Nilradical of a commutative Artinian ring
= Nilradical of a commutative Artinian ring
The <nilradical> $N$ of a commutative <Artinian ring> is a <nilpotent ideal>. Its powers stabilize, say $N^r=N^{r+1}=I$. If $I\ne0$, choose an ideal $J$ minimal subject to $IJ\ne0$, and then $x\in J$ with $Ix\ne0$. Minimality and $I^2=I$ give $Rx=Ix$, so $x=ax$ for some $a\in I$. Since $a$ is nilpotent, $1-a$ is a <unit>, contradicting $(1-a)x=0$ and $x\ne0$.