Reduced ring

Remarks

Example: $\Z/6\Z$ is reduced; if $r^n \equiv 0 \pmod 6$, then $2$ and $3$ divide $r$, so $r \equiv 0$. Non-example: $\Z/4\Z$ is not reduced, since $2^2 = 0$ and $2 \ne 0$. Equivalent forms: the nilradical of $R$ is zero; the zero ideal is a radical ideal; for all $r \in R$, $r^2 = 0$ implies $r = 0$. Every integral domain is reduced, and so is every product of integral domains. The same condition defines reduced rings that are not commutative.