I'm reviewing the structure theorem for finitely generated modules over a principal ideal domain, and am looking at the following lemma from Hungerford's Algebra IV 6.6.8:

Let $A$ be a (finitely generated, unitary) module over a PID $R$ such that $p^nA=0$ for some prime $p\in R$ and positive integer $n$, but $p^{n-1}A\neq 0$. Let $a\in A$ be an element with order $p^n$. Then
(i) if $A\neq Ra$, then there exists a nonzero $b\in A$ with $Ra \cap Rb = 0$.
(ii) There is a submodule $C$ of $A$ such that $A=Ra \bigoplus C $.

I understand the proof and how the lemma relates to the larger structure theorem, but I don't appreciate the result because I can't think of any examples where it wouldn't be true.

If I weaken the condition that $n$ has to be finite, then $\mathbb{Z}(p^\infty)$ is an example where (i) is false because all $\mathbb{Z}$-submodules intersect. Also, if I remove the condition that $p$ is a prime, then $(2,0)\in \mathbb{Z}_6^2$ seems to violate (ii), but I can't think of how (i) could ever be violated.

Can anyone provide some instructive counterexamples that illustrate how this theorem is nontrivial when we remove the condition that $R$ is a PID or that $p$ is prime? I tried to think of an example when $R$ was $\mathbb{R}[x,y]$, but I don't have the intuition to think of anything.

edit: I put linear algebra as a tag because I'm really hoping for an example from there.


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