
Example 10.31.2. Let $R = k[x_ n | n \in \mathbf{N}]$ be the polynomial ring in infinitely many variables over a field $k$. Let $I$ be the ideal generated by the elements $x_ n^ n$ for $n \in \mathbf{N}$ and $S = R/I$. Then the ideal $J \subset S$ generated by the images of $x_ n$, $n \in \mathbf{N}$ is locally nilpotent, but not nilpotent. Indeed, since $S$-linear combinations of nilpotents are nilpotent, to prove that $J$ is locally nilpotent it is enough to observe that all its generators are nilpotent (which they obviously are). On the other hand, for each $n \in \mathbf{N}$ it holds that $x_{n + 1}^ n \not\in I$, so that $J^ n \not= 0$. It follows that $J$ is not nilpotent.

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