Let $R$ be the integral closure of $\mathbb{Z}$ in the algebraic closure of $\mathbb{Q}$. What is an explicit example of an ascending chain of radical ideals in $R$ that does not stabilize?
Show your effort: if we drop the condition that the ideals have to be radical, I can give the chain $(2)\subset (2^{0.5})\subset (2^{0.25})\subset\dots$ but all of these ideals have the same radical, of course.
I also know that the chain of interest can not contain a non-zero prime ideal (since non-zero prime ideals are maximal, so the chain will stabilize).