0
$\begingroup$

Let $R:=k[x]/x^p$ with a field $k$ of characteristic $p$.

Then this local ring $R$ is regular or not?

  • 0
    @KeenanKidwell Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-12

1 Answers 1

5

The ring $R=k[X]/(X^p)$ is a finite local $k$-algebra with maximal ideal $\mathfrak{m}$ generated by the image of $X$, so if it is regular, then $\dim(\mathfrak{m}/\mathfrak{m}^2)=\dim(R)=0$. By Nakayama's lemma, this implies that $\mathfrak{m}=0$, so $R$ is a field. But clearly $R$ is not a field, so $R$ is not regular.