Elements in ring that satisfy $x^{m+n}=x^m$

Question

I am trying to recall a statement about ring which talks about the existence of an element $x$ that has the property $x^{m+n}=x^n$ for some positive integers $m$ and $n$, but I cannot remember where I read it before and cannot find any reference talking about this, and don't even remember whether this element or the ring has a specific name or not. Perhaps anyone can give me any reference or maybe explanation about this? I really appreciate the help. Thanks!