Example 10.32.2. 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.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: