= Prime radical of a noncommutative ring
{title2=$N(A)$}
= Prime radical
{synonym}
= Lower nilradical
{synonym}
The prime radical is the intersection of all <prime ideals of a noncommutative ring>. For a nonzero <ring> satisfying the <ascending chain condition> on two-sided <ideals>, there exist <prime ideals of a noncommutative ring> $P_1,\ldots,P_r$ with $P_1\cdots P_r=0$. Indeed, a maximal counterexample among two-sided <ideals> could not itself be a <prime ideal of a noncommutative ring>, and two larger witness <ideals> would contradict its maximality. Every <prime ideal of a noncommutative ring> contains one $P_i$, so the intersection equals $\bigcap_iP_i$, and its $r$th power is zero. This nilpotence assertion fails without a chain condition.
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