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

Problems and solutions of ring theory in abstract algebra

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 $\frakN(R/\frakN(R))=0$.

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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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