TITLE: Is this a property of an integral domain that is not a field?
QUESTION [6 upvotes]: I am working on a specific problem and I've almost got it solved. To solve it, however, I need to prove one last claim (if it is even true):

Consider an integral domain $R$ that is not a field. Let $a$ be an element of $R$ that is not a unit or $0$. I want to prove that there cannot exist nonzero elements $r_n$, $n \in \mathbb{N}$, of $R$ such that
$$ r_1 a=r_2a^2=r_3a^3=\cdots= r_na^n=\cdots$$

First of all, is this even true? My intuition says it is. And am I missing something obvious?  This is the last step in a long proof, so I hope it is true!
EDIT: It turns out my proof goes through without this being true (which is good since it's false), but definitely thanks for the answers. It has improved my understanding.

REPLY [16 votes]: This is false. As Zhen Lin points out, $R$ can't be Noetherian, so let $R$ be my favourite non-Noetherian domain, $\{f(x)\in \Bbb{Q}[x]\mid f(0)\in \Bbb{Z}\}$, and consider $a=2$. Then $$x=2\frac{x}{2}=4\frac{x}{4}=8\frac{x}{8}=\ldots$$ but $2$ is not a unit.