The Zero is the only Nilpotent Element of the Quotient Ring by its Nilradical

Problem 725

Prove that if $R$ is a commutative ring and $\frakN(R)$ is its nilradical, then the zero is the only nilpotent element of $R/\frakN(R)$. That is, show that $\frakN(R/\frakN(R))=0$.

Proof.

Let $r\in R$ and if $x:=r+\frakN(R) \in R/\frakN(R)$ is a nilpotent element of $R/\frakN(R)$, then there exists an integer $n$ such that
$x^n=(r+\frakN(R))^n=r^n+\frakN(R)=\frakN(R).$ Thus we have
$r^n\in \frakN(R).$

This means that $r^n$ is an nilpotent element of $R$, and hence there exists an integer $m$ such that
$r^{nm}=(r^n)^m=0.$

Therefore $r$ is an nilpotent element of $R$, that is $r\in \frakN(R)$ and we have $x=\frakN(R)$, which is the zero element in $R/\frakN(R)$.

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