# Integral Domain is Reduced Ring

## Theorem

Let $\left({D, +, \circ}\right)$ be an integral domain.

Then $D$ is reduced.

## Proof

Let $x \in D$ be a nilpotent element.

Then by Nilpotent Element is Zero Divisor, $x$ is a zero divisor in $D$.

By the definition of an integral domain, this means that $x = 0$.

Therefore the only nilpotent element of $D$ is $0$.

That is, $D$ is reduced.

$\blacksquare$