Nilradical that is not nilpotent
= Nilradical that is not nilpotent
{title2=$N(R)^j\ne0\quad(j\geq1)$}
In $R=k[t_1,t_2,\ldots]/(t_1^2,t_2^2,\ldots)$, the <nilradical> is generated by all $\bar t_i$. Every element uses finitely many variables and is nilpotent, while each square-free product $\bar t_1\cdots\bar t_j$ is nonzero. Thus this <nilradical> is not a <nilpotent ideal>. A <Noetherian ring> cannot have this behavior: finitely many nilpotent generators give a nilpotent <ideal>.