Definition 5

A ring $A$ is called reduced if its nilradical is reduced to 0, in other words if no element $\neq 0$ of $A$ is nilpotent.

If $\mathfrak{N}$ is the nilradical of a ring $A$, $A/\mathfrak{N}$ is reduced, for if the class mod. $\mathfrak{N}$ of an element $x \in A$ is nilpotent in $A/\mathfrak{N}$, this means that $x^h \in \mathfrak{N}$ for some integer $h$, hence $x^{hk} = 0$ for some integer $k$ and $x \in \mathfrak{N}$.