Example of a ring such that the nilradical of it is not nilpotent. Please help me.
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2$\begingroup$ 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. $\endgroup$– Zev ChonolesFeb 6, 2013 at 9:55
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$\begingroup$ @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. $\endgroup$– AliakbarFeb 6, 2013 at 9:59
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1$\begingroup$ 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. $\endgroup$– Zev ChonolesFeb 6, 2013 at 10:00
2 Answers
Take the polynomial ring $R=\mathbb K[x_i \mid i \in \mathbb N]$ in infinitely many variables. Then take the ideal $I=(x_i^i \mid i \in \mathbb N)$. The ring $R/I$ 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.
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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$\begingroup$ So consider $Z[x_{1},x_{2},...]$. Is it true? $\endgroup$– AliakbarFeb 6, 2013 at 10:08
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4$\begingroup$ 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. $\endgroup$ Feb 6, 2013 at 10:09