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.