Reduced ring

Symbol-free definition
A commutative unital ring is said to be reduced if it satisfies the following equivalent conditions:


 * The nilradical of the ring is the zero ideal
 * There are no nilpotents other than the zero element
 * The zero ideal is a radical ideal

Definition with symbols
A commutative unital ring $$R$$ is said to be reduced if it satisfies the following condition:

$$x^n = 0 \implies x = 0$$ for any $$x \in R$$

Stronger properties

 * Integral domain
 * Semisimple ring

Metaproperties
Any subring of a reduced ring is reduced. That's because an element in the subring that is nilpotent, is also nilpotent in the whole ring.