I'm trying to show that $(x)\cap(p^n)=(xp^n)$, where $x,p$ are elements of some ring $R$, $p$ is prime and $p\nmid x$.
The inclusion from right to left is obvious, but I can't make any progress in the other direction. Given an element $r\in(x)\cap(p^n)$ I get $r=ax=bp^n$ for certain $a,b\in R$ and hence that $p\mid a$ since by assumption $p\nmid x$ and $p$ is prime. So I have that $r=cxp=bp^n$ for some $c\in R$, but what now?