TITLE: Is there a ring such that the nilradical of it is not nilpotent?
QUESTION [5 upvotes]: Example of a ring such that the nilradical of it is not nilpotent. Please help me.

REPLY [9 votes]: Take the polynomial ring $R=\mathbb K[x_i \mid i \in \mathbb N]$ in infinitely many variables. Then take the ideal $I=(x_i^i \mid i \in \mathbb N)$. The ring $R/I$ has for each $i \in \mathbb N$ a nilpotent element $\bar x_i$ which has nilpotence index $i$. In this ring the nilradical contain for each $i \in \mathbb N$ the element $\bar x_i$, but for every $k \in \mathbb N$ we have that $\bar x^k_{k+1} \ne 0$ is an element of the $k$-th power of the nilradical. 
So the nilradical isn't nilpotent.