Definition 2

— Let $A$ be a commutative ring; then $A$ is said to be reduced if every nilpotent element (I, p. 98) of $A$ is zero.

If $A$ is a field, or an integral domain, or a product of reduced rings, it is a reduced ring. For a commutative ring $A$ to be reduced it is necessary and sufficient that $a^2 \neq 0$ for every $a \neq 0$ in $A$: for from this we obtain by induction on $n$, $a^{2^n} \neq 0$, hence $a^n \neq 0$ for any $a \neq 0$ in $A$.

An algebra is said to be reduced if its underlying ring is reduced.