathfrak{p}^{n}A$ but not $\mathfrak{p}^{n+1}A$?
Furthermore, for a torsion module $A$ where $\forall a \in A$, $\mathfrak{p}^n a = 0$ for some $n \in \mathbb{N}$, is the first Ulm submodule $A^1$ defined to be $\bigcap_{n \geq 0} \mathfrak{p}^n A$ ?
Any explanation or references would be quite helpful as I have combed through Fuch's Modules over non-Noetherian Domains and Modules over Valuation Domains as well as Kaplansky's paper Modules over Dedekind Rings and Valuation Rings for some time now. I found the introduction of Tilting Modules over Small Dedekind Domains by Trlifaj & Wallutis to be helpful but there is no mention of $\mathfrak{p}$-height in it.