# Is there a ring such that the nil radical of it is not nilpotent?

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What have you tried so far? To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. –  Zev Chonoles Feb 6 '13 at 9:55
@Zev Chonoles I know that in a Noetherian ring the nil radical is nilpotent. Also, in a Artinian ring. Indeed, the ring is commutative with identity. –  aliakbar Feb 6 '13 at 9:59
Do you know examples of non-Noetherian rings? Can you modify those examples so they have lots of nilpotents? Look at my answer for a hint. –  Zev Chonoles Feb 6 '13 at 10:00

Hint:

• The nilradical of $k[x]/(x^2)$ is the ideal $N=(x)$, and $N^2=(0)$, so $N$ is a nilpotent ideal.

• The nilradical of $k[x,y]/(x^2,y^2)$ is the ideal $N=(x,y)$, and $N^3=(0)$, so $N$ is a nilpotent ideal.

• ...

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So consider $Z[x_{1},x_{2},...]$. Is it true? –  aliakbar Feb 6 '13 at 10:08
The ring $\mathbb{Z}[x_1,x_2,\ldots]$ doesn't have any nilpotents; you need to take a quotient. Look again at how the rings were formed in my hint. –  Zev Chonoles Feb 6 '13 at 10:09

Take the polynomial ring $\mathbb K[x_i \mid i \in \mathbb N]$ in infinte variables. Then take the ideal $I=(x_i^i \mid i \in \mathbb N)$ the ring $\mathbb K[x_i \mid i \in \mathbb N]/I$ 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.

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+1, Good example! LaTeX tip: use \mid instead of | for a separator, the spacing is different. See tex.stackexchange.com/q/498/228 –  Zev Chonoles Feb 6 '13 at 10:11
@ZevChonoles did that, thank you very much for the tip :) –  Giorgio Mossa Feb 6 '13 at 10:14