This is in relation to the $p$-adic valuation on the field of fractions $F$ of an integral domain $D$. The idea is that for each $x \in F$ there is a unique maximal $k$ such that $x = p^k u v^{-1}$ where $u, v^{-1}$ are in $D$ and are not divisible by $p$. We then define the valuation $\nu _{\ p}$ by $\nu _{\ p}(p^k u v^{-1}) = k$.
My question is why is this $k$ well-defined? Naively we could take $x = (a,b)$ and start dividing $a$ by $p$ until we cannot do so any longer, do the same for $b$ to get $a = p^\alpha a' , b = p^\beta b' \implies (a,b) = (p^{\alpha - \beta} a' , b)$ and take $\alpha - \beta$ as our answer. But why do $\alpha, \beta$ need to be finite? Why can there not be some $x$ such that for any $k$ there is $a_k$ such that $a = p^k a_k$? If the Ring was Noetherian I can see why this is impossible but not for general integral domains.
That can happen in a general domain.
One way to produce a counterexample is through brute force. Let $F$ be a field and consider $F[a,p,y_1,y_2,y_3,\ldots]$ and mod out by the generators $\langle a-p^ky_k\mid k\in \Bbb N, k>0 \rangle$. At the end of the day, $a$ is divisible by every power of $p$.