Solution
= Solution
A <ring> is <reduced> if its only <nilpotent element> is zero. Equivalently,
$$
\boxed{x^n=0\text{ for some }n\geq1\ \Longrightarrow\ x=0,\qquad\text{or }\sqrt{(0)}=0.}
$$
The second expression says that its <nilradical> vanishes. A <reduced ring> may have <zero divisors>; for example, the <direct product of rings> $\mathbb R\times\mathbb R$ is <reduced> but $(1,0)(0,1)=0$.